A000123 -id:A000123 - cp's OEIS frontend

A162584 G.f.: A(x) = exp( 2Sum_{n>=1} sigma(n)A006519(n) * x^n/n ), where A006519(n) = highest power of 2 dividing n.

1, 2, 8, 16, 50, 96, 240, 448, 1024, 1858, 3888, 6896, 13696, 23776, 44960, 76608, 139970, 234432, 414904, 684336, 1181568, 1921472, 3242928, 5206208, 8623104, 13679490, 22268752, 34941120, 56039936, 87036576, 137686048, 211822976

Offset: 0

Paul D. Hanna, Jul 06 2009

nonn

Log of the g.f. A(x) is formed from the term-wise product of the log of the g.f.s of the partition numbers A000041 and the binary partitions A000123.

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 16*x^3 + 50*x^4 + 96*x^5 + 240*x^6 + ...
log(A(x))/2 = x + 6*x^2/2 + 4*x^3/3 + 28*x^4/4 + 6*x^5/5 + 24*x^6/6 + 8*x^7/7 + 120*x^8/8 + ... + sigma(n)*A006519(n)*x^n/n + ...
The log of the g.f. of the Partition numbers (A000041) is:
x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 6*x^5/5 + 12*x^6/6 + ... + sigma(n)*x^n/n + ...
The log of the g.f. of the binary partitions (A000123) is:
x + x^2/2 + x^3/3 + 4*x^4/4 + x^5/5 + 2*x^6/6 + x^7/7 + ... + A006519(n)*x^n/n + ...
From _Paul D. Hanna_, Jul 26 2009: (Start)
BISECTIONS begin:
B_0(q) = 1 + 8*q^2 + 50*q^4 + 240*q^6 + 1024*q^8 + 3888*q^10 + ...
B_1(q) = 2*q + 16*q^3 + 96*q^5 + 448*q^7 + 1858*q^9 + 6896*q^11 + ...
where 2*B_0(q)/B_1(q) = T16B(q):
T16B = 1/q + 2*q^3 - q^7 - 2*q^11 + 3*q^15 + 2*q^19 - 4*q^23 - 4*q^27 + ...
which is a g.f. of A029839. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)

Cf. A000203, A006519, A000041, A000123.

Cf. A163228 (B_0), A163229 (B_1), A029839 (T16B); variant: A163129. - Paul D. Hanna, Jul 26 2009

eta[q_]:= q^(1/24)*QPochhammer[q]; nmax = 250; a[n_]:=SeriesCoefficient[ Series[Exp[Sum[DivisorSigma[1, k]*2^(IntegerExponent[k, 2] + 1)*q^k/k, {k, 1, nmax}]], {q, 0, nmax}], n]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jul 03 2018 *)
nmax = 40; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*2^(IntegerExponent[k, 2] + 1)*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 20 2020 *)
nmax = 40; CoefficientList[Series[Product[1/EllipticTheta[4, 0, x^(2^k)]^(2^k), {k, 0, 1 + Log[2, nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 07 2023 *)

{a(n)=local(L=sum(m=1,n,2*sigma(m)*2^valuation(m,2)*x^m/m)+x*O(x^n));polcoeff(exp(L),n)}

From Paul D. Hanna, Jul 26 2009: (Start)
Define series BISECTIONS A(q) = B_0(q) + B_1(q), then
2*B_0(q)/B_1(q) = T16B(q) = q*eta(q^8)^6/(eta(q^4)^2*eta(q^16)^4), the McKay-Thompson series of class 16B for the Monster group (A029839). (End)
G.f.: 1/Product_{n>=0} Theta4(q^(2^n))^(2^n) = 1 / ( E(1)^2*E(2)^3*E(4)^6*E(8)^12* ... * E(2^n)^A042950(n) * ... ) where E(n) = Product_{k>=1} (1-q^(n*k)). - Joerg Arndt, Mar 20 2010
Compare to the previous formula: 1/Product_{n>=0} Theta3(q^(2^n))^(2^n) = Theta4(q). - Joerg Arndt, Aug 03 2011

