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A179991
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Nonhomogeneous three-term sequence a(n) = a(n-1) + a(n-2) + n.
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3
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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
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OFFSET
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1,1
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COMMENTS
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Extension of Fibonacci sequence, with the addition of natural number equal to the index.
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LINKS
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FORMULA
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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^(-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)
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EXAMPLE
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a(5) = a(4)+a(3)+5 = 8+15+5 = 28.
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PROG
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(PARI) Vec(x*(x-2)*(x^2-x+1) / ((x^2+x-1)*(x-1)^2) + O(x^50)) \\ Colin Barker, Mar 12 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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