# The sequence ledger

Eleven integer sequences fall out of the fractal work: cell counts, coprimality
counts, Euler characteristics, design counts. This page is the ledger they are
cited from. Every entry below meets the same standard, and nothing is listed
that does not.

- **Two independent generators.** The terms are produced twice, by programs that
  share no code and no method — a brute-force enumeration and a structural
  rewrite (digit automaton, Mobius inversion, Burnside over a different group
  representation). Where a generator merely paraphrases another it does not
  count, and independence is argued explicitly in each entry's `RUN.txt`.
- **Checked against the b-file.** Every term the stored b-file holds is diffed
  against a generator, and the two generators are diffed against each other and
  against the b-file over the whole range both can reach — which is the far end
  of the file where the cost allows it, and a long prefix rather than a token
  sample where a brute-force method runs out of room.
- **Adversarially re-verified.** A second reader re-ran the generators from the
  published directory, re-derived the constants, re-grepped the OEIS dump with
  extra windows of their own invention, and checked each claim against the
  files. Corrections landed in the entry's `RUN.txt`; they are noted here where
  they change what a reader should believe.

Novelty is tested by fixed-string search against a local copy of the OEIS
stripped dump (396480 lines, last modified 2026-06-11) at the full leading
window, at one- and two-term shifts, at interior windows, and at simple
transforms of the terms. A hit is a collision to be explained, not a failure.

Two entries carry A-numbers and are with the OEIS. The other nine are drafts
held in the lab; no A-number has been requested for them and nothing has been
submitted for them anywhere.

## A395241 - void subcubes of the odd sponge tile

The generalized Menger tile at odd subdivision `m = 2n+1`: `a(n)` counts the
removed subcubes, equivalently the cells of an `m^3` grid (coordinates from 1)
with at least two even coordinates. The b-file runs to `n = 10000` and the
closed form matches every one of its 10001 terms.

| field | value |
|---|---|
| status | submitted 2026-06-10 |
| terms | `0, 7, 44, 135, 304, 575, 972, 1519, 2240, 3159, 4300, 5687` |
| formula | `a(n) = n^2*(4*n + 3)`, offset 0 |
| g.f. | `x*(7 + 16*x + x^2)/(1-x)^4` |
| recurrence | `a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4)` for `n > 3` |
| crossrefs | `a(n) = A011934(2*n)`; partial sums of A154105; `a(n) + A103532(n) = (2*n+1)^3` |
| lab | `lab/sequences/A395241/` |

The independent generator bores `n^2` square channels along each axis and takes
the size of the union of drilled voxels: no closed form, no parity test, and
channel crossings resolved by the set rather than by inclusion-exclusion. All
six novelty greps returned nothing, as did four further windows tried during
re-verification. The sequence is the even bisection of A011934 (A103532 is the
odd bisection) — a declared crossref, not a duplicate. The dump postdates the
submission by a day but lists approved sequences only, and this entry was still
in draft, so its absence from the dump is expected and carries no information
either way.

## A396934 - coprime points of the Sierpinski triangle

Pairs `(i,j)` with `0 <= i,j < 2^n`, `i AND j = 0`, and `gcd(i,j) = 1`. The
condition `i AND j = 0` picks out exactly the `3^n` points of the n-th
Sierpinski step (odd `binomial(i+j,i)`, by Kummer); `a(n)` counts the coprime
ones. Note `gcd(0,k) = k`, so `(0,1)` and `(1,0)` count.

| field | value |
|---|---|
| status | submitted 2026-06-10; the A-number was confirmed against the live OEIS during verification, name and terms matching (the local dump stops at A396922, so it cannot show this entry) |
| terms | `0, 2, 4, 12, 34, 122, 362, 1130, 3406, 10506, 31550, 95260` |
| formula | none; no linear constant-coefficient recurrence of order 1..9 fits the 21 known terms |
| density | `a(n)/3^n -> 16/(3*Pi^2) = 0.5403796` conjectured; `a(20)/3^20 = 0.5403761` |
| lab | `lab/sequences/A396934/` |

The b-file reaches `n = 20`, every term of it produced by `fast.c` run in
eight chunks, with the independent generator reaching `n = 19` and the
brute-force scan `n = 14`. The independent generator indexes by the row
`m = i + j = i OR j` and uses `gcd(i,j) = gcd(i,m)`, enumerating only odd
submasks of odd rows and doubling by a fixed-point-free involution; it never
forms a complement and never evaluates `gcd(i,j)`. Two near misses were
checked and both diverge: A347825 agrees with `a(n)/2` for five terms, and
A004662 and A018413 agree with `a(n)-1` for five terms.

