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).

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, 19, 27, 14, 22, 30, 17, 25, 33, 54, 75, 41, 62, 83, 49, 70, 36, 57, 78, 44, 65, 86, 52, 73, 39, 60, 81, 47, 68, 34, 55, 76, 42, 63, 84, 50, 71, 37, 58, 79, 45, 66, 87, 53, 74, 40

Offset

0,3

Comments

Conjecture: sequence is its own inverse. - R. J. Mathar, May 08 2019

References

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

Links

Table of n, a(n) for n=0..70.

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

Example

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.

Maple

A035513 := proc(r::integer, c::integer)
    option remember;
    if c = 1 then
        A003622(r) ;
    elif c > 1 then
        A022342(1+procname(r, c-1)) ;
    elif c < 1 then
        procname(r, c+2)-procname(r, c+1) ;
    end if;
end proc:
# search in A035513 for row with consecutive w1, w2
A035513inv := proc(w1::integer, w2::integer)
    local r, c, W1, W2 ;
    for r from 1 do
        if A035513(r, 1) > w2 then
            return -1 ;
        end if;
        for c from 1 do
            W1 := A035513(r, c) ;
            W2 := A035513(r, c+1) ;
            if W1=w1 and W2=w2 then
                return r-1 ;
            elif W2 > w2 then
                break;
            end if;
        end do:
    end do:
end proc:
A269729 := proc(n)
    option remember;
    local c, W1, W2, r, n35513;
    n35513 := n+1 ;
    for c from 1 by -1 do
        W1 := A035513(n35513, c) ;
        W2 := A035513(n35513, c-1) ;
        if W1 < 0 and abs(W2) > abs(W1) then
            r :=  A035513inv(abs(W1), abs(W2)) ;
            if r >= 0 then
                return r;
            end if;
        end if;
    end do:
end proc:
seq(A269729(n), n=0..120) ; # R. J. Mathar, May 08 2019

Mathematica

W[n_, k_] := W[n, k] = Fibonacci[k+1] Floor[n*GoldenRatio] + (n-1)* Fibonacci[k];
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[]]]]]];
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]]]]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 120}] (* Jean-François Alcover, Aug 09 2023, after R. J. Mathar *)

Crossrefs

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

See A269733 for first differences.

Sequence in context: A078696 A256772 A274976 * A098115 A182833 A182832

Adjacent sequences: A269726 A269727 A269728 * A269730 A269731 A269732

Keyword

nonn,easy

Author

N. J. A. Sloane, Mar 08 2016

Extensions

Terms from a(18) on by R. J. Mathar, May 08 2019

Status

approved