A269729 -id:A269729 - cp's OEIS frontend

A269733 First differences of A269729.

1, 1, 1, 1, 3, 3, -5, 3, 3, -5, 3, 3, 8, 8, -13, 8, 8, -13, 8, -13, 8, 8, -13, 8, 8, -13, 8, -13, 8, 8, -13, 8, 8, 21, 21, -34, 21, 21, -34, 21, -34, 21, 21, -34, 21, 21, -34, 21, -34, 21, 21, -34, 21, -34, 21, 21, -34, 21, 21, -34, 21, -34, 21, 21, -34, 21, 21, -34, 21, -34, 21, 21, -34, 21, -34, 21, 21, -34, 21, 21

Offset: 0

N. J. A. Sloane, Mar 08 2016

John Conway remarks that he can explain why the terms of this sequence are (up to sign) Fibonacci numbers.

J. H. Conway, Postings to Math Fun Mailing List, Nov 25 1996 and Dec 02 1996.

Jean-François Alcover, Plot of differences
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

Cf. A269729.

[seq(A269729(n),n=0..120)] ;
DIFF(%) ; # R. J. Mathar, May 08 2019

More terms from R. J. Mathar, May 08 2019

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,... .

Original entry on oeis.org

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

N. J. A. Sloane, Mar 07 2016

			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.

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

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

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

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

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 *)

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

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A269733 First differences of A269729.

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