A192954 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
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
Keywords
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).
Programs
-
GAP
List([0..40], n-> 2*Lucas(1,-1,n+2)[2]-(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
-
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 *) -
PARI
vector(40, n, n--; f=fibonacci; 2*(f(n+3)+f(n+1))-(2*n+5)) \\ 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
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
Comments