A192966 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

0, 1, 2, 6, 14, 30, 59, 110, 197, 343, 585, 983, 1634, 2695, 4420, 7220, 11760, 19116, 31029, 50316, 81535, 132061, 213827, 346141, 560244, 906685, 1467254, 2374290, 3841922, 6216618, 10058975, 16276058, 26335529, 42612115, 68948205, 111560915

Offset

0,3

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

Vincenzo Librandi, Table of n, a(n) for n = 0..400

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 - 2*x + 3*x^2 - x^3)/((1-x-x^2)*(1-x)^3). - R. J. Mathar, May 11 2014

a(n) = Fibonacci(n+4) + 2*Fibonacci(n+2) - (n^2 + 5*n + 10)/2. - G. C. Greubel, Jul 11 2019

Mathematica

q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + n(n+1)/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}] (* A030119 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192966 *)
LinearRecurrence[{4, -5, 1, 2, -1}, {0, 1, 2, 6, 14}, 40] (* Vincenzo Librandi, Nov 16 2011 *)
Table[Fibonacci[n+4] +2*Fibonacci[n+2] -(n^2+5*n+10)/2, {n, 0, 40}] (* G. C. Greubel, Jul 11 2019 *)

Prog

(Magma) I:=[0, 1, 2, 6, 14]; [n le 5 select I[n] else 4*Self(n-1)-5*Self(n-2)+Self(n-3)+2*Self(n-4)-Self(n-5): n in [1..40]]; // Vincenzo Librandi, Nov 16 2011
(Magma) F:=Fibonacci; [F(n+4) +2*F(n+2) -(n^2+5*n+10)/2: n in [0..40]]; // G. C. Greubel, Jul 11 2019
(PARI) vector(40, n, n--; f=fibonacci; f(n+4)+2*f(n+2)-(n^2+5*n+10)/2) \\ G. C. Greubel, Jul 11 2019
(Sage) f=fibonacci; [f(n+4) +2*f(n+2) -(n^2+5*n+10)/2 for n in (0..40)] # G. C. Greubel, Jul 11 2019
(GAP) F:=Fibonacci;; List([0..40], n-> F(n+4) +2*F(n+2) -(n^2+5*n+10)/2); # G. C. Greubel, Jul 11 2019

Crossrefs

Cf. A000045, A192232, A192744, A192951.

Sequence in context: A072611 A284023 A366542 * A339668 A260058 A331699

Adjacent sequences: A192963 A192964 A192965 * A192967 A192968 A192969

Keyword

nonn,easy

Author

Clark Kimberling, Jul 13 2011

Status

approved