A001316 -id:A001316 - cp's OEIS frontend

A151930 First differences of A001316.

1, 0, 2, -2, 2, 0, 4, -6, 2, 0, 4, -4, 4, 0, 8, -14, 2, 0, 4, -4, 4, 0, 8, -12, 4, 0, 8, -8, 8, 0, 16, -30, 2, 0, 4, -4, 4, 0, 8, -12, 4, 0, 8, -8, 8, 0, 16, -28, 4, 0, 8, -8, 8, 0, 16, -24, 8, 0, 16, -16, 16, 0, 32, -62, 2, 0, 4, -4, 4, 0, 8, -12, 4, 0, 8, -8, 8, 0, 16, -28, 4, 0, 8, -8, 8, 0, 16, -24, 8, 0, 16, -16, 16, 0, 32

Offset: 0

N. J. A. Sloane, Aug 10 2009

sign

Net increase in number of ON cells at generation n of 1-D CA using Rule 90.

Index entries for sequences related to cellular automata

Cf. A001316, A220466.

nmax := 94: A001316 := n -> if n<=-1 then 0 else 2^add(i, i=convert(n, base, 2)) fi: for p from 0 to ceil(log[2](nmax))+1 do for n from 0 to nmax/(p+2)+1 do a((2*n+1)*2^p-1) := (2-2^p) * A001316(n) od: od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Jan 25 2013

a((2*n+1)*2^p-1) = (2-2^p) * A001316(n), p >= 0 and n >=0. - Johannes W. Meijer, Jan 25 2013

G.f.: -1/x + ((1 - x)/x)*Product_{k>=0} (1 + 2*x^(2^k)). - Ilya Gutkovskiy, Feb 28 2017

A167275 Row sums of triangle A167274 (a variant of Gould's sequence A001316).

Original entry on oeis.org

1, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 4, 8, 8, 16, 8, 16, 16, 32

Offset: 0

Gary W. Adamson & Mats Granvik, Oct 31 2009

nonn

			a(3) = 8 = 2*A001316 = 2*4. a(3) = 8 = (1 + 3 + 3 + 1); where (1, 3, 3, 1) = row 3 of triangle A167274.

Cf. A167274

Given Gould's sequence, A001316, (1, 2, 2, 4, 2, 4, 4, 8,...); for a(n)>1,

a(n) = 2*A001316(n).

A256257 6 times numbers of Gould's sequence A001316.

Original entry on oeis.org

6, 12, 12, 24, 12, 24, 24, 48, 12, 24, 24, 48, 24, 48, 48, 96, 12, 24, 24, 48, 24, 48, 48, 96, 24, 48, 48, 96, 48, 96, 96, 192, 12, 24, 24, 48, 24, 48, 48, 96, 24, 48, 48, 96, 48, 96, 96, 192, 24, 48, 48, 96, 48, 96, 96, 192, 48, 96, 96, 192, 96, 192, 192, 384, 12, 24, 24, 48, 24, 48, 48, 96, 24, 48, 48, 96, 48, 96, 96, 192

Offset: 0

Omar E. Pol, Mar 20 2015

Also, number of triangular cells turned ON at (n+1)-st stage in the structure of A256256.

First differences of A256256.

			Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
6;
12;
12, 24;
12, 24, 24, 48;
12, 24, 24, 48, 24, 48, 48, 96;
12, 24, 24, 48, 24, 48, 48, 96, 24, 48, 48, 96, 48, 96, 96, 192;
...

Cf. A001316, A011782, A047999, A048896, A117973, A151787, A160713, A160721, A256256.

a(n) = 6*A001316(n) = 3*A117973(n) = 2*A160713(n).

a(n) = 12*A048896(n-1), n >= 1.

A368655 Binomial transform of Gould's sequence (A001316).

