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A192979 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments. 3
1, 1, 4, 9, 19, 36, 65, 113, 192, 321, 531, 872, 1425, 2321, 3772, 6121, 9923, 16076, 26033, 42145, 68216, 110401, 178659, 289104, 467809, 756961, 1224820, 1981833, 3206707, 5188596, 8395361, 13584017, 21979440, 35563521, 57543027, 93106616 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 1 - n + n^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 (3,-2,-1,1).

FORMULA

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).

G.f.: (1-2*x+3*x^2)/((1-x)^2*(1-x-x^2)). - Colin Barker, May 11 2014

a(n) = Fibonacci(n+3) + Lucas(n+2) - 2*(n+2). - G. C. Greubel, Jul 24 2019

MATHEMATICA

(* First program *)

q = x^2; s = x + 1; z = 40;

p[0, x]:= 1;

p[n_, x_]:= x*p[n-1, x] +n^2-n+1;

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}] (* A192979 *)

u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192980 *)

(* Additional programs *)

Table[Fibonacci[n+3]+LucasL[n+2] -2*(n+2), {n, 0, 40}] (* G. C. Greubel, Jul 24 2019 *)

PROG

(PARI) vector(40, n, n--; f=fibonacci; 2*f(n+3)+f(n+1) -2*(n+2)) \\ G. C. Greubel, Jul 24 2019

(MAGMA) [Fibonacci(n+3)+Lucas(n+2)-2*(n+2): n in [0..40]]; // G. C. Greubel, Jul 24 2019

(Sage) f=fibonacci; [2*f(n+3)+f(n+1) -2*(n+2) for n in (0..40)] # G. C. Greubel, Jul 24 2019

(GAP) F:=Fibonacci;; List([0..40], n-> 2*F(n+3)+F(n+1) -2*(n+2)); # G. C. Greubel, Jul 24 2019

CROSSREFS

Cf. A000032, A000045, A192232, A192744, A192951, A192980.

Sequence in context: A008113 A008111 A023611 * A301080 A232623 A002804

Adjacent sequences:  A192976 A192977 A192978 * A192980 A192981 A192982

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 13 2011

STATUS

approved

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Last modified July 5 16:07 EDT 2020. Contains 335473 sequences. (Running on oeis4.)