OFFSET
1,1
COMMENTS
LINKS
Index entries for linear recurrences with constant coefficients, signature (5,-7,1,3,-1).
FORMULA
G.f.: -x*(3*x^4-7*x^3-x^2+6*x-2)/((x-1)*(x^2-3*x+1)*(x^2+x-1)). - Colin Barker, Nov 12 2012
a(n) = 1 - Fibonacci(n) + Fibonacci(1+n) - Fibonacci(2n) + Fibonacci(1+2n). - Friedjof Tellkamp, Nov 22 2021
EXAMPLE
MATHEMATICA
Remove["Global`*"];
q[x_] := x + 1; p[n_, x_] := 1 + x^n + x^(2 n);
Table[Simplify[p[n, x]], {n, 1, 5}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
(* A192464 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* 2*A051450 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]
(* A051450 *)
Table[1-Fibonacci[n]+Fibonacci[1+n]-Fibonacci[2n]+Fibonacci[1+2n], {n, 1, 29}]
(* Friedjof Tellkamp, Nov 22 2021 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 01 2011
STATUS
approved