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A192968
Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
3
0, 1, 1, 3, 7, 16, 33, 64, 118, 210, 364, 619, 1038, 1723, 2839, 4653, 7597, 12370, 20103, 32626, 52900, 85716, 138826, 224773, 363852, 588901, 953053, 1542279, 2495683, 4038340, 6534429, 10573204, 17108098, 27681798, 44790424, 72472783
OFFSET
0,4
COMMENTS
The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + n(n-1)/2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.
FORMULA
a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
G.f.: x*(1 -3*x +4*x^2 -x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014
a(n) = 3*Fibonacci(n+2) -(n^2+3*n+6)/2. - G. C. Greubel, Jul 11 2019
MATHEMATICA
Table[3*Fibonacci[n+2] -(n^2+3*n+6)/2, {n, 0, 40}] (* G. C. Greubel, Jul 11 2019 *)
PROG
(PARI) vector(40, n, n--; 3*fibonacci(n+2) -(n^2+3*n+6)/2) \\ G. C. Greubel, Jul 11 2019
(Magma) [3*Fibonacci(n+2) -(n^2+3*n+6)/2: n in [0..40]]; // G. C. Greubel, Jul 11 2019
(Sage) [3*fibonacci(n+2) -(n^2+3*n+6)/2 for n in (0..40)] # G. C. Greubel, Jul 11 2019
(GAP) List([0..40], n-> 3*Fibonacci(n+2) -(n^2+3*n+6)/2); # G. C. Greubel, Jul 11 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 13 2011
STATUS
approved