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 A179991 #17 Mar 12 2017 08:37:22 %S A179991 2,3,8,15,28,49,84,141,234,385,630,1027,1670,2711,4396,7123,11536, %T A179991 18677,30232,48929,79182,128133,207338,335495,542858,878379,1421264, %U A179991 2299671,3720964,6020665,9741660,15762357,25504050,41266441,66770526,108037003,174807566 %N A179991 Nonhomogeneous three-term sequence a(n) = a(n-1) + a(n-2) + n. %C A179991 Extension of Fibonacci sequence, with the addition of natural number equal to the index. %H A179991 Colin Barker, <a href="/A179991/b179991.txt">Table of n, a(n) for n = 1..1000</a> %H A179991 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1). %F A179991 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 A179991 G.f.: x*(x-2)*(x^2-x+1) / ( (x^2+x-1)*(x-1)^2 ). %F A179991 Limiting ratio a(n+1)/a(n) is Phi = 1.618034519... %F A179991 a(n) = 2*A000032(n+1)-n-3. - _R. J. Mathar_, Aug 09 2010 %F A179991 From _Colin Barker_, Mar 12 2017: (Start) %F A179991 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 A179991 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) for n>4. %F A179991 (End) %e A179991 a(5) = a(4)+a(3)+5 = 8+15+5 = 28. %o A179991 (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 A179991 Cf. A000045, A022319 (first differences). %K A179991 nonn,easy %O A179991 1,1 %A A179991 _Carmine Suriano_, Aug 05 2010