A107857 a(n) = floor[(phi + n mod 2)*a(n-1)], a(1)=1.

1, 1, 2, 3, 7, 11, 28, 45, 117, 189, 494, 799, 2091, 3383, 8856, 14329, 37513, 60697, 158906, 257115, 673135, 1089155, 2851444, 4613733, 12078909, 19544085, 51167078, 82790071, 216747219, 350704367, 918155952, 1485607537, 3889371025

Offset: 1

Author

Roger L. Bagula, Jun 12 2005

Keywords

• nonn
• easy

Comments

A switched sequence with alternating limits of the golden mean and its square. The sequence uses only one initial term. Note that lim_{n->oo} a(n)/a(n-1) does not exist.

The consecutive pairs (2,3), (7,11), (28,45) occur as pairs in columns 2 and 3 of the Wythoff array, A035513. Suppose (l(n)) and (u(n)) are the lower and upper Beatty sequences of positive irrational numbers rClark Kimberling, Nov 24 2010

Links

• Index entries for linear recurrences with constant coefficients, signature (1,4,-4,1,-1).

Crossrefs

Cf. A000045, A000201, A001950, A015448, A033887.

Programs

Magma

[ n eq 1 select 1 else Floor(((Sqrt(5)+1)/2+(n mod 2))*Self(n-1)): n in [1..35] ];

Mathematica

Phi = N[(Sqrt[5] + 1)/2] F[1] = 1; F[n__] := F[n] = If[Mod[n, 2] == 0, Floor[Phi*F[n - 1]], Floor[(Phi + 1)*F[n -1]]] a = Table[F[n], {n, 1, 50}]
LinearRecurrence[{1,4,-4,1,-1},{1,1,2,3,7},40] (* Harvey P. Dale, Mar 31 2023 *)

PARI

a(n)=if(n<2,1,floor((phi+n%2)*a(n-1)))

Formula

G.f.: -x*(-1+3*x^2-x^3+x^4) / ( (x-1)*(x^4+4*x^2-1) ). - R. J. Mathar, Sep 11 2011

a(2n+2) = (1/2)*(Fib(3n+2) + 1), a(2n+1) = (1/2)*(Fib(3n+1) + 1).

a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) + a(n-4) - a(n-5). - Wesley Ivan Hurt, May 04 2025

Extensions

Edited and better name by Ralf Stephan, Nov 24 2010