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
Examples
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.
References
- J. H. Conway, Postings to Math Fun Mailing List, Nov 25 1996 and Dec 02 1996.
Links
- 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
Crossrefs
Programs
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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
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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 *)
Extensions
Terms from a(18) on by R. J. Mathar, May 08 2019
Comments