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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,... .

%I A269725 #26 Sep 14 2022 13:37:39
%S A269725 0,2,3,4,15,18,21,24,27,30,33,96,104,112,120,128,136,144,152,160,168,
%T A269725 176,184,192,200,208,216,224,232,630,651,672,693,714,735,756,777,798,
%U A269725 819,840,861,882,903,924,945,966,987,1008,1029,1050,1071,1092,1113,1134,1155,1176,1197,1218,1239,1260
%N 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,... .
%D A269725 J. H. Conway, Posting to Math Fun Mailing List, Nov 25 1996.
%H A269725 J. H. Conway, Allan Wechsler, Marc LeBrun, Dan Hoey, N. J. A. Sloane, <a href="/A269725/a269725.txt">On Kimberling Sums and Para-Fibonacci Sequences</a>, Correspondence and Postings to Math-Fun Mailing List, Nov 1996 to Jan 1997
%F A269725 a(n) = A173027(n)-1. - _R. J. Mathar_, May 06 2017
%e A269725 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.
%p A269725 A269725 := proc(n)
%p A269725     local f,sl,r,c,wrks ;
%p A269725     f := [seq(n*combinat[fibonacci](i),i=2..30)] ;
%p A269725     for sl from 0 do
%p A269725         for r from 1 do
%p A269725             if A035513(r,1) = op(1+sl,f) then
%p A269725                 wrks := true;
%p A269725                 for c from 2 to 5 do
%p A269725                     if A035513(r,c) <> op(c+sl,f) then
%p A269725                         wrks := false;
%p A269725                     end if;
%p A269725                 end do:
%p A269725                 if wrks then
%p A269725                     print(n,f,r) ;
%p A269725                     return r-1 ;
%p A269725                 end if;
%p A269725             elif A035513(r,1) > op(1+sl,f) then
%p A269725                 break ;
%p A269725             end if;
%p A269725         end do:
%p A269725     end do:
%p A269725 end proc: # _R. J. Mathar_, May 06 2017
%t A269725 W[n_, k_] := Fibonacci[k+1] Floor[n*GoldenRatio] + (n-1) Fibonacci[k];
%t A269725 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[]]]]];
%t A269725 Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Sep 13 2022, after _R. J. Mathar_ *)
%Y A269725 Cf. A000045, A022413-A022423, A035513, A269726, A269729.
%K A269725 nonn,easy
%O A269725 1,2
%A A269725 _N. J. A. Sloane_, Mar 07 2016