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

Author

Clark Kimberling, Jul 09 2011

Keywords

• nonn

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).

Crossrefs

See A192750 for d_n.

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

Programs

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 *)

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

Extensions

Description corrected by Antti Karttunen, Dec 15 2015

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