A269725 a(n) = row number of extended Wythoff array (see A035513) which contains the sequence n times the Fibonacci numbers 1,2,3,5,8,13,21,... .

0, 2, 3, 4, 15, 18, 21, 24, 27, 30, 33, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 630, 651, 672, 693, 714, 735, 756, 777, 798, 819, 840, 861, 882, 903, 924, 945, 966, 987, 1008, 1029, 1050, 1071, 1092, 1113, 1134, 1155, 1176, 1197, 1218, 1239, 1260

Offset: 1

Author

N. J. A. Sloane, Mar 07 2016

Keywords

• nonn
• easy

Examples

			Take n=5: 5 times 1,2,3,5,8,13,... gives 5,10,15,25,40,65,.., which is row 15 of the extended Wythoff array (when extended to the left), so a(5) = 15.
		

References

• J. H. Conway, Posting to Math Fun Mailing List, Nov 25 1996.

Links

• J. H. Conway, Allan Wechsler, Marc LeBrun, Dan Hoey, N. J. A. Sloane, On Kimberling Sums and Para-Fibonacci Sequences, Correspondence and Postings to Math-Fun Mailing List, Nov 1996 to Jan 1997

Crossrefs

Cf. A000045, A022413-A022423, A035513, A269726, A269729.

Programs

Maple

A269725 := proc(n)
    local f,sl,r,c,wrks ;
    f := [seq(n*combinat[fibonacci](i),i=2..30)] ;
    for sl from 0 do
        for r from 1 do
            if A035513(r,1) = op(1+sl,f) then
                wrks := true;
                for c from 2 to 5 do
                    if A035513(r,c) <> op(c+sl,f) then
                        wrks := false;
                    end if;
                end do:
                if wrks then
                    print(n,f,r) ;
                    return r-1 ;
                end if;
            elif A035513(r,1) > op(1+sl,f) then
                break ;
            end if;
        end do:
    end do:
end proc: # R. J. Mathar, May 06 2017

Mathematica

W[n_, k_] := Fibonacci[k+1] Floor[n*GoldenRatio] + (n-1) Fibonacci[k];
a[n_] := Module[{f, sl, r, c, wrks}, f = Table[n*Fibonacci[i], {i, 2, 30}]; For[sl = 0, True, sl++, For[r = 1, True, r++, Which[W[r, 1] == f[[1 + sl]], wrks = True; For[c = 2, c <= 5, c++, If[W[r, c] != f[[c+sl]], wrks = False]]; If[wrks, Return[r-1]], W[r, 1] > f[[1+sl]], Break[]]]]];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Sep 13 2022, after R. J. Mathar *)

Formula

a(n) = A173027(n)-1. - R. J. Mathar, May 06 2017