A339479 -id:A339479 - cp's OEIS frontend

A343801 Total sum of the parts in all partitions counted by A339479(n).

0, 1, 5, 25, 121, 583, 2789, 13287, 63149, 299697, 1421107, 6735253, 31911985, 151174893, 716081551, 3391722505, 16064368343, 76084921797, 360353446761, 1706695118265, 8083167563465, 38283027343193, 181313615940197, 858725280497117, 4067034860337649

Offset: 0

Alois P. Heinz, Apr 29 2021

nonn

			a(3) = 25 = 3+4+6+5+7: [1,1,1], [1,1,2], [1,1,4], [1,2,2], [1,2,4].

Alois P. Heinz, Table of n, a(n) for n = 0..1481

Cf. A339479, A343799.

b:= proc(n, t) option remember; `if`(n=0, [1, 0],
     `if`(t=0, 0, (p-> p+[0, p[2]])(b(n, iquo(t, 2)))+
                  (p-> p+[0, p[1]])(b(n-1, t+2))))
    end:
a:= n-> b(n, 1)[2]:
seq(a(n), n=0..30);

b[n_, t_] := b[n, t] = If[n == 0, {1, 0}, If[t == 0, {0, 0},
     With[{p = b[n, Quotient[t, 2]]}, p + {0, p[[2]]}] +
     With[{p = b[n - 1, t + 2]}, p + {0, p[[1]]}]]];
a[n_] := b[n, 1][[2]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 02 2022, after Alois P. Heinz *)

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.

Original entry on oeis.org

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);

cp's OEIS Frontend

A343801 Total sum of the parts in all partitions counted by A339479(n).

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