OFFSET
0,3
COMMENTS
The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x), with p(0,x) = 1, p(1,x) = x + 1.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).
FORMULA
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3) for n>3.
G.f.: 1 + x*(1 - x - x^2)/((1 - x)*(1 - 3*x + x^2)). - R. J. Mathar, Jul 13 2011
a(n) = 2*Fibonacci(2*n-2) + 1 for n>0, a(0)=1. - Bruno Berselli, Dec 27 2016
a(n) = -1 + 3*a(n-1) - a(n-2) with a(1) = 1 and a(2) = 3. Cf. A055588 and A097136. - Peter Bala, Nov 12 2017
MATHEMATICA
u = 1; v = 1; a = 1; b = 1; c = 1; d = 1; e = 0; f = 1;
q = x^2; s = u*x + v; z = 26;
p[0, x_] := a; p[1, x_] := b*x + c
p[n_, x_] := d*(x^2)*p[n - 1, x] + e*x*p[n - 2, x] + f;
Table[Expand[p[n, x]], {n, 0, 8}]
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}];
u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192908 *)
u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A069403 *)
Simplify[FindLinearRecurrence[u0]] (* recurrence for 0-sequence *)
Simplify[FindLinearRecurrence[u1]] (* recurrence for 1-sequence *)
LinearRecurrence[{4, -4, 1}, {1, 1, 3, 7}, 30] (* G. C. Greubel, Jan 11 2019 *)
PROG
(PARI) vector(30, n, n--; if(n==0, 1, 1+2*fibonacci(2*n-2))) \\ G. C. Greubel, Jan 11 2019
(Magma) [1] cat [1+2*Fibonacci(2*(n-1)): n in [1..30]]; // G. C. Greubel, Jan 11 2019
(Sage) [1]+[1+2*fibonacci(2*(n-1)) for n in (1..30)] # G. C. Greubel, Jan 11 2019
(GAP) Concatenation([1], List([1..30], n -> 1+2*Fibonacci(2*(n-1)))); # G. C. Greubel, Jan 11 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 12 2011
STATUS
approved