## menger_void - subcubes removed from the sponge

`a(n) = 27^n - 20^n`, the cells drilled out in forming the level-n Menger
sponge. The one entry here with a fully elementary answer, kept because the
complement is not in the OEIS even though both parents are.

| field | value |
|---|---|
| status | draft |
| terms | `0, 7, 329, 11683, 371441, 11148907, 323420489, 9180353203, 256829536481, 7113597484987, 195651132094649, 5354260566555523` |
| formula | `a(n) = 27^n - 20^n`; `a(n) = 47*a(n-1) - 540*a(n-2)` for `n > 1`, `a(0)=0`, `a(1)=7` |
| g.f. | `7*x/((1 - 20*x)*(1 - 27*x))` |
| density | `a(n)/27^n -> 1` (`0.9999999999999074` at `n = 100`); the sponge has volume measure zero |
| crossrefs | A009964 (`20^n`, retained) and A009971 (`27^n`) |
| lab | `lab/sequences/menger_void/` |

All three formulas hold across the full 101-term b-file. The independent
generator writes down neither 20 nor 7: it derives them by enumerating
`{0,1,2}^3` against the drill predicate `|dx-1| + |dy-1| + |dz-1| <= 1`, then
counts void cells two ways (a direct `27^n` grid scan, and a two-state forward
accumulation). Re-verification corrected the cross-reference for `27^n`, which
the research notes gave as A001024; A001024 is `15^n`, and the correct entry is
A009971.

## menger_coprime - coprime triples in the sponge

Triples `(i,j,k)` below `3^n` whose base-3 digits are legal sponge digits (at
most one coordinate holding digit 1 in each position) and with
`gcd(i,j,k) = 1`. The `20^n` admissible triples are the cells of the level-n
sponge.

| field | value |
|---|---|
| status | draft, no A-number |
| terms | `0, 12, 270, 5916, 123504, 2538447, 51497040, 1038074187, 20860210527` (b-file, `n = 0..8`) |
| extension | `a(9) = 418429711224`, `a(10) = 8382927031902` — each confirmed by an independent third method; `a(11) = 167827226563374` rests on one method only |
| formula | none; no linear constant-coefficient recurrence of order 1..5 fits the 12 known terms |
| density | conjectured `a(n)/20^n -> (19/20)*(27/26)/zeta(3) = (513/520)/zeta(3) = 0.8207086195` |
| lab | `lab/sequences/menger_coprime/` |

The density is `1/zeta(3)` with the Euler factor `1 - 1/27` at `p = 3` replaced
by the exact sponge factor `1 - 1/20`. Ratios `0.8148519737`, `0.8172455297`,
`0.8186452180`, `0.8194688797` at `n = 8..11` close on it with gaps shrinking
by about `0.6` per level — support, not proof. The independent generator never
enumerates the `20^n` cells and never calls `gcd`: sponge membership becomes
pairwise bitmask disjointness, and coprimality is stripped by Mobius inversion
over the common divisor with a subset-sum transform.

## menger_pairwise - pairwise coprime triples in the sponge

The same sponge cells, but with the stronger condition
`gcd(i,j) = gcd(i,k) = gcd(j,k) = 1`. Ordered triples.

| field | value |
|---|---|
| status | draft, no A-number |
| terms | `0, 0, 60, 1434, 32268, 721524, 15141288, 309309138, 6288927012, 127139959548, 2556064988652` (b-file, `n = 0..10`) |
| formula | none claimed |
| density | conjectured `a(n)/20^n -> C_3 * (13/20)/(20/27) = 0.2516208685`, where `C_3 = Product_p (1 - 1/p)^2*(1 + 2/p) = 0.2867474284` is A065473, the strongly carefree constant |
| lab | `lab/sequences/menger_pairwise/` |

