|
%I #17 Mar 12 2017 08:37:22
%S 2,3,8,15,28,49,84,141,234,385,630,1027,1670,2711,4396,7123,11536,
%T 18677,30232,48929,79182,128133,207338,335495,542858,878379,1421264,
%U 2299671,3720964,6020665,9741660,15762357,25504050,41266441,66770526,108037003,174807566
%N Nonhomogeneous three-term sequence a(n) = a(n-1) + a(n-2) + n.
%C Extension of Fibonacci sequence, with the addition of natural number equal to the index.
%H Colin Barker, <a href="/A179991/b179991.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 (3,-2,-1,1).
%F a(n) = F(n)+sum(i; i=1 to n)+sum((F(k)*sum(j;j=1 to k-n-1); k=0 to n-4))+2F(n-3).
%F G.f.: x*(x-2)*(x^2-x+1) / ( (x^2+x-1)*(x-1)^2 ).
%F Limiting ratio a(n+1)/a(n) is Phi = 1.618034519...
%F a(n) = 2*A000032(n+1)-n-3. - _R. J. Mathar_, Aug 09 2010
%F From _Colin Barker_, Mar 12 2017: (Start)
%F a(n) = 2^(-n)*(-3*2^n+(1-sqrt(5))^(1+n) + (1+sqrt(5))^n + sqrt(5)*(1+sqrt(5))^n - 2^n*n).
%F a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) for n>4.
%F (End)
%e a(5) = a(4)+a(3)+5 = 8+15+5 = 28.
%o (PARI) Vec(x*(x-2)*(x^2-x+1) / ((x^2+x-1)*(x-1)^2) + O(x^50)) \\ _Colin Barker_, Mar 12 2017
%Y Cf. A000045, A022319 (first differences).
%K nonn,easy
%O 1,1
%A _Carmine Suriano_, Aug 05 2010
|
