A000123 -id:A000123 - cp's OEIS frontend

A339277 Number of partitions of 2*n into powers of 2 where every part appears at least 2 times.

1, 1, 2, 3, 5, 6, 9, 11, 16, 19, 25, 30, 39, 45, 56, 65, 81, 92, 111, 127, 152, 171, 201, 226, 265, 295, 340, 379, 435, 480, 545, 601, 682, 747, 839, 920, 1031, 1123, 1250, 1361, 1513, 1640, 1811, 1963, 2164, 2335, 2561, 2762, 3027, 3253, 3548, 3813, 4153, 4448, 4827, 5167

Offset: 0

Ilya Gutkovskiy, Nov 29 2020

nonn

			a(4) = 5 because we have [4, 4], [2, 2, 2, 2], [2, 2, 2, 1, 1], [2, 2, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

Index entries for sequences related to partitions

Cf. A000079, A000123, A007690, A018819, A339279.

nmax = 55; CoefficientList[Series[(1/(1 - x^3)) Product[1/(1 - x^(2^k)), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
A000123[0] = 1; A000123[n_] := A000123[n] = A000123[Floor[n/2]] + A000123[n - 1]; a[n_] := Sum[ChebyshevU[k, -1/2] A000123[n - k], {k, 0, n}]; Table[a[n], {n, 0, 55}]

G.f.: (1/(1 - x^3)) * Product_{k>=0} 1/(1 - x^(2^k)).
G.f.: (1/(1 - x)) * Product_{k>=0} (1 + x^(2^(k+1))/(1 - x^(2^k))).
a(n) = [x^(2*n)] Product_{k>=0} (1 + x^(2^(k+1))/(1 - x^(2^k))).
a(n) = Sum_{k=0..n} U(k,-1/2) * A000123(n-k), where U(k,x) is the Chebyshev U-polynomial.

A346912 a(0) = 1; a(n) = a(n-1) + a(floor(n/2)) + 1.

Original entry on oeis.org

1, 3, 7, 11, 19, 27, 39, 51, 71, 91, 119, 147, 187, 227, 279, 331, 403, 475, 567, 659, 779, 899, 1047, 1195, 1383, 1571, 1799, 2027, 2307, 2587, 2919, 3251, 3655, 4059, 4535, 5011, 5579, 6147, 6807, 7467, 8247, 9027, 9927, 10827, 11875, 12923, 14119, 15315

Offset: 0

Ilya Gutkovskiy, Aug 11 2021

nonn

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000123, A018819, A022907, A033485, A102378, A102379.

f:= proc(n) option remember; procname(n-1) + procname(floor(n/2)) + 1 end proc;
f(0):= 1:
map(f, [$1..50]); # Robert Israel, May 04 2025

a[0] = 1; a[n_] := a[n] = a[n - 1] + a[Floor[n/2]] + 1; Table[a[n], {n, 0, 47}]
nmax = 47; CoefficientList[Series[(1/(1 - x)) (-1 + 2 Product[1/(1 - x^(2^k)), {k, 0, Floor[Log[2, nmax]]}]), {x, 0, nmax}], x]

from itertools import islice
from collections import deque
def A346912_gen(): # generator of terms
    aqueue, f, b, a = deque([2]), True, 1, 2
    yield from (1, 3, 7)
    while True:
        a += b
        yield 4*a - 1
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A346912_list = list(islice(A346912_gen(),40)) # Chai Wah Wu, Jun 08 2022

G.f.: (1/(1 - x)) * (-1 + 2 * Product_{k>=0} 1/(1 - x^(2^k))).
a(n) = n + 1 + Sum_{k=1..n} a(floor(k/2)).
a(n) = 2 * A000123(n) - 1.
a(n) = 4 * A033485(n) - 1 for n > 0. - Hugo Pfoertner, Aug 12 2021
From Michael Tulskikh, Aug 12 2021: (Start)
2*a(2n) = a(2n-1) + a(2n+1).
a(2n) = a(2n-2) + a(n-1) + a(n) + 2.
a(2n) = 2*(Sum_{i=0..n} a(i)) - a(n) + 2n. (End)

A377556 E.g.f.: exp(Sum_{n>=1} A006519(n) * x^n).

Original entry on oeis.org

1, 1, 5, 19, 193, 1181, 13021, 117895, 1868609, 20980153, 348219541, 4940639771, 98898110785, 1632238421269, 34910480911853, 672959412044431, 16733065940227201, 359936040496423025, 9469928134781142949, 229631546862609396643, 6716832478519734558401, 178344294076141938008461

Offset: 0

Vaclav Kotesovec, Nov 01 2024

nonn

Cf. A000123, A006519, A038500, A161809, A377555.

nmax = 25; CoefficientList[Series[Exp[Sum[2^IntegerExponent[k, 2]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0,nmax]!

