A000123 -id:A000123 - cp's OEIS frontend

A342249 Number of partitions of n into nine powers of 2.

1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 5, 5, 5, 6, 6, 8, 7, 8, 8, 9, 8, 11, 10, 11, 10, 12, 9, 12, 10, 12, 10, 14, 11, 14, 12, 15, 13, 17, 14, 18, 14, 17, 13, 18, 14, 19, 15, 19, 15, 20, 15, 21, 17, 21, 14, 21, 15, 19, 13, 20, 14, 19, 14, 22, 16, 21, 16, 24, 18, 24, 18, 25

Offset: 9

Ilya Gutkovskiy, Mar 07 2021

nonn

Column k=9 of A089052.

Cf. A000079, A000123, A018819, A075897, A089048, A089049, A089050, A089051, A209229, A319922, A342247, A342248, A342252.

A343820 Number of partitions of 2n into powers of 2: p1 <= p2 <= ... <= p_k such that p_i <= 1 + Sum_{j=1..i-1} p_j.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 12, 15, 26, 32, 42, 50, 68, 80, 98, 113, 166, 192, 230, 262, 318, 360, 418, 468, 572, 640, 732, 812, 934, 1032, 1160, 1273, 1626, 1792, 2010, 2202, 2482, 2712, 3006, 3268, 3682, 4000, 4402, 4762, 5254, 5672, 6190, 6658, 7492, 8064, 8772, 9412

Offset: 0

Alois P. Heinz, Apr 30 2021

nonn

			a(2) = 2: [1,1,1,1], [1,1,2].
a(3) = 3: [1,1,1,1,1,1], [1,1,1,1,2], [1,1,2,2].
a(4) = 6: [1,1,1,1,1,1,1,1], [1,1,1,1,1,1,2], [1,1,1,1,2,2], [1,1,2,2,2], [1,1,1,1,4], [1,1,2,4].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000123, A000225, A002449, A018819, A343756, A343944.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0, (p->
      `if`(p>n or p>n-p+1, 0, b(n-p, i)))(2^i)+b(n, i-1)))
    end:
a:= n-> b(2*n, ilog2(n)+1):
seq(a(n), n=0..80);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 0, 0, Function[p, If[p > n || p > n - p + 1, 0, b[n - p, i]]][2^i] + b[n, i - 1]]];
a[n_] := b[2n, BitLength[n] + 1];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Feb 13 2023, after Alois P. Heinz *)

a(n) is odd <=> n in { A000225 }.

a(2^(n-1)) = A002449(n).

A347660 Number of partitions of n into at most 5 powers of 2.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Sep 09 2021

nonn

Cf. A000079, A000123, A018819, A089050, A089052, A347658, A347659, A347661.

a(n) = Sum_{k=0..5} A089052(n,k). - Alois P. Heinz, Sep 09 2021

A347661 Number of partitions of n into at most 6 powers of 2.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 5, 8, 7, 9, 7, 11, 8, 10, 7, 12, 8, 12, 8, 14, 10, 13, 8, 15, 9, 13, 8, 14, 8, 11, 5, 13, 8, 13, 8, 17, 11, 14, 8, 18, 11, 16, 9, 17, 9, 12, 5, 16, 9, 14, 8, 17, 9, 12, 5, 15, 8, 12, 5, 12, 5, 6, 1, 13, 8, 13, 8, 18, 11, 14, 8, 21, 12, 17

Offset: 0

Ilya Gutkovskiy, Sep 09 2021

nonn

Cf. A000079, A000123, A018819, A089051, A089052, A347658, A347659, A347660.

a(n) = Sum_{k=0..6} A089052(n,k). - Alois P. Heinz, Sep 09 2021

A088931 G.f.: Sum_{n >= 1} ( x^(2^n)/ ((1+x^(2^(n-1)))*Product_{j=0..n-1} (1-x^(2^j)) ) ).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 1, 4, 3, 5, 4, 7, 6, 8, 7, 12, 11, 15, 14, 20, 19, 24, 23, 31, 30, 37, 36, 45, 44, 52, 51, 64, 63, 75, 74, 90, 89, 104, 103, 124, 123, 143, 142, 167, 166, 190, 189, 221, 220, 251, 250, 288, 287, 324, 323, 369, 368, 413, 412, 465, 464, 516, 515, 580

Offset: 0

N. J. A. Sloane, Dec 02 2003

nonn

This is the g.f. for the number of non-squashing partitions with a repeated part, that is, A000123(n) - A088567(n).

			x^2/((1 + x)*(1 - x)) + x^4/((1 + x^2)*(1 - x)*(1 - x^2)) + x^8/((1 + x^4)*(1 - x)*(1 - x^2)*(1 - x^4)) + ...

