A192968 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

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.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

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

Cf. A000045, A192232, A192744, A192951, A192967.

Sequence in context: A084631 A219846 A229914 * A277968 A217942 A002936

Adjacent sequences: A192965 A192966 A192967 * A192969 A192970 A192971

Keyword

nonn

Author

Clark Kimberling, Jul 13 2011

Status

approved