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A269729 a(n) = row number of extended Wythoff array (see A035513) which contains the sequence obtained by reading the n-th row backwards (and adjusting signs).

%I A269729 #28 Aug 12 2023 00:56:38
%S A269729 0,1,2,3,4,7,10,5,8,11,6,9,12,20,28,15,23,31,18,26,13,21,29,16,24,32,
%T A269729 19,27,14,22,30,17,25,33,54,75,41,62,83,49,70,36,57,78,44,65,86,52,73,
%U A269729 39,60,81,47,68,34,55,76,42,63,84,50,71,37,58,79,45,66,87,53,74,40
%N A269729 a(n) = row number of extended Wythoff array (see A035513) which contains the sequence obtained by reading the n-th row backwards (and adjusting signs).
%C A269729 Conjecture: sequence is its own inverse. - _R. J. Mathar_, May 08 2019
%D A269729 J. H. Conway, Postings to Math Fun Mailing List, Nov 25 1996 and Dec 02 1996.
%H A269729 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
%e A269729 Take n=5: reading row 5 of A035513 backwards gives ... 23, 14, 9, 5, 4, 1, 3, -2, 5, -7, 12, -19, ..., which after adjusting the signs is row 7, so a(5) = 7.
%p A269729 A035513 := proc(r::integer, c::integer)
%p A269729     option remember;
%p A269729     if c = 1 then
%p A269729         A003622(r) ;
%p A269729     elif c > 1 then
%p A269729         A022342(1+procname(r, c-1)) ;
%p A269729     elif c < 1 then
%p A269729         procname(r,c+2)-procname(r,c+1) ;
%p A269729     end if;
%p A269729 end proc:
%p A269729 # search in A035513 for row with consecutive w1,w2
%p A269729 A035513inv := proc(w1::integer,w2::integer)
%p A269729     local r,c,W1,W2 ;
%p A269729     for r from 1 do
%p A269729         if A035513(r,1) > w2 then
%p A269729             return -1 ;
%p A269729         end if;
%p A269729         for c from 1 do
%p A269729             W1 := A035513(r,c) ;
%p A269729             W2 := A035513(r,c+1) ;
%p A269729             if W1=w1 and W2=w2 then
%p A269729                 return r-1 ;
%p A269729             elif W2 > w2 then
%p A269729                 break;
%p A269729             end if;
%p A269729         end do:
%p A269729     end do:
%p A269729 end proc:
%p A269729 A269729 := proc(n)
%p A269729     option remember;
%p A269729     local c,W1,W2,r,n35513;
%p A269729     n35513 := n+1 ;
%p A269729     for c from 1 by -1 do
%p A269729         W1 := A035513(n35513,c) ;
%p A269729         W2 := A035513(n35513,c-1) ;
%p A269729         if W1 < 0 and abs(W2) > abs(W1) then
%p A269729             r :=  A035513inv(abs(W1),abs(W2)) ;
%p A269729             if r >= 0 then
%p A269729                 return r;
%p A269729             end if;
%p A269729         end if;
%p A269729     end do:
%p A269729 end proc:
%p A269729 seq(A269729(n),n=0..120) ; # _R. J. Mathar_, May 08 2019
%t A269729 W[n_, k_] := W[n, k] = Fibonacci[k+1] Floor[n*GoldenRatio] + (n-1)* Fibonacci[k];
%t A269729 Winv[w1_, w2_] := Winv[w1, w2] = Module[{r, c, W1, W2}, For[r = 1, True, r++, If[W[r, 1] > w2, Return[-1]]; For[c = 1, True, c++, W1 = W[r, c]; W2 = W[r, c+1]; If[W1 == w1 && W2 == w2, Return[r-1], If[W2 > w2, Break[]]]]]];
%t A269729 a[n_] := a[n] = Module[{c, W1, W2, r, nw}, nw = n+1; For[c = 1, True, c--, W1 = W[nw, c]; W2 = W[nw, c-1]; If[W1 < 0 && Abs[W2] > Abs[W1], r = Winv[Abs[W1], Abs[W2]]; If[r >= 0, Return[r]]]]];
%t A269729 Table[Print[n, " ", a[n]]; a[n], {n, 0, 120}] (* _Jean-François Alcover_, Aug 09 2023, after _R. J. Mathar_ *)
%Y A269729 Cf. A000045, A022413-A022423, A035513, A269725, A269726, A269729.
%Y A269729 See A269733 for first differences.
%K A269729 nonn,easy
%O A269729 0,3
%A A269729 _N. J. A. Sloane_, Mar 08 2016
%E A269729 Terms from a(18) on by _R. J. Mathar_, May 08 2019