N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.

Apart from initial terms, same as A088980.

max = 65; Sum[x^(2^n)/((1+x^(2^(n-1))) Product[1-x^(2^j), {j, 0, n-1}]), {n, 1, Log[2, max] // Ceiling}] + O[x]^(max) // CoefficientList[#, x]& (* Jean-François Alcover, Oct 05 2018 *)

A089055 Solution to the non-squashing boxes problem (version 2).

Original entry on oeis.org

2, 4, 8, 16, 28, 46, 72, 108, 156, 218, 298, 398, 524, 678, 868, 1096, 1372, 1698, 2086, 2538, 3070, 3684, 4398, 5214, 6156, 7226, 8450, 9830, 11400, 13162, 15152, 17372, 19868, 22642, 25742, 29170, 32986, 37192, 41850, 46962, 52606, 58784, 65576, 72984, 81106

Offset: 0

N. J. A. Sloane, Dec 04 2003

nonn

Given n+1 boxes labeled 0..n, such that box i weighs i grams and can support a total weight of i grams; a(n) = number of stacks of boxes that can be formed such that no box is squashed.

Amanda Folsom, Youkow Homma, Jun Hwan Ryu, and Benjamin Tong, On a general class of non-squashing partitions, Discrete Mathematics 339 (2016) 1482-1506.
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, arXiv:math/0312418 [math.CO], 2003; Discrete Math., 294 (2005), 259-274.

Cf. A000123, A088567. Equals 2*A089054. Row sums of A089239.

See A089054 for g.f.

A089300 Number of planar partitions of n (A000219) that are non-squashing along rows and down columns (cf. A018819).

Original entry on oeis.org

1, 1, 3, 4, 9, 13, 25, 33, 56, 76, 122, 170, 264, 360, 538, 707, 1002, 1332, 1853, 2409, 3294

Offset: 0

N. J. A. Sloane, Dec 25 2003

			E.g. a(4) = 9:
4.31.3.22.2.211.21..2..11
.....1....2.....1...1..11
....................1....

N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.

Cf. A000219, A000123, A018819, A008763, A001970, A089292.

More terms from Wouter Meeussen, Dec 30 2003

A089688 Table T(n,k), n>=0 and k>=1, read by antidiagonals; the k-th row is defined by : partitions of k*n into powers of k (with T(0,k) = 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 6, 3, 2, 1, 1, 10, 5, 3, 2, 1, 1, 14, 7, 4, 3, 2, 1, 1, 20, 9, 6, 4, 3, 2, 1, 1, 26, 12, 8, 5, 4, 3, 2, 1, 1, 36, 15, 10, 7, 5, 4, 3, 2, 1, 1, 46, 18, 12, 9, 6, 5, 4, 3, 2, 1, 1, 60, 23, 15, 11, 8, 6, 5, 4, 3, 2, 1

Offset: 0

Philippe Deléham, Jan 05 2004

			Row k = 1 : 1, 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1, ... (see A000012).
Row k = 2 : 1, 2, 4, 6, 10, 14, 20, 26, 36, 46, 60, 74, ... (see A000123).
Row k = 3 : 1, 2, 3, 5,  7,  9, 12, 15, 18, 23, 28, 33, ... (see A005704).
Row k = 4 : 1, 2, 3, 4,  6,  8, 10, 12, 15, 18, 21, 24, ... (see A005705).
Row k = 5 : 1, 2, 3, 4,  5,  7,  9, 11, 13, 15, 18, 21, ... (see A005706).

