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Number of binary partitions of n in which exactly one of the powers of 2 is used an odd number of times.
2

%I #21 Dec 18 2023 08:26:41

%S 0,1,1,1,2,2,3,2,5,4,7,4,10,6,12,6,17,10,21,10,28,14,32,14,42,20,48,

%T 20,60,26,66,26,83,36,93,36,114,46,124,46,152,60,166,60,198,74,212,74,

%U 254,94,274,94,322,114,342,114,402,140,428,140,494,166,520

%N Number of binary partitions of n in which exactly one of the powers of 2 is used an odd number of times.

%C If someone extends this, the analogs L(m, n) = numbers of binary partitions of n in which exactly m of the powers of 2 are used an odd number of times for m>2 could also be added (A307149 is the case m=2).

%H Vaclav Kotesovec, <a href="/A307148/b307148.txt">Table of n, a(n) for n = 0..10000</a>

%H George E. Andrews and Jim Lawrence, <a href="https://georgeandrews1.github.io/pdf/321.pdf">Binary partitions and binary partition polytopes</a>, preprint.

%H George E. Andrews and Jim Lawrence, <a href="https://doi.org/10.1007/s00010-017-0493-8">Binary partitions and binary partition polytopes</a>, Aequationes mathematicae 91.5 (2017): 859-869.

%t Clear[L]; L[m_, n_] := L[m, n] = If[n == 0, If[m == 0, 1, 0], If[EvenQ[n] && n >= 2, L[m, n - 2] + L[m, n/2], If[m >= 1, L[m - 1, n - 1], 0]]]; Table[L[1, n], {n, 0, 100}] (* _Vaclav Kotesovec_, Mar 29 2019 *)

%Y Cf. A000123, A307149.

%K nonn

%O 0,5

%A _N. J. A. Sloane_, Mar 28 2019

%E More terms from _Vaclav Kotesovec_, Mar 29 2019