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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. 3

%I #13 Nov 16 2022 08:53:08

%S 0,1,2,5,7,12,15,29,35,50,58,86,98,128,143,225,251,318,350,453,495,

%T 603,653,846,914,1092,1172,1419,1517,1773,1886,2521,2687,3130,3322,

%U 3917,4147,4759,5021,5909,6227,7082,7442,8537,8955,10076,10544,12326,12898,14452

%N 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 Alois P. Heinz, <a href="/A343944/b343944.txt">Table of n, a(n) for n = 0..20000</a>

%e a(5) = 12 = 5+4+3: [1,1,1,1,1], [1,1,1,2], [1,2,2].

%e a(6) = 15 = 6+5+4: [1,1,1,1,1,1], [1,1,1,1,2], [1,1,2,2].

%e 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 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<0, 0, (p-> `if`(

%p p>n or p>n-p+1, 0, (h-> h+[0, h[1]])(b(n-p, i))))(2^i)+b(n, i-1)))

%p end:

%p a:= n-> b(n, ilog2(n))[2]:

%p seq(a(n), n=0..60);

%t 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 a[n_] := b[n, Floor@Log2[n]][[2]];

%t Table[a[n], {n, 0, 60}] (* _Jean-Fran├žois Alcover_, Nov 16 2022, after _Alois P. Heinz_ *)

%Y Cf. A281688, A343820.

%K nonn

%O 0,3

%A _Alois P. Heinz_, May 04 2021

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Last modified June 17 05:47 EDT 2024. Contains 373432 sequences. (Running on oeis4.)