Alois P. Heinz, Table of n, a(n) for n = 0..10010

A102379 a(n) is the minimal number of nodes in a binary tree of height n.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 12, 17, 22, 29, 36, 46, 56, 69, 82, 100, 118, 141, 164, 194, 224, 261, 298, 345, 392, 449, 506, 576, 646, 729, 812, 913, 1014, 1133, 1252, 1394, 1536, 1701, 1866, 2061, 2256, 2481, 2706, 2968, 3230, 3529, 3828, 4174, 4520, 4913

Offset: 1

Mitch Harris, Jan 05 2005

nonn

Conjecture: Let b(n) be the number of fixed points of the set of binary partitions of n under Glaisher's function that proves Euler's odd-distinct theorem. Then b(1) = 1 and for n > 1, b(2*n) = b(2*n+1) = 2*a(n). - George Beck, Jul 23 2022

de Bruijn, N. G., On Mahler's partition problem. Nederl. Akad. Wetensch., Proc. 51, (1948) 659-669 = Indagationes Math. 10, 210-220 (1948).
Gonnet, Gaston H.; Olivie, Henk J.; and Wood, Derick, Height-ratio-balanced trees. Comput. J. 26 (1983), no. 2, 106-108.
Mahler, Kurt On a special functional equation. J. London Math. Soc. 15, (1940). 115-123.
Nievergelt, J.; Reingold, E. M., Binary search trees of bounded balance, SIAM J. Comput. 2 (1973), 33-43.

Cf. A000123, A033485, A102378.

Essentially partial sums of A040039.

from functools import cache
@cache
def a(n: int) -> int:
    return a(n - 1) + a(n // 2) + 1 if n > 1 else 0
print([a(n) for n in range(1, 51)])  # Peter Luschny, Jul 24 2022

a(n) = a(n-1) + a(floor(n/2)) + 1, a(1) = 0.
a(n) - a(n-1) = A018819(n+1).
G.f. A(x) satisfies (1-x)*A(x) = 2*(1 + x)*B(x^2), where B(x) is the g.f. of A033485.

A145523 Least integer k > 0 such that A000041(k) is divisible by 2^n.

Original entry on oeis.org

1, 2, 11, 11, 15, 66, 66, 96, 96, 96, 96, 96, 3693, 15005, 18978, 18978, 18978, 43002, 55943, 972190, 1151214, 2799146, 15519397, 15519397, 15519397, 122101417, 210553237, 289585489, 473093534

Offset: 0

M. F. Hasler, Oct 12 2008

The requirement a(n) > 0 is somewhat arbitrary, chosen for agreement with A046641; a(n) >= 0 would have been possible, too, yielding a(0)=0.

a(29) > 10^9.

Cf. A000041, A046641, A000123, A046063.

i=1; c=1; while [ $c -le 21 ]; do echo -n `./A046641 $i`", "; c=`expr $c + 1`; i=`expr $i + $i`; done # M. F. Hasler, Oct 18 2008

a(n) = A046641(2^n).

More terms from M. F. Hasler, Oct 18 2008

a(22)-a(28) from Max Alekseyev, Dec 16 2011

cp's OEIS Frontend

A342249 Number of partitions of n into nine powers of 2.

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A343820 Number of partitions of 2n into powers of 2: p1 <= p2 <= ... <= p_k such that p_i <= 1 + Sum_{j=1..i-1} p_j.

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A347660 Number of partitions of n into at most 5 powers of 2.

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A347661 Number of partitions of n into at most 6 powers of 2.

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A088931 G.f.: Sum_{n >= 1} ( x^(2^n)/ ((1+x^(2^(n-1)))*Product_{j=0..n-1} (1-x^(2^j)) ) ).

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A089055 Solution to the non-squashing boxes problem (version 2).

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A089300 Number of planar partitions of n (A000219) that are non-squashing along rows and down columns (cf. A018819).

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A089688 Table T(n,k), n>=0 and k>=1, read by antidiagonals; the k-th row is defined by : partitions of k*n into powers of k (with T(0,k) = 1).

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A102379 a(n) is the minimal number of nodes in a binary tree of height n.

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A145523 Least integer k > 0 such that A000041(k) is divisible by 2^n.

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