A030119 a(n) = a(n-1) + a(n-2) + n, a(0) = a(1) = 1.
1, 1, 4, 8, 16, 29, 51, 87, 146, 242, 398, 651, 1061, 1725, 2800, 4540, 7356, 11913, 19287, 31219, 50526, 81766, 132314, 214103, 346441, 560569, 907036, 1467632, 2374696, 3842357, 6217083, 10059471, 16276586, 26336090, 42612710, 68948835, 111561581
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..400
- Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).
Programs
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GAP
F:=Fibonacci;; List([0..40], n-> F(n+3)+2*F(n+1)-n-3); # G. C. Greubel, Jul 24 2019
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Magma
[Lucas(n+2) + Fibonacci(n+1) - (n+3) : n in [0..40]]; // Vincenzo Librandi, Nov 16 2011
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Mathematica
RecurrenceTable[{a[0]==a[1]==1,a[n]==a[n-1]+a[n-2]+n},a,{n,40}] (* or *) LinearRecurrence[{3,-2,-1,1},{1,1,4,8},40] (* Harvey P. Dale, Nov 06 2011 *) -
PARI
Vec((1-2*x+3*x^2-x^3)/((1-x-x^2)*(1-x)^2) + O(x^40)) \\ Colin Barker, Mar 11 2017
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PARI
vector(40, n, n--; f=fibonacci; f(n+3)+2*f(n+1)-n-3) \\ G. C. Greubel, Jul 24 2019
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Sage
f=fibonacci; [f(n+3)+2*f(n+1)-n-3 for n in (0..40)] # G. C. Greubel, Jul 24 2019
Formula
Periodic mod 6.
G.f.: (1 - 2*x + 3*x^2 - x^3) / ((1 - x - x^2)*(1-x)^2).
a(n) = Lucas(n+2) + Fibonacci(n+1) - (n+3).
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4); a(0)=1, a(1)=1, a(2)=4, a(3)=8. - Harvey P. Dale, Nov 06 2011
a(n) = -3 + (2^(-n)*((1-sqrt(5))^n*(-3+2*sqrt(5)) + (1+sqrt(5))^n*(3+2*sqrt(5)))) / sqrt(5) - n. - Colin Barker, Mar 11 2017
Extensions
Description corrected and sequence extended by Erich Friedman