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 A179991 Nonhomogeneous three-term sequence a(n) = a(n-1) + a(n-2) + n. 3
 2, 3, 8, 15, 28, 49, 84, 141, 234, 385, 630, 1027, 1670, 2711, 4396, 7123, 11536, 18677, 30232, 48929, 79182, 128133, 207338, 335495, 542858, 878379, 1421264, 2299671, 3720964, 6020665, 9741660, 15762357, 25504050, 41266441, 66770526, 108037003, 174807566 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Extension of Fibonacci sequence, with the addition of natural number equal to the index. LINKS Colin Barker, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1). FORMULA 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). G.f.: x*(x-2)*(x^2-x+1) / ( (x^2+x-1)*(x-1)^2 ). Limiting ratio a(n+1)/a(n) is Phi = 1.618034519... a(n) = 2*A000032(n+1)-n-3. - R. J. Mathar, Aug 09 2010 From Colin Barker, Mar 12 2017: (Start) 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). a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) for n>4. (End) EXAMPLE a(5) = a(4)+a(3)+5 = 8+15+5 = 28. PROG (PARI) Vec(x*(x-2)*(x^2-x+1) / ((x^2+x-1)*(x-1)^2) + O(x^50)) \\ Colin Barker, Mar 12 2017 CROSSREFS Cf. A000045, A022319 (first differences). Sequence in context: A249357 A291400 A056802 * A026698 A099428 A099920 Adjacent sequences:  A179988 A179989 A179990 * A179992 A179993 A179994 KEYWORD nonn,easy AUTHOR Carmine Suriano, Aug 05 2010 STATUS approved

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Last modified September 20 22:20 EDT 2019. Contains 327252 sequences. (Running on oeis4.)