cp's OEIS Frontend

A126714 Dual Wythoff array read along antidiagonals.

%I A126714 #18 Aug 19 2025 21:05:17
%S A126714 1,2,4,3,6,7,5,10,11,9,8,16,18,14,12,13,26,29,23,19,15,21,42,47,37,31,
%T A126714 24,17,34,68,76,60,50,39,27,20,55,110,123,97,81,63,44,32,22,89,178,
%U A126714 199,157,131,102,71,52,35,25,144,288,322,254,212,165,115,84,57,40,28
%N A126714 Dual Wythoff array read along antidiagonals.
%C A126714 The dual Wythoff array is the dispersion of the sequence w given by w(n)=2+floor(n*x), where x=(golden ratio), so that w=2+A000201(n). For a discussion of dispersions, see A191426. - _Clark Kimberling_, Jun 03 2011
%D A126714 Clark Kimberling, "Stolarsky Interspersions," Ars Combinatoria 39 (1995) 129-138. See page 135 for the dual Wythoff array and other dual arrays. - _Clark Kimberling_, Oct 29 2009
%H A126714 P. Hegarty, U. Larsson, <a href="https://www.emis.de/journals/INTEGERS/papers/g3/g3.Abstract.html">Permutations of the natural numbers with prescribed difference multisets</a>, Electr. J. Combin. Numb. Theory 6 (2006) #A03.
%e A126714 Array starts
%e A126714 1 2 3 5 8 13 21 34 55 89 144
%e A126714 4 6 10 16 26 42 68 110 178 288 466
%e A126714 7 11 18 29 47 76 123 199 322 521 843
%e A126714 9 14 23 37 60 97 157 254 411 665 1076
%e A126714 12 19 31 50 81 131 212 343 555 898 1453
%e A126714 15 24 39 63 102 165 267 432 699 1131 1830
%e A126714 17 27 44 71 115 186 301 487 788 1275 2063
%e A126714 20 32 52 84 136 220 356 576 932 1508 2440
%e A126714 22 35 57 92 149 241 390 631 1021 1652 2673
%e A126714 25 40 65 105 170 275 445 720 1165 1885 3050
%e A126714 28 45 73 118 191 309 500 809 1309 2118 3427
%p A126714 Tn1 := proc(T,nmax,row) local n,r,c,fnd; n := 1; while true do fnd := false; for r from 1 to row do for c from 1 to nmax do if T[r,c] = n then fnd := true; fi; od; if T[r,nmax] < n then RETURN(-1); fi; od; if fnd then n := n+1; else RETURN(n); fi; od; end; Tn2 := proc(T,nmax,row,ai1) local n,r,c,fnd; for r from 1 to row do for c from 1 to nmax do if T[r,c]+1 = ai1 then RETURN(T[r,c+1]+1); fi; od; od; RETURN(-1); end; T := proc(nmax) local a,col,row; a := array(1..nmax,1..nmax); for col from 1 to nmax do a[1,col] := combinat[fibonacci](col+1); od; for row from 2 to nmax do a[row,1] := Tn1(a,nmax,row-1); a[row,2] := Tn2(a,nmax,row-1,a[row,1]); for col from 3 to nmax do a[row,col] := a[row,col-2]+a[row,col-1]; od; od; RETURN(a); end; nmax := 12; a := T(nmax); for d from 1 to nmax do for row from 1 to d do printf("%d, ",a[row,d-row+1]); od; od;
%t A126714 (* program generates the dispersion array T of the complement of increasing sequence f[n] *)
%t A126714 r = 40; r1 = 12;  (* r=# rows of T, r1=# rows to show *)
%t A126714 c = 40; c1 = 12;   (* c=# cols of T, c1=# cols to show *)
%t A126714 x = GoldenRatio; f[n_] := Floor[n*x + 2]
%t A126714 (* f(n) is complement of column 1 *)
%t A126714 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A126714 rows = {NestList[f, 1, c]};
%t A126714 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A126714 t[i_, j_] := rows[[i, j]];  (* the array T *)
%t A126714 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1,10}]]
%t A126714 (* Dual Wythoff array, A126714 *)
%t A126714 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* array as a sequence *)
%t A126714 (* Program by _Peter J. C. Moses_, Jun 01 2011; added here by _Clark Kimberling_, Jun 03 2011 *)
%Y A126714 First three rows identical to A035506. First column is A007066. First row is A000045. 2nd row is essentially A006355. 3rd row is essentially A000032. 4th row essentially A000285. 5th row essentially A013655 or A001060. 6th row essentially A022086 or A097135. 7th row essentially A022120. 8th row essentially A022087. 9th row essentially A022130. 10th row essentially A022088. 11th row essentially A022095. 12th row essentially A022089 etc.
%Y A126714 Cf. A035513 (Wythoff array).
%K A126714 easy,nonn,tabl
%O A126714 1,2
%A A126714 _R. J. Mathar_, Feb 12 2007