A381810 Array read by downward antidiagonals: A(n,k) is a generalization of odd columns of A125790 defined in Comments for n > 0, k >= 0.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 8, 36, 20, 10, 10, 64, 42, 84, 14, 12, 100, 72, 286, 100, 20, 14, 144, 110, 680, 322, 120, 26, 16, 196, 156, 1330, 744, 364, 140, 36, 18, 256, 210, 2300, 1430, 816, 406, 656, 46, 20, 324, 272, 3654, 2444, 1540, 888, 3396, 740, 60, 22, 400, 342, 5456, 3850, 2600, 1650, 10816, 3682, 840, 74

Offset: 1

Mikhail Kurkov, May 05 2025

This is generalization in the sense that first column of A125790 is A000123(2^(n-1)) while in this square array column zero is conjecturally A000123(n).
A(n,k) = v_{A001511(n)} where we start with vector v of fixed length L(n) = A070939(n) with elements v_i = A125790(i,2*k+1), pre-calculate A078121 up to L(n)-th row, reserve t as an empty vector of fixed length L(n) and for i=1..A119387(n+1), for j=1..L(n)-i+1 apply t := v (at the beginning of each cycle for i) and also apply v_j := Sum_{k=1..j+1} A078121(j,k-1)*t_k if R(n,L(n)-i) = 1, otherwise v_j := Sum_{k=1..j+1} A078121(j,k-1)*t_k*(-1)^(j+k+1). Here R(n,k) = floor(n/(2^k)) mod 2 is the (k+1)-th bit in the binary expansion of n.
Conjecture: sequence A(n,k) for fixed n is a polynomial of degree A070939(n).

			Array begins:
===========================================================
n\k|  0    1     2      3      4      5       6       7 ...
---+-------------------------------------------------------
1  |  2,   4,    6,     8,    10,    12,     14,     16 ...
2  |  4,  16,   36,    64,   100,   144,    196,    256 ...
3  |  6,  20,   42,    72,   110,   156,    210,    272 ...
4  | 10,  84,  286,   680,  1330,  2300,   3654,   5456 ...
5  | 14, 100,  322,   744,  1430,  2444,   3850,   5712 ...
6  | 20, 120,  364,   816,  1540,  2600,   4060,   5984 ...
7  | 26, 140,  406,   888,  1650,  2756,   4270,   6256 ...
8  | 36, 656, 3396, 10816, 26500, 55056, 102116, 174336 ...
  ...

Cf. A000123, A001511, A007814, A053645, A062383, A070939, A078121, A106400, A119387, A125790, A236206.

upto1(n) = my(v1); v1 = vector(n+1, i, vector(i, j, j==1 || j==i)); for(i=2, n, for(j=1, i-1, v1[i+1][j+1] = sum(k=j-1, i-1, v1[i][k+1]*v1[k+1][j]))); v1
A(n,m) = my(L = logint(n,2), A = valuation(n,2), B = logint(n>>A,2), v1, v2, v3); v1 = upto1(L+2); v2 = vector(L+2, i, vecsum(v1[i])); for(i=1, 2*m, v2 = vector(L+2, i, sum(j=1, i, v1[i][j]*v2[j]))); for(i=1, B, v3 = v2; for(j=1, L-i+1, v2[j+1] = sum(k=1, j+1, v1[j+1][k]*v3[k+1]*if(!bittest(n,L-i+1), (-1)^(j+k+1), 1)))); v2[A+2]

upto1(n) = my(v1); v1 = vector(n+1, i, vector(i, j, j==1 || j==i)); for(i=2, n, for(j=1, i-1, v1[i+1][j+1] = sum(k=j-1, i-1, v1[i][k+1]*v1[k+1][j]))); v1
upto2(n,m) = my(L = logint(n,2), A = valuation(n,2), B = logint(n>>A,2), v1, v2, v3, v4, v5); v1 = upto1(L+2); v2 = vector(L+2, i, 1); v3 = vector(m+1, i, 0); for(s=0, m, for(i=1, min(s+1,2), v2 = vector(L+2, i, sum(j=1, i, v1[i][j]*v2[j]))); v4 = v2; for(i=1, B, v5 = v4; for(j=1, L-i+1, v4[j+1] = sum(k=1, j+1, v1[j+1][k]*v5[k+1]*if(!bittest(n,L-i+1), (-1)^(j+k+1), 1)))); v3[s+1] = v4[A+2]); v3 \\ slightly modified version of the first program, some kind of memoization; generates A(n,k) for k=0..m