A174868 Partial sums of Stern's diatomic series A002487.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 10, 13, 14, 18, 21, 26, 28, 33, 36, 40, 41, 46, 50, 57, 60, 68, 73, 80, 82, 89, 94, 102, 105, 112, 116, 121, 122, 128, 133, 142, 146, 157, 164, 174, 177, 188, 196, 209, 214, 226, 233, 242, 244, 253, 260, 272, 277, 290, 298, 309, 312, 322, 329, 340, 344, 353, 358, 364, 365, 372, 378, 389, 394, 408, 417, 430, 434, 449, 460, 478, 485, 502, 512, 525, 528, 542, 553, 572, 580, 601, 614, 632, 637, 654, 666, 685

Offset: 0

Jonathan Vos Post, Dec 01 2010

After the initial 0, identical to A007729.

			a(16) = 0 + 1 + 1 + 2 + 1 + 3 + 2 + 3 + 1 + 4 + 3 + 5 + 2 + 5 + 3 + 4 + 1 = 41.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Michael J. Collins and David Wilson, Equivalence of OEIS A007729 and A174868, arXiv:1812.11174 [math.CO], 2018.
Clemens Heuberger, Daniel Krenn and Gabriel F. Lipnik, Asymptotic Analysis of q-Recursive Sequences, Algorithmica, Vol. 84 (2022), pp. 2480-2532; arXiv preprint, arXiv:2105.04334 [math.CO], 2021-2022.

Cf. A002487, A007729, A084091, A049456, A020946, A020950, A046815, A070871, A070872, A071883, A064881-A064886, A072170, A001045, A002083, A073459, A000123, A126606, A049455.

Cf. A001622, A242208.

a[n_] := a[n] = If[EvenQ[n], 2*a[n/2] + a[n/2 - 1], 2*a[(n - 1)/2] + a[(n + 1)/2]]; a[0] = 0; a[1] = 1; Array[a, 100, 0] (* Amiram Eldar, May 18 2023 *)

from itertools import accumulate, count, islice
from functools import reduce
def A174868_gen(): # generator of terms
    return accumulate((sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0))) for n in count(1)),initial=0)
A174868_list = list(islice(A174868_gen(),30)) # Chai Wah Wu, May 07 2023

a(n) = Sum_{i=0..n} A002487(i).
G.f.: (x/(1 - x))*Product_{k>=0} (1 + x^(2^k) + x^(2^(k+1))). - Ilya Gutkovskiy, Feb 27 2017
a(2k) = 2*a(k) + a(k-1); a(2k+1) = 2*a(k) + a(k+1). - Michael J. Collins, Dec 25 2018
a(n) = n^log_2(3) + Psi_D(log_2(n)) + O(n^log_2(phi)), where phi is the golden ratio (A001622) and Psi_D is a 1-periodic continuous function which is Hölder continuous with any exponent smaller than log_2(3/phi) (Heuberger et al., 2022). - Amiram Eldar, May 18 2023

A277904 Irregular table: row n (n >= 0) is obtained by listing numbers 0 .. A018819(n)-1.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19

Offset: 1

Antti Karttunen, Nov 14 2016

			A018819 -> range -> terms on row n
  1        [0,0]:    0;
  1        [0,0]:    0;
  2        [0,1]:    0, 1;
  2        [0,1]:    0, 1;
  4        [0,3]:    0, 1, 2, 3;
  4        [0,3]:    0, 1, 2, 3;
  6        [0,5]:    0, 1, 2, 3, 4, 5;
etc.

Antti Karttunen, Table of n, a(n) for n = 1..9828

Cf. A000123, A018819, A277903.
Used for constructing A277905.
Retaining only every second row gives A278164.

(define (A277904 n) (if (= 1 n) 0 (- n (A000123 (+ -1 (A277903 n))) 1)))

a(1) = 0; for n > 1, a(n) = n - A000123(A277903(n)-1) - 1.

