OFFSET
0,2
COMMENTS
See A192872.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).
FORMULA
a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
a(n) = A192883(n+1).
G.f.: x*(5-2*x)/((1+x)*(1-3*x+x^2)). - R. J. Mathar, Aug 01 2011
a(n) = (A005248(n+2) - 7*(-1)^n)/5. - R. J. Mathar, Aug 01 2011
a(n) = Fibonacci(n+2)^2 - (-1)^n. - G. C. Greubel, Feb 06 2019
Sum_{n>=1} 1/a(n) = 7/18. - Amiram Eldar, Oct 05 2020
MATHEMATICA
(* First program *)
q = x^2; s = x + 1; z = 28;
p[0, x_]:= 3; p[1, x_]:= 5 x;
p[n_, x_]:= p[n-1, x]*x + p[n-2, x]*x^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}] (* A192919 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192920 *)
(* Second program *)
LinearRecurrence[{2, 2, -1}, {0, 5, 8}, 30] (* G. C. Greubel, Feb 06 2019 *)
PROG
(PARI) vector(30, n, n--; fibonacci(n+2)^2 -(-1)^n) \\ G. C. Greubel, Feb 06 2019, modified Jul 28 2019
(Magma) [Fibonacci(n+2)^2 -(-1)^n: n in [0..30]]; // G. C. Greubel, Feb 06 2019, modified Jul 28 2019
(Sage) [fibonacci(n+2)^2 -(-1)^n for n in (0..30)] # G. C. Greubel, Feb 06 2019, modified Jul 28 2019
(GAP) a:=[0, 5, 8];; for n in [4..30] do a[n]:=2*a[n-1]+2*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Feb 06 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 12 2011
STATUS
approved