Original entry on oeis.org

1, 3, 7, 17, 39, 85, 181, 387, 839, 1829, 3953, 8391, 17461, 35759, 72559, 146921, 298631, 611733, 1265185, 2641351, 5555729, 11735571, 24798755, 52219493, 109213269, 226322799, 464125219, 941694917, 1891879215, 3769497853, 7465462669, 14735667195, 29070011399

Offset: 0

Joseph M. Shunia, Jan 02 2024

Consider the multivariate polynomial quotient ring K'n = Z[x_1, x_2, x_3, ..., x_n]/I where I = <x_1^2 - P_1, x_2^2 - P_2, ..., x_n^2 - P_n> is an ideal in Z[x_1, x_2, x_3, ..., x_n]. Here, each polynomial P_i = -2x_i + x{i+1} for 0 < i <= n, with x_{n+1} assumed to be 1. In this ring, every variable x_i for 0 < i <= n satisfies the recursive relation x_i^2 = -2x_i + x_{i+1}. The n-th term of this sequence is obtained by expanding the polynomial (2 + x_1)^n within the ring K'_n and evaluating at x_1 = x_2 = ... = x_n = 1. For a detailed explanation and proof, refer to Shunia's paper under links.

Joseph M. Shunia, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Plot of a(n)/a(n-1) for n = 2..2000
Joseph M. Shunia, A Polynomial Ring Connecting Central Binomial Coefficients and Gould's Sequence, arXiv:2312.00302 [math.GM], 2023.

Cf. A000120, A001316 (Gould's sequence).

Table[Sum[Binomial[n, k] * 2^DigitCount[k, 2, 1], {k, 0, n}], {n, 0, 32}] (* Vaclav Kotesovec, Apr 02 2024 *)

{a(n) = sum(k=0, n, binomial(n,k) * 2^hammingweight(k))};

def a(n):
    R = PolynomialRing(ZZ, n, 'x')
    x = R.gens()
    I_list = [x[i]^2 - (-2*x[i] + x[i+1]) if i < n-1 else x[i]^2 for i in range(n)]
    I = R.ideal(I_list)
    K_n = R.quotient(I, 'x')
    p_n = K_n((x[0]+2)^n)
    subs_dict = {x[i]: 1 for i in range(n)}
    a_n = p_n.lift().subs(subs_dict)
    return a_n # Joseph M. Shunia, Mar 22 2024

a(n) = Sum_{k=0..n} binomial(n,k)*A001316(k).

a(n) = Sum_{k=0..n} binomial(n,k)*2^(A000120(k)).

A130831 Irregular triangle read by rows: row(n) contains the first A001146(n) terms of A001316.

Original entry on oeis.org

1, 2, 1, 2, 2, 4, 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 4, 8, 8, 16, 8, 16, 16, 32, 8, 16, 16, 32, 16, 32, 32, 64, 2, 4, 4, 8, 4, 8, 8, 16

Offset: 1

Roger L. Bagula, Aug 20 2007

The n-th row of the triangle consists of the first A001146(n) terms of A001316. - Benjamin Heiland, Dec 12 2011

			Triangle begins:
{1, 2},
{1, 2, 2, 4},
{1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16}
...

Cf. A001146, A060803, A130830.

Edited by N. J. A. Sloane, Jun 07 2008

New name using Benjamin Heiland's comment, Joerg Arndt, May 11 2023

A160110 Numerators of |Bernoulli(n)*Gould(n)| for even n, (Gould A001316).

Original entry on oeis.org

1, 1, 1, 2, 1, 10, 1382, 28, 3617, 87734, 349222, 3418052, 472728182, 34212412, 94997844116, 68926730208040, 7709321041217, 5155375716734, 52630543106106954746, 11719975655366236, 522165436992898244102

Offset: 0

Peter Luschny, May 02 2009

A001897 give the denominators of |Bernoulli(n)*Gould(n)| for even n, also the denominators of the cosecant numbers.

G. C. Greubel, Table of n, a(n) for n = 0..314

Cf. A001897.

b := n -> bernoulli(n)*2^add(i,i=convert(n,base,2));
a := n -> numer(abs(b(2*n)));

G[n_] := Sum[Mod[Binomial[n, k], 2], {k, 0, n}]; (* A001316 *) Table[Abs[BernoulliB[n]*G[n]], {n, 0, 20}][[1 ;; -1 ;; 2]]//Numerator (* G. C. Greubel, Sep 25 2018 *)

A160713 3 times numbers of Gould's sequence: a(n) = A001316(n)*3.

