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A192954
Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
3
1, 1, 5, 11, 23, 43, 77, 133, 225, 375, 619, 1015, 1657, 2697, 4381, 7107, 11519, 18659, 30213, 48909, 79161, 128111, 207315, 335471, 542833, 878353, 1421237, 2299643, 3720935, 6020635, 9741629, 15762325, 25504017, 41266407, 66770491
OFFSET
0,3
COMMENTS
The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 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.
FORMULA
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
From R. J. Mathar, May 08 2014: (Start)
G.f.: (1 -2*x +4*x^2 -x^3)/((1-x-x^2)*(1-x)^2).
a(n) - a(n-1) = A168674(n-1). (End)
a(n) = 2*Lucas(n+2) - (2*n+5). - G. C. Greubel, Jul 12 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;
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}] (* A192954 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192955 *)
(* Second program *)
Table[2*LucasL[n+2]-(2*n+5), {n, 0, 40}] (* G. C. Greubel, Jul 12 2019 *)
LinearRecurrence[{3, -2, -1, 1}, {1, 1, 5, 11}, 40] (* Harvey P. Dale, Jan 13 2022 *)
PROG
(PARI) vector(40, n, n--; f=fibonacci; 2*(f(n+3)+f(n+1))-(2*n+5)) \\ G. C. Greubel, Jul 12 2019
(Magma) [2*Lucas(n+2)-(2*n+5): n in [0..40]]; // G. C. Greubel, Jul 12 2019
(Sage) [2*lucas_number2(n+2, 1, -1)-(2*n+5) for n in (0..40)] # G. C. Greubel, Jul 12 2019
(GAP) List([0..40], n-> 2*Lucas(1, -1, n+2)[2]-(2*n+5)); # G. C. Greubel, Jul 12 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 13 2011
STATUS
approved