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

0,1

Comments

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

Links

Hans J. H. Tuenter, Table of n, a(n) for n = 0..100

Eric Weisstein's World of Mathematics, Incomplete Gamma Function, Golden Ratio.

Formula

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

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

a(n)/n! ~ exp(phi) + exp(1-phi) = 2*exp(1/2)*cosh(sqrt(5)/2) = 5.582168726... = A328344. - Vaclav Kotesovec, Oct 10 2016 [Corrected and extended by Hans J. H. Tuenter, Mar 24 2026]

From Hans J. H. Tuenter, Mar 19 2026: (Start)

a(n) = n!*Sum_{k=0..n} L(k)/k!, where L(n)=A000032(n) are the Lucas numbers.

a(n) = L(n) + n*a(n-1). (End)

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 *)
f[n_] := n!*Sum[LucasL[k]/k!, {k, 0, n}]; Table[ f[n], {n, 0, 100}] (* Hans J. H. Tuenter, Mar 19 2026 *)

Crossrefs

Cf. A000032, A111139, A263823, A328344.

Sequence in context: A281270 A322752 A386925 * A259943 A296263 A064020

Adjacent sequences: A277342 A277343 A277344 * A277346 A277347 A277348

Keyword

nonn

Author

Vladimir Reshetnikov, Oct 09 2016

Status

approved