`n = 6..10` were computed here and carry the provenance the b-file rests on:
`n = 6` agrees across all three generators, `n = 7` and `n = 8` across the
independent generator and `fast.c`, `n = 9` and `n = 10` from `fast.c` with the
chunk decomposition cross-checked. The whole b-file was afterwards reproduced
term for term by a clean recompile of `fast.c`, `n = 10` again matching its
chunk decomposition — verified. Because both halves of the condition are
pairwise, the independent generator builds a graph on `0..3^n-1` and counts
triangles by bitset row intersection, with `a(n) = 6*T`; divisibility by 6
comes free.
The `p = 3` factor `13/20` is proved and was re-derived — 13 of the 20 legal
digit patterns have no coordinate pair both 0 — while the limit itself is
conjecture, with ratios `0.2254762`, `0.2456612`, `0.2496157` at `n = 5, 8, 10`.
`C_3` was corrected here: it had been read off a truncated Euler product over
the primes below `3*10^6`, whose tail is larger than the digits it appeared to
settle, giving `0.2867474465` against the true `0.2867474284...` and a limit of
`0.2516208843` against `0.2516208685`. The product equals
`Product_p (1 - (3p-2)/p^3)`, which is A065473, and recomputing it from prime
zeta values — with `p = 2` handled exactly, since the log series sits on its
radius of convergence there — reproduces the OEIS digits. The empirical ratios
are unaffected; the gap at `n = 10` is `0.002`, a hundred thousand times the
correction.
The one dump hit, `15141288`, is a substring inside a term of A141148, not a
match.

## carpet_coprime - coprime pairs in the Sierpinski carpet

Pairs `(i,j)` below `3^n` with no base-3 position where both hold digit 1, and
`gcd(i,j) = 1`. The `8^n` admissible pairs are the cells of the level-n carpet.

| field | value |
|---|---|
| status | draft, no A-number |
| terms | `0, 4, 32, 274, 2320, 19178, 155392, 1248416, 10013432, 80226680, 642182854, 5138968090, 41117712068` |
| extension | `a(13) = 328967430388`, `a(14) = 2631822322616`, both confirmed by an independent second method |
| formula | none; no linear constant-coefficient recurrence of order 1..6 fits the 13 terms |
| density | conjectured `a(n)/8^n -> 189/(32*Pi^2) = 0.5984282409`, i.e. `6/Pi^2 * (1 - 1/8)/(1 - 1/9)` |
| lab | `lab/sequences/carpet_coprime/` |

Ratios `0.5980794`, `0.5983415`, `0.5984071` at `n = 10, 12, 14` with the gap
halving each level. The independent generator costs `3^n` rather than `8^n`:
carpet membership becomes bitmask disjointness and coprimality is removed by
`a(n) = sum_{d=1}^{3^n-1} mu(d)*(C(d) - 1)` with a zeta transform for `C(d)`.
Both scripts need numpy and are run through the project environment, not a
system python.

## vicsek_coprime - coprime pairs in the Vicsek fractal

Pairs `(i,j)` below `3^n` where every base-3 position has at least one
coordinate holding digit 1, and `gcd(i,j) = 1`. The admissible pairs are the
`5^n` cells of the level-n plus-shaped fractal. `a(0) = 0` because the only
pair is `(0,0)` and `gcd(0,0) = 0`.

| field | value |
|---|---|
| status | draft |
| terms | `0, 5, 16, 90, 418, 2178, 10560, 54120, 266478, 1338422, 6658480, 33439772` |
| formula | none; no constant-coefficient linear recurrence of order 1..6 fits the 15 terms |
| density | conjectured `a(n)/5^n -> Product_{p prime, p <> 3} (1 - 1/p^2) = 27/(4*Pi^2) = 0.6839179896` |
| lab | `lab/sequences/vicsek_coprime/` |

The b-file reaches `n = 14`, the last term summed from five parallel ranges of
`fast.c`. Support for the density is weak and non-monotonic — `0.6840242012`,
`0.6844535652`, `0.6836273784` at `n = 12, 13, 14` — but the fact underneath it
is exact and verified: no admissible pair has both coordinates divisible by 3,
so the Euler factor at `p = 3` is absent. The independent generator fixes `i`,
builds its partners as a coset of size `3^|S(i)|` about `(3^n-1)/2`, and
decides coprimality by disjointness of prime-divisor sets, never calling `gcd`.

