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.

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.

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.

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.

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.

n = 6..10 are computed here and are not in the b-file: 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. 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. 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.

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.

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.

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.

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.

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).

Only five terms exist, so all five were compared rather than a prefix, and brute-force orbit enumeration independently confirms n = 1 and n = 2. 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.

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.