A248924 Sequence derived from arithmetic relations between powers of phi (A001622): a(n) = phi^n - (-1)^n * (n - phi^-n).

2, 2, 1, 7, 3, 16, 12, 36, 39, 85, 113, 210, 310, 534, 829, 1379, 2191, 3588, 5760, 9368, 15107, 24497, 39581, 64102, 103658, 167786, 271417, 439231, 710619, 1149880, 1860468, 3010380, 4870815, 7881229, 12752009, 20633274, 33385246, 54018558, 87403765

Offset

0,1

Links

Table of n, a(n) for n=0..38.

Eric Weisstein's World of Mathematics, Phi Number System

Index entries for linear recurrences with constant coefficients, signature (-1,2,3,1).

Formula

a(n) = phi^n - (-1)^n * (n - phi^-n), phi = (1 + sqrt(5))/2 = A001622.

G.f.: (2*x+1)*(x^2-2)/((x^2+x-1)*(x+1)^2). - Alois P. Heinz, Oct 17 2014

a(n) = A000032(n) - (-1)^n*n. - Alois P. Heinz, Oct 17 2014

Example

a(7) = phi^7 + (n - phi^-7) = 36; a(10) = phi^10 - (n - phi^-10) = 113.

Mathematica

LinearRecurrence[{-1, 2, 3, 1}, {2, 2, 1, 7}, 40] (* Harvey P. Dale, Sep 21 2023 *)

Prog

(PARI) a(n)=fibonacci(n-1) + fibonacci(n+1) - n*(-1)^n \\ Charles R Greathouse IV, Oct 28 2014

Crossrefs

Cf. A000032, A001622.

Sequence in context: A108338 A021455 A271460 * A307455 A136502 A144502

Adjacent sequences: A248921 A248922 A248923 * A248925 A248926 A248927

Keyword

nonn,easy

Author

Gustavo Mendoza, Oct 16 2014

Status

approved