A(2^(n-1),k) = A125790(n,2*k+1) for n > 0, k >= 0.
Conjectured formulas: (Start)
A(n,0) = A000123(n) for n > 0.
A(n,k) = Sum_{j=0..k} A000123(A062383(n)*j+n)*A106400(k-j) for n > 0, k >= 0.
If we change v_i = A125790(i,2*k+1) to v_i = A125790(i,2*k) to get similar generalization of even columns, then for resulting array B(n,k) we have B(n,k) = Sum_{j=0..k} A000123(A062383(n)*j+A053645(n))*A106400(k-j) for n > 0, k >= 0.
2*(k+1) divides A(n,k) for n > 0 if (k+1) is a term of A236206.
G.f. for n-th row is f(A070939(n)+1,n) for n > 0 where f(n,k) = (Sum_{(c_0 + c_1 + ... + c_{n-1}) == 2*k (mod 2^n), 0 <= c_i < 2^n, 2^i divides c_i} x^((c_0 + c_1 + ... + c_{n-1} - 2*k)/2^n))/(1-x)^n for n > 0, k >= 0. Similarly, g.f. for n-th row of B(n,k) is f(A070939(n)+1,A053645(n)).
G.f. for n-th row is (Sum_{i=0..L(n)-1} x^i * Sum_{j=0..i} binomial(L(n)+1,j)*A(n,i-j)*(-1)^j)/(1-x)^(L(n)+1) for n > 0 where L(n) = A070939(n).
s(4*n+1) = 1 for n >= 0, s(4*n) = s(4*n+2) = 1 if A010060(n) = 1 for n > 0 where s(n) = A007814(Sum_{k=0..n-1} A(k+1,n-k-1)). (End)

A099729 a(n) = a(n-1)^2 + a(floor(n/2))^2; a(0) = 1.

Original entry on oeis.org

1, 2, 8, 68, 4688, 21977408, 483006462403088, 233295242723145661671791940368, 54426670277251448804881298598338033107084297639402489952768

Offset: 0

Reinhard Zumkeller, Nov 08 2004

nonn

Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A000123.

a = ConstantArray[0, 10]; a[[1]]=2; Do[a[[n]] = a[[n-1]]^2 + a[[Floor[n/2]]]^2,{n,2,10}]; Flatten[{1,a}] (* Vaclav Kotesovec, Dec 18 2014 *)

a(n) ~ c^(2^n), where c = 1.6960426326108078452611100055735661068469295099996784988263600567400352165... . - Vaclav Kotesovec, Dec 18 2014

A152560 a(n) = A131205(A005578(n)).

Original entry on oeis.org

1, 1, 3, 7, 37, 225, 2707, 47991, 1566965, 85002865, 8346008131, 1441358958439, 452666387604933, 257384373709193473, 268828878795481175283, 516988085848458847554135, 1844678455777198460320221077

Offset: 0

Paul D. Hanna, Dec 22 2008

nonn

Related to binary partitions (A000123); A131205 is defined by a(n) = a(n-1) + a(floor(n/2)) + a(ceiling(n/2)) and A005578(n) = (2^(n+1)+3+(-1)^n)/6.

Cf. A131205, A005578, A000123.

{a(n)=local(N=(2^(n+1)+3+(-1)^n)/6,X=x+O(x^N));polcoeff(1/((1-X)^2*prod(m=0,n-1,1-X^(2^m))),N-1)}

A161803 G.f.: A(x) = exp( Sum_{n>=1} A162552(n) * 2A006519(n) x^n/n ).

Original entry on oeis.org

1, 2, 0, -2, 6, 12, 0, -8, 24, 44, 0, -30, 54, 104, 0, -60, 238, 466, 0, -402, 924, 1892, 0, -1228, 3264, 6006, 0, -4052, 6688, 13052, 0, -7452, 16536, 32140, 0, -24828, 39660, 85744, 0, -53592, 114336, 212406, 0, -141090, 190754, 386956, 0, -216572, 136078

Offset: 0

Paul D. Hanna, Jul 19 2009

sign

A162552 forms the l.g.f. of log[ Sum_{n>=0} x^(n^2) ], while
2*A006519 forms the l.g.f. of binary partitions (A000123) and
A006519(n) is the highest power of 2 dividing n.

			G.f.: 1 + 2*x - 2*x^3 + 6*x^4 + 12*x^5 - 8*x^7 + 24*x^8 + 44*x^9 +...

Cf. A161800, A162552.

