A192750 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 d_n.

1, 6, 11, 21, 36, 61, 101, 166, 271, 441, 716, 1161, 1881, 3046, 4931, 7981, 12916, 20901, 33821, 54726, 88551, 143281, 231836, 375121, 606961, 982086, 1589051, 2571141, 4160196, 6731341, 10891541, 17622886, 28514431, 46137321, 74651756

Offset: 0

Author

Clark Kimberling, Jul 09 2011

Keywords

• nonn

Comments

Old definition was: constant term of the reduction by 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
• Feng-Zhen Zhao, The log-behavior of some sequences related to the generalized Leonardo numbers, Integers (2024) Vol. 24, Art. No. A82.
• Index entries for linear recurrences with constant coefficients, signature (2, 0, -1).

Crossrefs

See A192751 for c_n.

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

Programs

Mathematica

q = x^2; s = x + 1; z = 40;
p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 4 n + 2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}] :=
FixedPoint[(s PolynomialQuotient @@ #1 +
       PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
  (* A192750 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192751 *)
LinearRecurrence[{2,0,-1},{1,6,11},40] (* Harvey P. Dale, Dec 03 2023 *)

Formula

G.f.: ( 1+4*x-x^2 ) / ( (x-1)*(x^2+x-1) ). The first differences are in A022088. - R. J. Mathar, May 04 2014

a(n) = 5*Fibonacci(n+2)-4. - Gerry Martens, Jul 04 2015

a(n) = A265752(A265750(n)). - Antti Karttunen, Dec 15 2015

Extensions

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