A166966 -id:A166966 - cp's OEIS frontend

A166967 Triangle read by rows, (Sierpinski's gasket, A047999) * A166966 (diagonalized as a lower triangular matrix).

1, 1, 1, 1, 0, 2, 1, 1, 2, 3, 1, 0, 0, 0, 7, 1, 1, 0, 0, 7, 8, 1, 0, 2, 0, 7, 0, 17, 1, 1, 2, 3, 7, 8, 17, 27, 1, 0, 0, 0, 0, 0, 0, 0, 66, 1, 1, 0, 0, 0, 0, 0, 0, 66, 67, 1, 0, 2, 0, 0, 0, 0, 0, 66, 0, 135, 1, 1, 2, 3, 0, 0, 0, 0, 66, 67, 135, 204

Offset: 0

Gary W. Adamson, Oct 25 2009

An eigentriangle (a given triangle * its own eigensequence); in this case A047999 * A166966.

Triangle A166967 has the properties of: row sums = the eigensequence, A166966 and sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
1, 1;
1, 1, 2, 3;
1, 0, 0, 0, 7;
1, 1, 0, 0, 7, 8;
1, 0, 2, 0, 7, 0, 17;
1, 1, 2, 3, 7, 8, 17, 27;
1, 0, 0, 0, 0, 0,..0,..0, 66;
1, 1, 0, 0, 0, 0,..0,..0, 66, 67;
1, 0, 2, 0, 0, 0,..0,..0, 66,..0, 135;
1, 1, 2, 3, 0, 0,..0,..0, 66, 67, 135, 204;
1, 0, 0, 0, 7, 0,..0,..0, 66,..0,...0,...0, 479;
1, 1, 0, 0, 7, 8,..0,..0, 66, 67,...0,...0, 479, 553
1, 0, 2, 0, 7, 0, 17,..0, 66,..0, 135,...0, 479,...0, 1182;
1, 1, 2, 3, 7, 8, 17, 27, 66, 67, 135, 204, 479, 553, 1182, 1189;
...

Cf. A047999, A166966.

Let Sierpinski's gasket, A047999 = S; and Q = the eigensequence of A047999 prefaced with a 1: (1, 1, 2, 3, 7, 8, 17,...) then diagonalized as an infinite lower triangular matrix: [1; 0,1; 0,0,2; 0,0,0,3; 0,0,0,0,7,...].

Triangle A166967 = S * Q.

A380652 Shifts left one place under the inverse modulo 2 binomial transform.

Original entry on oeis.org

1, 1, 0, -1, -1, -2, -1, 1, 4, 3, -1, -4, -1, -3, 0, 5, 5, 4, -1, -5, -2, -5, -1, 6, 9, 1, -6, -5, 13, 14, 11, 1, -38, -39, -1, 38, 41, 81, 42, -37, -163, -128, 37, 167, 56, 143, 11, -214, -253, -219, 36, 257, 149, 328, 105, -303, -624, -247, 313, 490, -455, -387, -476, -417, 1251, 1250

Offset: 0

Ilya Gutkovskiy, Jan 29 2025

sign

Cf. A000587, A010060, A047999, A166966.

a[0] = 1; a[n_] := a[n] = Sum[(-1)^ThueMorse[k] Mod[Binomial[n - 1, k], 2] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 65}]

from functools import lru_cache
@lru_cache(maxsize=None)
def A380652(n): return sum((-A380652(n-k-1) if k.bit_count()&1 else A380652(n-k-1)) for k in range(n) if not (k&~(n-1))) if n else 1 # Chai Wah Wu, Feb 11 2025

a(0) = 1; a(n) = Sum_{k=0..n-1} (-1)^A010060(k) * (binomial(n-1,k) mod 2) * a(n-k-1).

A331519 a(0) = 1; a(n) = Sum_{k=1..n} (binomial(n,k) mod 2) * a(k-1) * a(n-k).

Original entry on oeis.org

1, 1, 1, 3, 3, 9, 15, 63, 63, 189, 315, 1323, 1701, 6237, 12285, 59535, 59535, 178605, 297675, 1250235, 1607445, 5893965, 11609325, 56260575, 63761985, 213790185, 393824025, 1811590515, 2531725875, 10025634465, 21210236775, 109876902975, 109876902975, 329630708925

Offset: 0

Ilya Gutkovskiy, Jan 19 2020

nonn

Cf. A000108, A001147, A047749, A047999, A062177, A166966.

a[0] = 1; a[n_] := a[n] = Sum[Mod[Binomial[n, k], 2] a[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]

A331520 a(0) = a(1) = 1; a(n+2) = Sum_{k=0..n} (binomial(n,k) mod 2) * a(k).

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 3, 9, 7, 24, 8, 33, 17, 77, 27, 134, 66, 351, 67, 419, 135, 908, 204, 1469, 479, 3643, 553, 4572, 1182, 10227, 1889, 17125, 4641, 43640, 4642, 48283, 9285, 101211, 13929, 158786, 32504, 384441, 37153, 465259, 78957, 1020640, 125414, 1675453

Offset: 0

Ilya Gutkovskiy, Jan 19 2020

nonn

Shifts 2 places left under the modulo 2 binomial transform.

Cf. A007476, A010060, A047999, A166966.

a[0] = a[1] = 1; a[n_] := a[n] = Sum[Mod[Binomial[n - 2, k], 2] a[k], {k, 0, n - 2}]; Table[a[n], {n, 0, 47}]

a(n) = Sum_{k=0..n} (-1)^A010060(n-k) * (binomial(n, k) mod 2) * a(k+2).

cp's OEIS Frontend

A166967 Triangle read by rows, (Sierpinski's gasket, A047999) * A166966 (diagonalized as a lower triangular matrix).

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A380652 Shifts left one place under the inverse modulo 2 binomial transform.

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Formula

A331519 a(0) = 1; a(n) = Sum_{k=1..n} (binomial(n,k) mod 2) * a(k-1) * a(n-k).

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Mathematica

A331520 a(0) = a(1) = 1; a(n+2) = Sum_{k=0..n} (binomial(n,k) mod 2) * a(k).

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