A007728 5th binary partition function.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 5, 4, 8, 6, 9, 7, 12, 8, 12, 9, 17, 12, 18, 14, 23, 15, 22, 16, 28, 19, 27, 20, 32, 20, 29, 21, 38, 26, 38, 29, 47, 30, 44, 32, 55, 37, 52, 38, 60, 37, 53, 38, 66, 44, 63, 47, 74, 46, 66, 47, 79, 52, 72, 52, 81, 49, 70, 50, 88, 59, 85, 64

Offset: 0

N. J. A. Sloane

The number of ways of writing n as a sum of powers of 2, each power being used at most four times. - Dmitry Kamenetsky, Jul 14 2023

Alois P. Heinz, Table of n, a(n) for n = 0..10000
B. Reznick, Some binary partition functions, in "Analytic number theory" (Conf. in honor P. T. Bateman, Allerton Park, IL, 1989), 451-477, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990.

A column of A072170.

Cf. A000123, A018819.

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<0, 0, add(`if`(n-j*2^i<0, 0,
         b(n-j*2^i, i-1)), j=0..4)))
    end:
a:= n-> b(n, ilog2(n)):
seq(a(n), n=0..70);  # Alois P. Heinz, Jun 21 2012

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 0, 0, Sum[If[n-j*2^i < 0, 0, b[n-j*2^i, i-1, k]], {j, 0, k-1}]]]; a[n_] := b[n, Log[2, n] // Floor, 5]; Table[a[n], {n, 0, 70} ] (* Jean-François Alcover, Jan 17 2014, after Alois P. Heinz *)

G.f.: Product_{k>=0} (1 - x^(5*2^k))/(1 - x^(2^k)). - Ilya Gutkovskiy, Jul 09 2019

More terms from Vladeta Jovovic, May 06 2004

A058039 a(n) = a(n-1) + 2*a(floor(n/2)) if n > 0, otherwise 1.

Original entry on oeis.org

1, 3, 9, 15, 33, 51, 81, 111, 177, 243, 345, 447, 609, 771, 993, 1215, 1569, 1923, 2409, 2895, 3585, 4275, 5169, 6063, 7281, 8499, 10041, 11583, 13569, 15555, 17985, 20415, 23553, 26691, 30537, 34383, 39201, 44019, 49809, 55599, 62769, 69939, 78489, 87039

Offset: 0

Reinhard Zumkeller, Feb 16 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000123, A002061, A067868, A347027.

[1] cat [n le 2 select 3^n else Self(n-1) + 2*Self(Floor(n/2)): n in [1..51]]; // G. C. Greubel, Feb 10 2021

a[n_]:= a[n] = If[n==0, 1, a[n-1] + 2*a[Floor[n/2]]];
Table[a[n], {n, 0, 50}] (* G. C. Greubel, Feb 10 2021 *)

a(n) = if (n==0, 1, a(n-1)+2*a(n\2)); \\ Michel Marcus, Feb 04 2021

def a(n): return 1 if n == 0 else a(n-1) + 2*a(n//2)
print([a(n) for n in range(44)]) # Michael S. Branicky, Feb 04 2021

a(n) = 1 + 2 * Sum_{k=1..n} a(floor(k/2)). - Ilya Gutkovskiy, Aug 15 2021

Name corrected by and more terms from Michael S. Branicky, Feb 04 2021

A067868 a(n) = a(n-1) + a(floor(n/2))^2 for n > 0, a(0) = 1.

Original entry on oeis.org

1, 2, 6, 10, 46, 82, 182, 282, 2398, 4514, 11238, 17962, 51086, 84210, 163734, 243258, 5993662, 11744066, 32120262, 52496458, 178789102, 305081746, 627715190, 950348634, 3560128030, 6169907426, 13261231526, 20352555626, 47161378382, 73970201138, 133144655702, 192319110266

Offset: 0

Reinhard Zumkeller, Feb 16 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000123, A056520, A058039.