Original entry on oeis.org

3, 6, 6, 12, 6, 12, 12, 24, 6, 12, 12, 24, 12, 24, 24, 48, 6, 12, 12, 24, 12, 24, 24, 48, 12, 24, 24, 48, 24, 48, 48, 96, 6, 12, 12, 24, 12, 24, 24, 48, 12, 24, 24, 48, 24, 48, 48, 96, 12, 24, 24, 48, 24, 48, 48, 96, 24, 48, 48, 96, 48, 96, 96, 192, 6, 12, 12, 24, 12, 24, 24, 48

Offset: 0

Omar E. Pol, May 25 2009

nonn

			From _Omar E. Pol_, Aug 09 2009: (Start)
If written as a triangle:
3;
6;
6,12;
6,12,12,24;
6,12,12,24,12,24,24,48;
6,12,12,24,12,24,24,48,12,24,24,48,24,48,48,96;
6,12,12,24,12,24,24,48,12,24,24,48,24,48,48,96,12,24,24,48,24,48,48,96,24,...
(End)

Cf. A001316.

read("transforms3") ; L := BFILETOLIST("b001316.txt") ; for n from 1 to 120 do printf("%d,",3*op(n,L)) ; od: # R. J. Mathar, Jul 13 2009

More terms from R. J. Mathar, Jul 13 2009

A165641 A091137(n) / A001316(n) .

Original entry on oeis.org

1, 1, 6, 6, 360, 360, 15120, 15120, 1814400, 1814400, 119750400, 119750400, 653837184000, 653837184000, 3923023104000, 3923023104000, 16005934264320000, 16005934264320000, 12772735542927360000, 12772735542927360000, 8430005458332057600000, 8430005458332057600000

Offset: 0

Paul Curtz, Sep 23 2009

nonn

Cf. A165636, A036281, A002445.

Edited and extended by R. J. Mathar, Sep 25 2009

A268433 a(n) = A106184(n) / A001316(n).

Original entry on oeis.org

1, 1, 5, 7, 59, 95, 377, 655, 10163, 18459, 71099, 132641, 1021455, 1937515, 7477505, 14335423, 443971523, 857241875, 3328921191, 6459762413, 50311588373, 97986366561, 382518036575, 747066030569, 11690129046071, 22881444619663, 89673873841559, 175837008468485

Offset: 0

Peter Luschny, Feb 24 2016

nonn

Cf. A000120, A000984, A001316, A106184.

H := n -> hypergeom([1/2,-n/2,-n/2+1/2],[-n/2+3/4,-n/2+1/4],1/2):
A000984 := n -> 4^n*hypergeom([-n,1/2],[1],1):
A001316 := n -> 2^(add(i, i = convert(n, base, 2))):
a := n -> H(n)*A000984(n)/A001316(n): seq(simplify(a(n)),n=0..27);

a(n) = 2^(2*n-A000120(n))*hypergeometric([-n,1/2],[1],1)*hypergeometric([1/2,-n/2,-n/2+1/2],[-n/2+3/4,-n/2+1/4],1/2).

A102377 Gould's sequence A001316 in binary.

Original entry on oeis.org

1, 10, 10, 100, 10, 100, 100, 1000, 10, 100, 100, 1000, 100, 1000, 1000, 10000, 10, 100, 100, 1000, 100, 1000, 1000, 10000, 100, 1000, 1000, 10000, 1000, 10000, 10000, 100000, 10, 100, 100, 1000, 100, 1000, 1000, 10000, 100, 1000, 1000, 10000, 1000

Offset: 0

Paul Barry, Jan 05 2005

Cf. A001316, A048883, A102376.

a(n) = 10^hammingweight(n); \\ Kevin Ryde, Jan 11 2024

Formulas due to Paul D. Hanna:
a(n) = 10^A000120(n).
a(n) = Product_{k=0..log_2(n)} 10^b(n,k) where b(n,k) = coefficient of 2^k in binary expansion of n.
a(n) = Sum_{k=0..n} (C(n,k) mod 2)*9^A000120(n-k).
G.f.: Product_{k>=0} 1 + 10*x^(2^k).

cp's OEIS Frontend

A151930 First differences of A001316.

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A167275 Row sums of triangle A167274 (a variant of Gould's sequence A001316).

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A256257 6 times numbers of Gould's sequence A001316.

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A368655 Binomial transform of Gould's sequence (A001316).

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A130831 Irregular triangle read by rows: row(n) contains the first A001146(n) terms of A001316.

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A160110 Numerators of |Bernoulli(n)*Gould(n)| for even n, (Gould A001316).

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A160713 3 times numbers of Gould's sequence: a(n) = A001316(n)*3.

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A165641 A091137(n) / A001316(n) .

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A268433 a(n) = A106184(n) / A001316(n).

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