A307148 Number of binary partitions of n in which exactly one of the powers of 2 is used an odd number of times.

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, 20, 60, 26, 66, 26, 83, 36, 93, 36, 114, 46, 124, 46, 152, 60, 166, 60, 198, 74, 212, 74, 254, 94, 274, 94, 322, 114, 342, 114, 402, 140, 428, 140, 494, 166, 520

Offset: 0

Author

N. J. A. Sloane, Mar 28 2019

Keywords

• nonn

Comments

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).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• George E. Andrews and Jim Lawrence, Binary partitions and binary partition polytopes, preprint.
• George E. Andrews and Jim Lawrence, Binary partitions and binary partition polytopes, Aequationes mathematicae 91.5 (2017): 859-869.

Crossrefs

Cf. A000123, A307149.

Programs

Mathematica

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 *)

Extensions

More terms from Vaclav Kotesovec, Mar 29 2019