[1] cat [n le 2 select Factorial(n+1) else Self(n-1) + (Self(Floor(n/2)))^2: n in [1..51]]; // G. C. Greubel, Feb 10 2021

a[n_]:= a[n] = If[n==0, 1, a[n-1] + (a[Floor[n/2]])^2];
Table[a[n], {n, 0, 50}] (* G. C. Greubel, Feb 10 2021 *)

PARI
```
a(n) = if (n==0, 1, a(n-1)+a(n\2)^2);
```

def a(n): return 1 if n == 0 else a(n-1) + (a(n//2))^2
[a(n) for n in range(50)] # G. C. Greubel, Feb 10 2021

Corrected name and more terms from Michel Marcus, Feb 05 2021

A073469 Expansion of x/B(x) where B(x) is the g.f. for A002487.

Original entry on oeis.org

1, -1, -1, 2, -2, 0, 4, -4, -2, 6, -4, -2, 10, -8, -6, 14, -10, -4, 20, -16, -8, 24, -18, -6, 34, -28, -14, 42, -34, -8, 56, -48, -18, 66, -52, -14, 86, -72, -30, 102, -80, -22, 126, -104, -40, 144, -110, -34, 178, -144, -62, 206, -158, -48, 248, -200, -82, 282, -208, -74, 338, -264, -122, 386, -282, -104, 452, -348, -156, 504

Offset: 0

N. J. A. Sloane, Aug 26 2002

sign

a(n) is the Euler transform of a sequence b(n) = [-1,-1,1,-1,0,1,0,-1,0,0,0,1 ...] that has (for n > 0, k > 0) b(2^k-1) = -1, b(3*2^k-1) = 1, and b(n) = 0 otherwise. - Georg Fischer, Aug 24 2020

Georg Fischer, Table of n, a(n) for n = 0..1000
P. Dumas and P. Flajolet, Asymptotique des récurrences mahlériennes: le cas cyclotomique, Journal de Théorie des Nombres de Bordeaux 8 (1996), pp. 1-30.

terms = 70; A[x_] = 1/Product[1 + x^(2^k) + x^(2^(k + 1)), {k, 0, Ceiling[ Log[2, terms]]}] + O[x]^terms; CoefficientList[A[x], x] (* Jean-François Alcover, Jun 30 2011, updated Jan 15 2018 *)

This sequence grows asymptotically roughly like exp(log(n)^2), but with a complicated pattern of oscillations: see the article by Dumas-Flajolet, page 4, for a complete expansion that is related to A000123 and methods of de Bruijn. - Philippe Flajolet, Sep 06 2008

G.f. A(x) satisfies: A(x) = A(x^2) / (1 + x + x^2). - Ilya Gutkovskiy, Jul 09 2019

A162580 G.f.: A(x) = exp( 2Sum_{n>=1} 2^[A007814(n)^2] x^n/n ), where A007814(n) = exponent of highest power of 2 dividing n.

Original entry on oeis.org

1, 2, 4, 6, 16, 26, 44, 62, 240, 418, 756, 1094, 2544, 3994, 6556, 9118, 32352, 55586, 99492, 143398, 330000, 516602, 845900, 1175198, 3452112, 5729026, 9953556, 14178086, 31076592, 47975098, 77547580, 107120062, 298608832, 490097602

Offset: 0

Paul D. Hanna, Jul 06 2009

nonn

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 6*x^3 + 16*x^4 + 26*x^5 + 44*x^6 + ...
log(A(x))/2 = 2^0*x + 2^1*x^2 + 2^0*x^3/3 + 2^4*x^4/4 + 2^0*x^5/5 + 2^1*x^6/6 + 2^0*x^7/7 + 2^9*x^8/8 + ... + 2^[A007814(n)^2]*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A155200, A007814, A000123, A162581, A162582.

nmax = 500; a[n_]:= SeriesCoefficient[Series[Exp[ Sum[2^(IntegerExponent[k, 2]^2 + 1)*q^k/k, {k, 1, nmax}]], {q,0,nmax}], n]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jul 04 2018 *)

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

A162582 G.f.: A(x) = exp( 2Sum_{n>=1} A006519(n)^n x^n/n ), where A006519(n) = highest power of 2 dividing n.

