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A192973 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments. 3
1, 3, 10, 23, 47, 88, 157, 271, 458, 763, 1259, 2064, 3369, 5483, 8906, 14447, 23415, 37928, 61413, 99415, 160906, 260403, 421395, 681888, 1103377, 1785363, 2888842, 4674311, 7563263, 12237688, 19801069, 32038879, 51840074, 83879083 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

Vincenzo Librandi, Table of n, a(n) for n = 1..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.: x*(1+3*x^2)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, May 11 2014

a(n) = Lucas(n+4) - Fibonacci(n-1) - 2*(2*n+3). - Ehren Metcalfe, Jul 13 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}] (* A192973 *)

u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192974 *)

(* Additional programs *)

LinearRecurrence[{3, -2, -1, 1}, {1, 3, 10, 23}, 50] (* Vincenzo Librandi, Jul 14 2019 *)

With[{F = Fibonacci}, Table[F[n+4]+3*F[n+2] -2*(2*n+3), {n, 40}]] (* G. C. Greubel, Jul 24 2019 *)

PROG

(MAGMA) [Lucas(n+4)-Fibonacci(n-1)-2*(2*n+3): n in [1..40]]; // Vincenzo Librandi, Jul 14 2019

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

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

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

CROSSREFS

Cf. A000032, A000045, A192232, A192744, A192951, A192974.

Sequence in context: A145069 A293350 A256525 * A294503 A080204 A115982

Adjacent sequences:  A192970 A192971 A192972 * A192974 A192975 A192976

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 13 2011

STATUS

approved

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Last modified July 24 06:30 EDT 2021. Contains 346273 sequences. (Running on oeis4.)