A294199 Number of partitions of n into powers of 2 such that 1 and 2 cannot both be parts of a particular partition, and 4 and 8 cannot both be parts of a particular partition, and 16 and 32, and so on.

1, 1, 2, 1, 3, 2, 4, 2, 6, 4, 8, 4, 9, 5, 10, 5, 13, 8, 16, 8, 18, 10, 20, 10, 24, 14, 28, 14, 30, 16, 32, 16, 38, 22, 44, 22, 48, 26, 52, 26, 60, 34, 68, 34, 72, 38, 76, 38, 85, 47, 94, 47, 99, 52, 104, 52, 114, 62, 124, 62, 129, 67, 134, 67, 147, 80, 160

Offset: 0

Author

Vaclav Kotesovec, Oct 24 2017

Keywords

• nonn

Examples

			a(10) = 8 where the partitions are the following: 8+2, 8+1+1, 4+4+2, 4+2+2+2, 4+4+1+1, 4+1+1+1+1+1+1, 2+2+2+2+2, 1+1+1+1+1+1+1+1+1+1.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Bin Lan and James A. Sellers, Properties of a Restricted Binary Partition Function a la Andrews and Lewis, #A23 INTEGERS 15 (2015), p.2.

Crossrefs

Cf. A070047.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Product[(1-x^(3*2^(2*k-2)))/(1-x^(2^(k-1))), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[1] = 1; a[2] = 2; a[3] = 1; Flatten[{1, 1, 2, 1, Table[If[EvenQ[n], a[n] = a[n-2] + a[Floor[n/4]], a[n] = a[n-1] - a[n-2]], {n, 4, 100}]}]

Formula

G.f.: Product_{k>=1} (1 - x^(2^(2*k-2) + 2^(2*k-1))) / ((1 - x^(2^(2*k-2))) * (1 - x^(2^(2*k-1)))).

G.f.: Product_{k>=1} (1 - x^(3*2^(2*k-2))) / (1 - x^(2^(k-1))).

For n>=1 a(2*n) = a(2*n-2) + a([n/2]).

For n>=1 a(2*n+1) = a(2*n) - a(2*n-1).