A136376 - OEIS

%I #29 Aug 01 2023 08:07:04

%S 1,3,8,18,37,73,139,259,474,856,1529,2707,4757,8307,14428,24942,42941,

%T 73661,125951,214739,365166,619508,1048753,1771943,2988457,5031843,

%U 8459504,14201994,23811349,39873841,66695539,111440227,186016962

%N a(n) = n*F(n) + (n-1)*F(n-1).

%C For n>2, mod 2 = (0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, ...), i.e., two evens followed by four odds (repeating).

%C Inverse binomial transform of A117202: (1, 4, 15, 52, ...). - _Gary W. Adamson_, Sep 03 2008

%H Alois P. Heinz, <a href="/A136376/b136376.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).

%F a(n) = n*F(n) + (n-1)*F(n-1). Equals the matrix product A128064 (unsigned) * A000045.

%F From _R. J. Mathar_, Jul 13 2009: (Start)

%F a(n) = A045925(n) + A045925(n-1).

%F a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4).

%F G.f.: x*(1+x)*(1+x^2)/(x^2+x-1)^2. (End)

%F a(n) = A238344(3n-2,n-1). - _Alois P. Heinz_, Apr 11 2014

%F From _Vladimir Reshetnikov_, Oct 28 2015: (Start)

%F a(n) = ((n+1)*F(n)+(n-1)*L(n))/2, where L(n) are Lucas numbers (A000032).

%F E.g.f.: (exp(phi*x)*(phi^3*x-1)-exp(-x/phi)*(phi^3+x)/phi)/(sqrt(5)*phi)+1, where phi=(1+sqrt(5))/2.

%F (End)

%e a(5) = 37 = a(n)*F(n) + (n-1)*F(n-1) = 5*5 + 4*3 = 25 + 12.

%t Table[n*Fibonacci[n] + (n - 1)*Fibonacci[n - 1], {n, 1, 50}] (* _Stefan Steinerberger_, Dec 28 2007 *)

%o (PARI) a(n)=n*fibonacci(n)+(n-1)*fibonacci(n-1) \\ _Charles R Greathouse IV_, Oct 07 2015

%o (PARI) Vec(x*(1+x)*(1+x^2)/(x^2+x-1)^2 + O(x^100)) \\ _Altug Alkan_, Oct 28 2015

%Y Cf. A000032, A000045, A045925, A128064, A117202, A238344.

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_, Dec 28 2007

%E More terms from _Stefan Steinerberger_, Dec 28 2007