This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A277345 #11 Feb 16 2025 08:33:36
%S A277345 2,3,9,31,131,666,4014,28127,225063,2025643,20256553,222822282,
%T A277345 2673867706,34760280699,486643930629,7299658960799,116794543374991,
%U A277345 1985507237378418,35739130272817302,679043475183538087,13580869503670776867,285198259577086338683
%N A277345 a(n) = Gamma(n+1, phi)*exp(phi) + Gamma(n+1, 1-phi)*exp(1-phi), where phi=(1+sqrt(5))/2.
%C A277345 Gamma(a, x) is the upper incomplete Gamma function.
%H A277345 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/IncompleteGammaFunction.html">Incomplete Gamma Function</a>, <a href="https://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>.
%F A277345 E.g.f: (exp(phi*x) + exp((1-phi)*x))/(1-x).
%F A277345 Recurrence: n*(a(n) + a(n-1)) = (n+3)*a(n+1) - a(n+2).
%F A277345 a(n) ~ 2*exp(1/2)*cosh(sqrt(5)/2) * (n-1)!. - _Vaclav Kotesovec_, Oct 10 2016
%t A277345 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 *)
%t A277345 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 *)
%Y A277345 Cf. A263823.
%K A277345 nonn
%O A277345 0,1
%A A277345 _Vladimir Reshetnikov_, Oct 09 2016