A269729 - OEIS

%I #28 Aug 12 2023 00:56:38

%S 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 19,27,14,22,30,17,25,33,54,75,41,62,83,49,70,36,57,78,44,65,86,52,73,

%U 39,60,81,47,68,34,55,76,42,63,84,50,71,37,58,79,45,66,87,53,74,40

%N 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 Conjecture: sequence is its own inverse. - _R. J. Mathar_, May 08 2019

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

%H 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 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 A035513 := proc(r::integer, c::integer)

%p     option remember;

%p     if c = 1 then

%p         A003622(r) ;

%p     elif c > 1 then

%p         A022342(1+procname(r, c-1)) ;

%p     elif c < 1 then

%p         procname(r,c+2)-procname(r,c+1) ;

%p     end if;

%p end proc:

%p # search in A035513 for row with consecutive w1,w2

%p A035513inv := proc(w1::integer,w2::integer)

%p     local r,c,W1,W2 ;

%p     for r from 1 do

%p         if A035513(r,1) > w2 then

%p             return -1 ;

%p         end if;

%p         for c from 1 do

%p             W1 := A035513(r,c) ;

%p             W2 := A035513(r,c+1) ;

%p             if W1=w1 and W2=w2 then

%p                 return r-1 ;

%p             elif W2 > w2 then

%p                 break;

%p             end if;

%p         end do:

%p     end do:

%p end proc:

%p A269729 := proc(n)

%p     option remember;

%p     local c,W1,W2,r,n35513;

%p     n35513 := n+1 ;

%p     for c from 1 by -1 do

%p         W1 := A035513(n35513,c) ;

%p         W2 := A035513(n35513,c-1) ;

%p         if W1 < 0 and abs(W2) > abs(W1) then

%p             r :=  A035513inv(abs(W1),abs(W2)) ;

%p             if r >= 0 then

%p                 return r;

%p             end if;

%p         end if;

%p     end do:

%p end proc:

%p seq(A269729(n),n=0..120) ; # _R. J. Mathar_, May 08 2019

%t W[n_, k_] := W[n, k] = Fibonacci[k+1] Floor[n*GoldenRatio] + (n-1)* Fibonacci[k];

%t 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 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 Table[Print[n, " ", a[n]]; a[n], {n, 0, 120}] (* _Jean-François Alcover_, Aug 09 2023, after _R. J. Mathar_ *)

%Y Cf. A000045, A022413-A022423, A035513, A269725, A269726, A269729.

%Y See A269733 for first differences.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Mar 08 2016

%E Terms from a(18) on by _R. J. Mathar_, May 08 2019