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A107857 a(n) = floor[(phi + n mod 2)*a(n-1)], a(1)=1. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

A switched sequence with alternating limits of the golden mean and its square. The sequence uses only one initial term. Note that Limit[a[n]/a[n-1],n->Infinity] 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 r<s, where 1/r+1/s=1. Write l for (l(n)), lu for (l(u(n))), ulu for u(l(u(n)))), etc. Then this sequence is (u, lu, ulu, lulu, ...) = ([s], [r[s]], [s[r[s]]], ...), where [ ] denotes the floor function. For this sequence, r is the golden mean. -- Clark Kimberling, Nov 24 2010

LINKS

Table of n, a(n) for n=1..33.

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

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}]

PROG

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

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

CROSSREFS

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

Sequence in context: A049454 A095055 A305750 * A107858 A214938 A143926

Adjacent sequences:  A107854 A107855 A107856 * A107858 A107859 A107860

KEYWORD

nonn,easy

AUTHOR

Roger L. Bagula, Jun 12 2005

EXTENSIONS

Edited and better name by Ralf Stephan, Nov 24 2010

STATUS

approved

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Last modified September 30 23:44 EDT 2020. Contains 337440 sequences. (Running on oeis4.)