{a(n)=local(SQ=sum(m=0, sqrtint(n+1), x^(m^2))+x*O(x^n), L=sum(m=1,n,2*2^valuation(m,2)*polcoeff(log(SQ),m)*x^m)+x*O(x^n)); polcoeff(exp(L),n)}

A318296 Number of conjugacy classes of the Sylow 2-subgroup of the alternating group on n letters.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 7, 7, 9, 9, 11, 11, 18, 18

Offset: 1

Richard Locke Peterson, Aug 23 2018

nonn

Also number of partitions of n containing only powers of 2 and having an even number of even elements.

These partitions form a semiring. The semiring uses the following binary operations *,+: Let A=(a1,a2,..,aj) be a partition of k that has j parts with i of those j being powers of 2 greater than 1, written in nonincreasing order. Let B be a partition of y that has x parts, with w of the x being powers of 2 greater than 1, arranged in descending order. Then A+B = (a1,a2,...,aj,b1,b2,...,bx), and A*B=AB=(a1,a2,...,aj)*(b1,b2,...,bx) is defined to be the partition (a1b1,a2b1,...,a1bx,a2b1,...,a2bx,...,ajbx) of ky. Since i and w are even by assumption, the numbers of powers of two in A+B (= i + w) and AB (= ix + jw - iw) must also be even, and both are members of the semiring. In addition, if C = (c1,...cm) is a partition of k into m parts, n of which are powers of two, (AB)C = A(BC) = (a1b1c1,a2b1c1,...,ajbxcm), and (A+B)C = (a1,a2,...,aj,b1,b2,...,bx)(c1,...cm) = (a1c1,a2c1,...,ajcm,b1c1,...) = (a1c1,a2c1,...,ajcm) + (b1c1,b2c1,...,bxcm) = AC + BC, so the necessary criteria for a semiring hold. [Missing parts added by Charlie Neder, Feb 09 2019]

Cf. A018819, A000123, A000041.

a(2n) = a(2n+1) for all n. - Charlie Neder, Feb 09 2019

A322010 Inverse permutation to A322000.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 14, 20, 26, 36, 3, 5, 7, 11, 15, 21, 27, 37, 46, 59, 8, 12, 16, 22, 28, 38, 47, 60, 72, 90, 17, 23, 29, 39, 48, 61, 73, 91, 108, 130, 30, 40, 49, 62, 74, 92, 109, 131, 152, 182, 50, 63, 75, 93, 110, 132, 153, 183, 212, 248, 76, 94, 111, 133, 154, 184, 213, 249

Offset: 0

M. F. Hasler, Feb 19 2019

nonn

a(n) is the position of n in the list A322000 of "decibinary numbers", i.e., integers sorted according to their decibinary value A028897(n) = Sum d[i]*2^i, where d[i] are the decimal digits of n.
For 0 <= m <= 9, we have a(n) = A322003(n) = A000123(n-1), because 1..9 are the first few terms of A322000 where the decibinary value increases.
We see that a(10..19) = a(2..9)+1 concatenated with (46, 49). Then, a(20..29) = a(12..19)+1 concatenated with (72, 90). Then, a(30..39) = a(22..29)+1 concatenated with (108, 130), and so on. This yields an alternate way to compute the sequence.

Cf. A322000, A028897, A322003, A072170(.,10), A000123.

vec_A322010=vecsort(A,,1)[1..vecmin(setminus([1..#A],Set(A)))-1] \\ Assumes the vector A = A322000(1..N) has been computed for some N. Exclude initial 0's to have correct (1-based) indices of the vectors.

cp's OEIS Frontend

A339277 Number of partitions of 2*n into powers of 2 where every part appears at least 2 times.

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A346912 a(0) = 1; a(n) = a(n-1) + a(floor(n/2)) + 1.

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A377556 E.g.f.: exp(Sum_{n>=1} A006519(n) * x^n).

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A381810 Array read by downward antidiagonals: A(n,k) is a generalization of odd columns of A125790 defined in Comments for n > 0, k >= 0.

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PARI

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A099729 a(n) = a(n-1)^2 + a(floor(n/2))^2; a(0) = 1.

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Mathematica

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A152560 a(n) = A131205(A005578(n)).

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PARI

A161803 G.f.: A(x) = exp( Sum_{n>=1} A162552(n) * 2*A006519(n) * x^n/n ).

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PARI

A318296 Number of conjugacy classes of the Sylow 2-subgroup of the alternating group on n letters.

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A322010 Inverse permutation to A322000.

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A161803 G.f.: A(x) = exp( Sum_{n>=1} A162552(n) * 2A006519(n) x^n/n ).