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A076739 Number of compositions of n into Fibonacci numbers (1 counted as single Fibonacci number). 6
1, 1, 2, 4, 7, 14, 26, 49, 94, 177, 336, 637, 1206, 2288, 4335, 8216, 15574, 29515, 55943, 106030, 200959, 380889, 721906, 1368251, 2593291, 4915135, 9315811, 17656534, 33464955, 63427148, 120215370, 227847814, 431846824, 818492263 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

From Gary W. Adamson, Sep 12 2008: (Start)

Equals right border of triangle A144172 and row sums with offset 1.

Equals INVERT transform of the characteristic function of the Fibonacci numbers starting with offset 1: (1, 1, 1, 0, 1,...) (if the first "1" is retained: = 1, 1, 2, 4, 7, 14,...). (End)

REFERENCES

A. Knopfmacher & N. Robbins, On binary and Fibonacci compositions, Annales Univ. Sci. Budapest, Sect. Comp. 22 (2003) 193-206. - Neville Robbins, Mar 06 2010

LINKS

T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..3600 (first 301 terms from T. D. Noe)

FORMULA

G.f.: 1/(1-Sum_{k>1} x^Fibonacci(k)). - Vladeta Jovovic, Jun 20 2003

a(n) ~ c * d^n, where d=1.8953300920998046150867311236880760382884608526935119695..., c=0.5615834114640436146286049301387868479914202616794427372... - Vaclav Kotesovec, May 01 2014

EXAMPLE

a(4) = 7 since 3+1 = 2+2 = 2+1+1 = 1+3 = 1+2+1 = 1+1+2 = 1+1+1+1.

MAPLE

a:= proc(n) option remember; local r, f;

      if n=0 then 1 else r, f:= 0, [1$2];

        while f[2] <= n do r:= r+a(n-f[2]);

          f:= [f[2], f[1]+f[2]]

        od; r

      fi

    end:

seq(a(n), n=0..35);  # Alois P. Heinz, Feb 20 2017

MATHEMATICA

max=40; 1/(1-Total[x^Fibonacci[Range[2, Ceiling[Sqrt[max]]+2]]]) + O[x]^max // CoefficientList[#, x]& (* Jean-Fran├žois Alcover, Mar 29 2017, after Vladeta Jovovic *)

CROSSREFS

Cf. A080888.

Cf. A144172, A010056. - Gary W. Adamson, Sep 12 2008

Sequence in context: A257792 A079975 A253511 * A017996 A287154 A024502

Adjacent sequences:  A076736 A076737 A076738 * A076740 A076741 A076742

KEYWORD

nonn

AUTHOR

David W. Wilson, Jun 19 2003

STATUS

approved

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Last modified June 6 01:16 EDT 2020. Contains 334858 sequences. (Running on oeis4.)