## vicsek_perimeter - perimeter of the Vicsek fractal

The level-n plus fractal drawn as a polyomino on a `3^n x 3^n` grid: `a(n)` is
the number of unit edges separating a filled cell from an empty one or from the
exterior. Level 0 is a single cell, so `a(0) = 4`; level 1 is the plus
pentomino, `a(1) = 12`.

| field | value |
|---|---|
| status | draft; terms, b-file and formula all verified, but the draft's own PROGRAM block is off by one and must be fixed before submission |
| terms | `4, 12, 52, 252, 1252, 6252, 31252, 156252, 781252, 3906252, 19531252, 97656252` |
| formula | `a(n) = 2*5^n + 2`; `a(n) = 5*a(n-1) - 8` for `n > 0`; `a(n) = 6*a(n-1) - 5*a(n-2)` for `n > 1` |
| g.f. | `(4 - 12*x)/((1 - x)*(1 - 5*x))` |
| crossrefs | `a(n) = 2*A034474(n)` (`5^n + 1`); first differences hit A128625; family neighbours A381517 (carpet perimeter), A332705 (Menger surface), A000351 (`5^n`) |
| lab | `lab/sequences/vicsek_perimeter/` |

All three formulas hold on all 201 b-file terms and neither generator assumes
them. This is the one entry whose novelty was settled against the live OEIS
api as well as the dump: null for the sequence and for both shifts, with a
Fibonacci control returning results. The defect is in the draft text only —
its snippet seeds the grid with the level-1 tile and then applies `n` further
Kronecker steps, so it skips `a(1) = 12`; the fix is to seed with the 1x1
all-true matrix, as `terms.py` does.

## euler_xyz - Euler characteristic of the AND-zero solid

`V - E + F - C` of the cubical complex made of the unit cubes at all
`x, y, z` in `[0, 2^n)` with `x AND y AND z = 0`. That solid has `7^n` cubes at
level n, so it is the natural three-dimensional Sierpinski analogue.

| field | value |
|---|---|
| status | draft, not submitted |
| terms | `1, 2, 18, 157, 1211, 8832, 62908, 443627, 3115221, 21836062, 152940998, 1070852697` |
| formula | `a(n) = (13*7^(n-1) - 27*3^(n-1) + 38)/24`, offset 1 |
| recurrences | `a(n) = 7*a(n-1) + (3^n - 19)/2` for `n > 1`; `a(n) = 11*a(n-1) - 31*a(n-2) + 21*a(n-3)` for `n > 3` |
| g.f. | `x*(1 - 9*x + 27*x^2)/((1 - x)*(1 - 3*x)*(1 - 7*x))`, verified by series expansion |
| dimension | `log(7)/log(2) = 2.8073549220576...`, from the `7^n` cubes (A000420) at scale `2^n` |
| lab | `lab/sequences/euler_xyz/` |

The closed form matches all 60 b-file terms. The independent generator never
enumerates a cube: a cell is filled iff no binary position has all three bits
set, so each count is the accepting count of an 8-state automaton read over the
digit positions, with inclusion-exclusion turning each Minkowski dilation into
intersections of translates — linear in `n` instead of `7^n`. Beyond the
b-file, `a(61)` through `a(64)` are computed and reproduced digit-for-digit by
the closed form; a third, fully independent brute-force census confirms
`n = 1..6` and the `7^n` cube count.

## euler_xyorz - Euler characteristic of the AND-OR solid

The same construction with the condition `x AND (y OR z) = 0`, which gives a
`5^n`-cube solid and, unlike its sibling, a negative and rapidly diverging
characteristic.

| field | value |
|---|---|
| status | draft, not submitted |
| terms | `1, -2, -17, -92, -467, -2342, -11717, -58592, -292967, -1464842, -7324217, -36621092` |
| formula | `a(n) = (7 - 3*5^(n-1))/4`, offset 1 |
| recurrences | `a(n) = 5*a(n-1) - 7` for `n > 1`; `a(n) = 6*a(n-1) - 5*a(n-2)` for `n > 2` |
| g.f. | `x*(1 - 8*x)/((1 - x)*(1 - 5*x))` |
| crossrefs | A000351 (`5^n`, the cube count); `a(n) = 1 - A125833(n-1)` |
| lab | `lab/sequences/euler_xyorz/` |

All three formulas hold on all 100 b-file terms, and an earlier claimed form
`2 - 5^(n-1)` is disproved — it fails at `n = 2`. The independent generator
builds each x-slab from submasks of `~x`, never using the level `n-1` solid,
and computes the characteristic by inclusion-exclusion over the cubes with a
256-entry 2x2x2 window table, so it never counts a vertex, edge or face. Of
twelve novelty greps, eleven returned nothing; the twelfth found A125833 under
the transform `1 - a(n)`, which is the crossref above rather than a collision.

