A274983 a(n) = [n]phi! + [n]{1-phi}!, where [n]_q! is the q-factorial, phi = (1+sqrt(5))/2.

2, 2, 3, 14, 130, 2120, 58120, 2636360, 196132320, 23805331920, 4698862837680, 1505416321070640, 781888977967152000, 657866357975539785600, 896265744457831561756800, 1976607903479486428467148800, 7055269158071576119808840371200

Offset: 0

Author

Vladimir Reshetnikov, Sep 23 2016

Keywords

• nonn

Examples

			For n = 3, [3]_phi! = 1060 + 474*sqrt(5), so a(5) = 2*1060 = 2120 and A274985(5) = 2*474 = 948.
		

Links

• Eric Weisstein's World of Mathematics, q-Factorial, Golden Ratio.

Crossrefs

Cf. A274985, A005329, A275706, A276474, A276688, A276987.

Programs

Mathematica

Round@Table[QFactorial[n, GoldenRatio] + QFactorial[n, 1 - GoldenRatio], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)

Formula

a(n) ~ c * phi^(n*(n+3)/2), where c = QPochhammer(phi-1) = A276987 = 0.1208019218617061294237231569887920563043992516794... . - Vaclav Kotesovec, Sep 24 2016

From Vladimir Reshetnikov, Sep 24 2016 (Start)

[n]_phi! = (a(n) + A274985(n)*sqrt(5))/2.

[n]_{1-phi}! = (a(n) - A274985(n)*sqrt(5))/2. (End)