A343820 -id:A343820 - cp's OEIS frontend

A343756 Number of nondecreasing sequences s1, s2, ..., s_n of powers of 2 such that s_i <= 1 + Sum_{j=1..i-1} s_j.

1, 1, 2, 4, 9, 20, 45, 102, 232, 528, 1203, 2742, 6252, 14258, 32519, 74173, 169189, 385929, 880337, 2008141, 4580803, 10449377, 23836364, 54373853, 124033933, 282937887, 645419092, 1472287344, 3358484832, 7661154644, 17476122569, 39865383333, 90938296121

Offset: 0

Alois P. Heinz, Apr 27 2021

nonn

			a(0) = 1: [].
a(1) = 1: [1].
a(2) = 2: [1,1], [1,2].
a(3) = 4: [1,1,1], [1,1,2], [1,2,2], [1,2,4].
a(4) = 9 : [1,1,1,1], [1,1,1,2], [1,1,1,4], [1,1,2,2], [1,1,2,4], [1,2,2,2], [1,2,2,4], [1,2,4,4], [1,2,4,8].
a(5) = 20: [1,1,1,1,1], [1,1,1,1,2], [1,1,1,1,4], [1,1,1,2,2], [1,1,1,2,4], [1,1,1,4,4], [1,1,1,4,8], [1,1,2,2,2], [1,1,2,2,4], [1,1,2,4,4], [1,1,2,4,8], [1,2,2,2,2], [1,2,2,2,4], [1,2,2,2,8], [1,2,2,4,4], [1,2,2,4,8], [1,2,4,4,4], [1,2,4,4,8], [1,2,4,8,8], [1,2,4,8,16].

Alois P. Heinz, Table of n, a(n) for n = 0..2793

Cf. A000079, A339479, A343799, A343820.

b:= proc(n, t) option remember; `if`(n=0, 1,
     `if`(t=0, 0, b(n, iquo(t, 2))+b(n-1, t+1)))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..38);

A343944 Total number of parts in all partitions of n 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

0, 1, 2, 5, 7, 12, 15, 29, 35, 50, 58, 86, 98, 128, 143, 225, 251, 318, 350, 453, 495, 603, 653, 846, 914, 1092, 1172, 1419, 1517, 1773, 1886, 2521, 2687, 3130, 3322, 3917, 4147, 4759, 5021, 5909, 6227, 7082, 7442, 8537, 8955, 10076, 10544, 12326, 12898, 14452

Offset: 0

Alois P. Heinz, May 04 2021

nonn

			a(5) = 12 = 5+4+3: [1,1,1,1,1], [1,1,1,2], [1,2,2].
a(6) = 15 = 6+5+4: [1,1,1,1,1,1], [1,1,1,1,2], [1,1,2,2].
a(7) = 29 = 7+6+5+4+4+3: [1,1,1,1,1,1,1], [1,1,1,1,1,2], [1,1,1,2,2], [1,2,2,2], [1,1,1,4], [1,2,4].

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A281688, A343820.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<0, 0, (p-> `if`(
       p>n or p>n-p+1, 0, (h-> h+[0, h[1]])(b(n-p, i))))(2^i)+b(n, i-1)))
    end:
a:= n-> b(n, ilog2(n))[2]:
seq(a(n), n=0..60);

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 0, {0, 0}, Function[p, If[p > n || p > n - p + 1, {0, 0}, Function[h, h + {0, h[[1]]}][b[n - p, i]]]][2^i] + b[n, i - 1]]];
a[n_] := b[n, Floor@Log2[n]][[2]];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 16 2022, after Alois P. Heinz *)

cp's OEIS Frontend

A343756 Number of nondecreasing sequences s1, s2, ..., s_n of powers of 2 such that s_i <= 1 + Sum_{j=1..i-1} s_j.

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