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A192975 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments. 3
1, 1, 8, 19, 41, 78, 141, 245, 416, 695, 1149, 1886, 3081, 5017, 8152, 13227, 21441, 34734, 56245, 91053, 147376, 238511, 385973, 624574, 1010641, 1635313, 2646056, 4281475, 6927641, 11209230, 18136989, 29346341, 47483456, 76829927 (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 + 2*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+7*x^2-2*x^3)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, May 11 2014

a(n) = Fibonacci(n+3) + 3*Lucas(n+2) - 2*(2*n+5). - 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] +2*n^2 -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}] (* A192975 *)

u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192976 *)

(* Additional programs *)

Table[Fibonacci[n+3]+3*LucasL[n+2] -2*(2*n+5), {n, 0, 40}] (* G. C. Greubel, Jul 24 2019 *)

PROG

(PARI) vector(40, n, n--; f=fibonacci; 4*f(n+3)+3*f(n+1) -2*(2*n+5)) \\ G. C. Greubel, Jul 24 2019

(MAGMA) [Fibonacci(n+3)+3*Lucas(n+2)-2*(2*n+5): n in [0..40]]; // G. C. Greubel, Jul 24 2019

(Sage) f=fibonacci; [4*f(n+3)+3*f(n+1) -2*(2*n+5) for n in (0..40)] # G. C. Greubel, Jul 24 2019

(GAP) F:=Fibonacci;; List([0..40], n-> 4*F(n+3)+3*F(n+1) -2*(2*n+5)); # G. C. Greubel, Jul 24 2019

CROSSREFS

Cf. A000032, A000045, A192232, A192744, A192951, A192976.

Sequence in context: A289877 A089111 A127873 * A156198 A278947 A153026

Adjacent sequences:  A192972 A192973 A192974 * A192976 A192977 A192978

KEYWORD

nonn

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.)