A277345 a(n) = Gamma(n+1, phi)*exp(phi) + Gamma(n+1, 1-phi)*exp(1-phi), where phi=(1+sqrt(5))/2.

2, 3, 9, 31, 131, 666, 4014, 28127, 225063, 2025643, 20256553, 222822282, 2673867706, 34760280699, 486643930629, 7299658960799, 116794543374991, 1985507237378418, 35739130272817302, 679043475183538087, 13580869503670776867, 285198259577086338683

Offset: 0

Author

Vladimir Reshetnikov, Oct 09 2016

Keywords

• nonn

Comments

Gamma(a, x) is the upper incomplete Gamma function.

Links

• Eric Weisstein's MathWorld, Incomplete Gamma Function, Golden Ratio.

Crossrefs

Cf. A263823.

Programs

Mathematica

RecurrenceTable[{a[0] == 2, a[1] == 3, a[2] == 9, n (a[n] + a[n - 1]) == (n + 3) a[n + 1] - a[n + 2]}, a[n], {n, 0, 20}] (* or *)
Round@Table[Gamma[n + 1, GoldenRatio] Exp[GoldenRatio] + Gamma[n + 1, 1 - GoldenRatio] Exp[1 - GoldenRatio], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster *)

Formula

E.g.f: (exp(phi*x) + exp((1-phi)*x))/(1-x).

Recurrence: n*(a(n) + a(n-1)) = (n+3)*a(n+1) - a(n+2).

a(n) ~ 2*exp(1/2)*cosh(sqrt(5)/2) * (n-1)!. - Vaclav Kotesovec, Oct 10 2016