A339371 Number of partitions of n into an even number of Fibonacci parts (with a single type of 1).

1, 0, 1, 1, 3, 2, 5, 4, 8, 7, 13, 12, 18, 18, 27, 27, 39, 38, 53, 53, 72, 73, 96, 98, 126, 128, 165, 168, 209, 216, 266, 274, 334, 345, 416, 430, 514, 533, 628, 655, 766, 797, 929, 966, 1115, 1164, 1336, 1395, 1590, 1661, 1885, 1969, 2226, 2326, 2611, 2734

Offset: 0

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

• nonn

Examples

			a(7) = 4 because we have [5, 2], [3, 2, 1, 1], [2, 2, 2, 1] and [2, 1, 1, 1, 1, 1].
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A000045, A003107, A027187, A093997, A093998, A298949, A339372.

Programs

Mathematica

nmax = 55; CoefficientList[Series[(1/2) (Product[1/(1 - x^Fibonacci[k]), {k, 2, 26}] + Product[1/(1 + x^Fibonacci[k]), {k, 2, 26}]), {x, 0, nmax}], x]

Formula

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

a(n) = (A003107(n) + A298949(n)) / 2.