Original entry on oeis.org

1, 2, 6, 10, 146, 282, 826, 1370, 4204986, 8408602, 25223066, 42037530, 615687706, 1189337882, 3483938586, 5778539290, 2305851850537847066, 4611703695297154842, 13835111074334385946, 23058518453371617050

Offset: 0

Paul D. Hanna, Jul 06 2009

nonn

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 10*x^3 + 146*x^4 + 282*x^5 + 826*x^6 + ...
log(A(x))/2 = 2^0*x + 2^2*x^2 + 2^0*x^3/3 + 2^8*x^4/4 + 2^0*x^5/5 + 2^6*x^6/6 + 2^0*x^7/7 + 2^24*x^8/8 + ... + A006519(n)^n*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..450

Cf. A162580, A162581, A155200, A006519, A000123.

nmax = 200; a[n_]:= SeriesCoefficient[Series[Exp[ Sum[2^(k*IntegerExponent[k, 2] + 1)*q^k/k, {k, 1, nmax}]], {q,0,nmax}], n]; Table[a[n], {n,0,50}] (* G. C. Greubel, Jul 04 2018 *)

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

A171370 Sequence generated from Lim:_{n..inf.} M^n, M = an infinite lower triangular matrix with (1,3,3,3,...) in every column, shifted down twice.

Original entry on oeis.org

1, 3, 6, 12, 18, 30, 42, 66, 84, 120, 150, 210, 252, 336, 402, 534, 618, 786, 906, 1146, 1296, 1596, 1806, 2226, 2478, 2982, 3318, 3990, 4392, 5196, 5730, 6798, 7416, 8652, 9438, 11010, 11916, 13728, 14874, 17166, 18462, 21054, 22650, 25842, 27648, 31260

Offset: 0

Gary W. Adamson, Dec 06 2009

nonn

A000123 can be generated through an analogous procedure replacing (1,3,3,3,...) with (1,2,2,2,...).
A171370 has the property that (1, 3, 6, 12, 18,...) / (1, 3, 3, 3,..) generates an aerated variant: (1, 0, 3, 0, 6, 0, 12,...).
Similarly, given A000123; (1, 2, 4, 6, 10, 14,...) / (1, 2, 2, 2,...) generates an aerated variant: (1, 0, 2, 0, 6, 0, 10,...).
Row sums of the generating triangle = A032766 starting with 1. - Gary W. Adamson, Feb 15 2010

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000123.

Cf. A032766. - Gary W. Adamson, Feb 15 2010

a:= n-> (Matrix(n+1, (i, j)-> `if`(i=2*j-1, 1,
        `if`(i>2*j-1, 3, 0)))^n)[n+1, 1]:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 16 2014

Let M = an infinite lower triangular matrix with (1,3,3,3,...) in every column shifted down twice:
1;
3;
3, 1;
3, 3;
3, 3, 1;
3, 3, 3;
...
Sequence A171370 = Lim:_{n..inf.} M^n, the left-shifted vector considered as a sequence.

a(20)-a(45) from Alois P. Heinz, Apr 16 2014

cp's OEIS Frontend

A162584 G.f.: A(x) = exp( 2*Sum_{n>=1} sigma(n)*A006519(n) * x^n/n ), where A006519(n) = highest power of 2 dividing n.

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A174868 Partial sums of Stern's diatomic series A002487.

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A277904 Irregular table: row n (n >= 0) is obtained by listing numbers 0 .. A018819(n)-1.

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A007728 5th binary partition function.

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A058039 a(n) = a(n-1) + 2*a(floor(n/2)) if n > 0, otherwise 1.

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A067868 a(n) = a(n-1) + a(floor(n/2))^2 for n > 0, a(0) = 1.

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A073469 Expansion of x/B(x) where B(x) is the g.f. for A002487.

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A162584 G.f.: A(x) = exp( 2Sum_{n>=1} sigma(n)A006519(n) * x^n/n ), where A006519(n) = highest power of 2 dividing n.

A162580 G.f.: A(x) = exp( 2Sum_{n>=1} 2^[A007814(n)^2] x^n/n ), where A007814(n) = exponent of highest power of 2 dividing n.

A162582 G.f.: A(x) = exp( 2Sum_{n>=1} A006519(n)^n x^n/n ), where A006519(n) = highest power of 2 dividing n.