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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.

%I A343944 #13 Nov 16 2022 08:53:08
%S A343944 0,1,2,5,7,12,15,29,35,50,58,86,98,128,143,225,251,318,350,453,495,
%T A343944 603,653,846,914,1092,1172,1419,1517,1773,1886,2521,2687,3130,3322,
%U A343944 3917,4147,4759,5021,5909,6227,7082,7442,8537,8955,10076,10544,12326,12898,14452
%N 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.
%H A343944 Alois P. Heinz, <a href="/A343944/b343944.txt">Table of n, a(n) for n = 0..20000</a>
%e A343944 a(5) = 12 = 5+4+3: [1,1,1,1,1], [1,1,1,2], [1,2,2].
%e A343944 a(6) = 15 = 6+5+4: [1,1,1,1,1,1], [1,1,1,1,2], [1,1,2,2].
%e A343944 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].
%p A343944 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<0, 0, (p-> `if`(
%p A343944        p>n or p>n-p+1, 0, (h-> h+[0, h[1]])(b(n-p, i))))(2^i)+b(n, i-1)))
%p A343944     end:
%p A343944 a:= n-> b(n, ilog2(n))[2]:
%p A343944 seq(a(n), n=0..60);
%t A343944 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]]];
%t A343944 a[n_] := b[n, Floor@Log2[n]][[2]];
%t A343944 Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Nov 16 2022, after _Alois P. Heinz_ *)
%Y A343944 Cf. A281688, A343820.
%K A343944 nonn
%O A343944 0,3
%A A343944 _Alois P. Heinz_, May 04 2021