A192751 Define a pair of sequences c_n, d_n by c_0=0, d_0=1 and thereafter c_n = c_{n-1}+d_{n-1}, d_n = c_{n-1}+4*n+2; sequence here is c_n.

0, 1, 7, 18, 39, 75, 136, 237, 403, 674, 1115, 1831, 2992, 4873, 7919, 12850, 20831, 33747, 54648, 88469, 143195, 231746, 375027, 606863, 981984, 1588945, 2571031, 4160082, 6731223, 10891419, 17622760, 28514301, 46137187, 74651618, 120788939

Offset

0,3

Comments

Old definition was: coefficient of x in the reduction under x^2->x+1 of the polynomial p(n,x) defined recursively by p(n,x) = x*p(n-1,x) + 4n+2 for n>0, with p(0,x)=1.

For discussions of polynomial reduction, see A192232 and A192744.

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Formula

G.f.: x*(x^2-4*x-1)/((x-1)^2*(x^2+x-1)). First differences are in A192750. - Colin Barker, Nov 13 2012

a(n) = 5*Fibonacci(n+3) - (4*n+10). - N. J. A. Sloane, Dec 15 2015

a(n) = A265753(A265750(n)). - Antti Karttunen, Dec 15 2015

Mathematica

(See A192750.)
CoefficientList[Series[x (x^2-4x-1)/((x-1)^2(x^2+x-1)), {x, 0, 40}], x] (* or *) LinearRecurrence[{3, -2, -1, 1}, {0, 1, 7, 18}, 40] (* Harvey P. Dale, Feb 23 2022 *)

Prog

(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; 1, -1, -2, 3]^n*[0; 1; 7; 18])[1, 1] \\ Charles R Greathouse IV, May 20 2026

Crossrefs

See A192750 for d_n.

Cf. A000045, A192744, A192232, A022088, A265750, A265752.

Sequence in context: A320281 A169677 A263876 * A272459 A133673 A023166

Adjacent sequences: A192748 A192749 A192750 * A192752 A192753 A192754

Keyword

nonn,easy

Author

Clark Kimberling, Jul 09 2011

Extensions

Description corrected by Antti Karttunen, Dec 15 2015

Entry revised by N. J. A. Sloane, Dec 15 2015

Status

approved