A070990 -id:A070990 - cp's OEIS frontend

A281048 Expansion of x(1 - x)Product_{k>=0} (1 + x^(2^k) - x^(2^(k+1))).

1, 0, -1, 1, -2, 1, 1, 0, -3, 1, 2, -1, 1, 0, -1, 1, -4, 1, 3, -2, 3, -1, -2, 1, 1, 0, -1, 1, -2, 1, 1, 0, -5, 1, 4, -3, 5, -2, -3, 1, 4, -1, -3, 2, -3, 1, 2, -1, 1, 0, -1, 1, -2, 1, 1, 0, -3, 1, 2, -1, 1, 0, -1, 1, -6, 1, 5, -4, 7, -3, -4, 1, 7, -2, -5, 3, -4, 1, 3, -2, 5, -1, -4, 3, -5, 2, 3, -1, -4, 1, 3, -2, 3, -1, -2, 1, 1, 0, -1, 1

Offset: 1

Ilya Gutkovskiy, Feb 27 2017

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First differences of A005590.

Michael Gilleland, Some Self-Similar Integer Sequences
Ilya Gutkovskiy, Extended graphical example
R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.

Cf. A005590, A070990, A182093.

Rest[CoefficientList[Series[x (1 - x) Product[1 + x^2^k - x^2^(k + 1), {k, 0, 15}], {x, 0, 100}], x]]
Differences[a[0] = 0; a[1] = 1; a[n_] := a[n] = If[OddQ[n], a[(n-1)/2 + 1] - a[(n-1)/2], a[n/2]]; Table[a[n], {n, 0, 100}]]

G.f.: x*(1 - x)*Product_{k>=0} (1 + x^(2^k) - x^(2^(k+1))).

A283104 Second differences of Stern's diatomic sequence (A002487).

Original entry on oeis.org

-1, 1, -2, 3, -3, 2, -3, 5, -4, 3, -5, 6, -5, 3, -4, 7, -5, 4, -7, 9, -8, 5, -7, 10, -7, 5, -8, 9, -7, 4, -5, 9, -6, 5, -9, 12, -11, 7, -10, 15, -11, 8, -13, 15, -12, 7, -9, 14, -9, 7, -12, 15, -13, 8, -11, 15, -10, 7, -11, 12, -9, 5, -6, 11, -7, 6, -11, 15, -14, 9, -13, 20, -15, 11, -18, 21, -17, 10, -13, 21, -14, 11, -19

Offset: 0

Ilya Gutkovskiy, Feb 28 2017

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Michael Gilleland, Some Self-Similar Integer Sequences
Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Stern's Diatomic Series
Index entries for sequences related to Stern's sequences

Cf. A002487, A070990.

Differences[a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 84}], 2]
CoefficientList[Series[-1/x + ((1 - x)^2/x) Product[1 + x^2^k + x^2^(k + 1), {k, 0, 20}], {x, 0, 82}], x]

G.f.: -1/x + ((1 - x)^2/x)*Product_{k>=0} (1 + x^(2^k) + x^(2^(k+1))).

A309014 a(n) = Sum_{k=0..n} (-1)^(n-k) * (Stirling2(n,k) mod 2).

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 1, 2, 3, 1, 2, 3, 3, 2, 1, 3, 4, 1, 3, 4, 5, 3, 2, 5, 5, 2, 3, 5, 4, 3, 1, 4, 5, 1, 4, 5, 7, 4, 3, 7, 8, 3, 5, 8, 7, 5, 2, 7, 7, 2, 5, 7, 8, 5, 3, 8, 7, 3, 4, 7, 5, 4, 1, 5, 6, 1, 5, 6, 9, 5, 4, 9, 11, 4, 7, 11, 10, 7, 3, 10, 11, 3, 8, 11, 13, 8, 5, 13, 12, 5

Offset: 0

Ilya Gutkovskiy, Jul 06 2019

nonn

Cf. A001316, A002487, A007306, A008277, A048993, A060632, A070990.

Table[Sum[(-1)^(n - k) Mod[StirlingS2[n, k], 2], {k, 0, n}], {n, 0, 90}]
nmax = 90; CoefficientList[Series[1 + x (1 + x^3) Product[(1 + x^(2^k) + x^(2^(k + 1))), {k, 1, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

a(n) = sum(k=0, n, (-1)^(n-k) * (stirling(n,k,2) % 2)); \\ Michel Marcus, Jul 06 2019

G.f.: 1 + x * (1 + x^3) * Product_{k>=1} (1 + x^(2^k) + x^(2^(k+1))).

a(0) = 1; a(2*k+1) = A002487(k+1); a(2*k+2) = A002487(k).

cp's OEIS Frontend

A281048 Expansion of x(1 - x)Product_{k>=0} (1 + x^(2^k) - x^(2^(k+1))).

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A283104 Second differences of Stern's diatomic sequence (A002487).

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Mathematica

Formula

A309014 a(n) = Sum_{k=0..n} (-1)^(n-k) * (Stirling2(n,k) mod 2).

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cp's OEIS Frontend

A281048 Expansion of x*(1 - x)*Product_{k>=0} (1 + x^(2^k) - x^(2^(k+1))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A283104 Second differences of Stern's diatomic sequence (A002487).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A309014 a(n) = Sum_{k=0..n} (-1)^(n-k) * (Stirling2(n,k) mod 2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A281048 Expansion of x(1 - x)Product_{k>=0} (1 + x^(2^k) - x^(2^(k+1))).