## design_D3 - three-dimensional design classes

Number of `n x n x n` binary arrays up to symmetry, the group being independent
cyclic rotation and reflection of the layers along each axis together with all
permutations of the three axes: `D_n^3` semidirect `S_3`, of order `48*n^3`.
Burnside gives `a(n) = (1/|G|) * Sum_{g in G} 2^c(g)`.

| field | value |
|---|---|
| status | draft |
| terms | `2, 22, 111618, 6005363762644688, 7089215977519836239803174210135872, 10157435539019790383692007859901914095646506996125324171134976, ...` (b-file, `n = 1..8`; `a(7)` has 100 digits and `a(8)` has 150) |
| formula | none claimed, none found |
| crossrefs | A255016, the two-dimensional toroidal parent (`1, 2, 6, 26, 805, 172112, ...`); A000616, where `A000616(3) = 22 = a(2)` |
| lab | `lab/sequences/design_D3/` |

Only eight terms exist, so all eight were compared rather than a prefix, and
brute-force orbit enumeration independently confirms `n = 1` and `n = 2`. A
third generator, written from the definition for the OEIS submission and
sharing no code with the other two, reproduces all eight; it is what promoted
`a(7)` and `a(8)` out of single-method status and into the b-file. A fourth
route, a Burnside-free flood fill over the orbits of all `2^27` colourings,
confirms `a(3) = 111618` without using Burnside's lemma at all. The
independent generator carries a group element as an affine map
`x -> M*x + t` with `M` a signed permutation matrix and gets the cycle count
from `c(g) = (1/m) * Sum_{k=1..m} |Fix(g^k)|`, so it never builds a cell-image
array and never walks a cycle. Recomputing the two-dimensional analogue under
the identical convention reproduces A255016 exactly, which validates the group
convention as the true three-dimensional promotion of that parent. Note that
`a(3) = 111618` is the same number the census reports as the full base-3,
dimension-3 design space.

## Established entries this builds on

Two existing OEIS sequences anchor the census work rather than the sequence
work, and research pages should cite them in that role.

| entry | counts | where it lands |
|---|---|---|
| A000616 | designs up to symmetry, as the dimension grows at base 2: `3, 6, 22, 402, 1228158, 400507806843728` | the shape-class column of the fill-class census; `A000616(3) = 22` is also `design_D3(2)` |
| A129824 | fill classes, `Prod_{k=0}^{D} (1 + C(D,k))`: `2, 4, 12, 64, 700, 17424, ...` at offset 0 | the count of distinct base-2 fractals in dimension D |

A129824 is an identity, not a numerical coincidence. Two base-2 designs are the
same fractal — equal fill at every side `n` and level `L`, hence equal fractal
dimension — exactly when they share a popcount profile, the number of filled
corners of each Hamming weight; A129824 counts the possible shapes
`(k_0..k_n)` of a collection of subsets of an n-set, which is precisely such a
profile. The proof is the linear independence of the `D+1` functions
`E^(D-w) * O^w` in `n`. The closed form reproduces all 16 published terms with
no shift; two independent fill generators agree on every design at `D = 2` and
`D = 3`, and the class count matches A129824 for `D = 1..4` with zero
profile collisions. A000616 is quoted from the research notes and was not
recomputed; its offset is `-1`, so `A000616(3) = 22`.

Both live in `lab/census/`. The fill-class census is `lab/census/fillclasses/`:
58 canonical designs across base 2 and 3 in dimensions 2 and 3, each validated
cell by cell against an independently rendered array, with orbit counts checked
against a Burnside average. Two caveats travel with it — the group includes
translations, which are symmetries of the infinite tiling but not of the
truncation, so the level-1 fill sequence is not constant on an orbit (what is
invariant is the tail: degree and leading coefficient, hence the fractal
dimension); and at base 2 the carpet and the net are one orbit, so only one
label survives.

The coprimality census is `lab/census/coprime/`: 763 designs across bases 2, 3
and 6, each measured against the predicted density
`delta = B(F) * (1/zeta(D)) * Prod_{p | q} (1 - p^(-D))^(-1)`. Zero designs are
flagged. Read the status honestly, though — the `OK` verdict is numerical
agreement at a shallow level within a flat tolerance of `0.06`, not a proof, and
the deepest level is capped at 200000 points, so the sponge only reaches
`n = 4`. The solid half is the exact base-local identity
`#{x in S_n : e | x_i for all i} = k_e * k^(n-1)`, which is a theorem and passes
on all 763 lines. The density formula itself is proved prime by prime but not in
the interchange-of-limits step; the census tests it and does not close it.