A277345 - OEIS

%I #10 Oct 10 2016 02:47:13

%S 2,3,9,31,131,666,4014,28127,225063,2025643,20256553,222822282,

%T 2673867706,34760280699,486643930629,7299658960799,116794543374991,

%U 1985507237378418,35739130272817302,679043475183538087,13580869503670776867,285198259577086338683

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

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

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html">Incomplete Gamma Function</a>, <a href="http://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>.

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

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

%F a(n) ~ 2*exp(1/2)*cosh(sqrt(5)/2) * (n-1)!. - _Vaclav Kotesovec_, Oct 10 2016

%t 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 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 Cf. A263823.

%K nonn

%O 0,1

%A _Vladimir Reshetnikov_, Oct 09 2016