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# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a035506 Showing 1-1 of 1 %I A035506 #58 Mar 22 2023 13:11:05 %S A035506 1,2,4,3,6,7,5,10,11,9,8,16,18,15,12,13,26,29,24,19,14,21,42,47,39,31, %T A035506 23,17,34,68,76,63,50,37,28,20,55,110,123,102,81,60,45,32,22,89,178, %U A035506 199,165,131,97,73,52,36,25,144,288,322,267,212,157,118,84,58,40,27,233,466,521,432,343,254,191,136,94,65,44,30 %N A035506 Stolarsky array read by antidiagonals. %C A035506 Inverse of sequence A064357 considered as a permutation of the positive integers. - _Howard A. Landman_, Sep 25 2001 %C A035506 The PARI/GP script gives a general solution for the Stolarsky array in square array form by row, column. Increase the default precision to compute large values in the array. - _Randall L Rathbun_, Jan 25 2002 %C A035506 The Stolarsky array is the dispersion of the sequence s given by s(n)=(integer nearest n*x), where x=(golden ratio). For a discussion of dispersions, see A191426. %C A035506 See A098861 for the row in which is a given number. - _M. F. Hasler_, Nov 05 2014 %C A035506 Named after the American mathematician Kenneth Barry Stolarsky. - _Amiram Eldar_, Jun 11 2021 %D A035506 C. Kimberling, "Stolarsky interspersions," Ars Combinatoria 39 (1995) 129-138. %H A035506 Alois P. Heinz, Antidiagonals n = 0..140, flattened %H A035506 Clark Kimberling, Interspersions. %H A035506 Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, Vol. 117 (1993), pp. 313-321. %H A035506 David R. Morrison, A Stolarsky array of Wythoff pairs, A collection of manuscripts related to the Fibonacci sequence, Santa Clara, CA: Fibonacci Association, 1980, pp. 134-136. %H A035506 N. J. A. Sloane, Classic Sequences. %H A035506 Eric Weisstein's World of Mathematics, Stolarsky arrays. %H A035506 Index entries for sequences that are permutations of the natural numbers %F A035506 T(1,k) = 2*T(0,k+1); T(3,k) = 3*T(0,k+2). - _M. F. Hasler_, Nov 05 2014 %e A035506 Top left corner of the array is: %e A035506 1 2 3 5 8 13 21 34 55 %e A035506 4 6 10 16 26 42 68 110 178 %e A035506 7 11 18 29 47 76 123 119 322 %e A035506 9 15 24 39 63 102 165 267 432 %e A035506 12 19 31 50 81 131 212 343 555 %e A035506 14 23 37 60 97 157 254 411 665 %p A035506 A:= proc(n, k) local t, a, b; t:= (1+sqrt(5))/2; a:= floor(n*(t+1)+1 +t/2); b:= round(a*t); (Matrix([[b, a]]). Matrix([[1, 1], [1, 0]])^k) [1, 2] end: seq(seq(A (n, d-n), n=0..d), d=0..10); # _Alois P. Heinz_, Aug 17 2008 %t A035506 (* program generates the dispersion array T of the complement of increasing sequence f[n] *) %t A035506 r = 40; r1 = 12; (* r=# rows of T, r1=# rows to show *) %t A035506 c = 40; c1 = 12; (* c=# cols of T, c1=# cols to show *) %t A035506 x = GoldenRatio; f[n_] := Floor[n*x + 1/2] %t A035506 (* f(n) is complement of column 1 *) %t A035506 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]] %t A035506 rows = {NestList[f, 1, c]}; %t A035506 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}]; %t A035506 t[i_, j_] := rows[[i, j]]; %t A035506 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] %t A035506 (* t=Stolarsky array, A035506 *) %t A035506 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] %t A035506 (* Stolarsky array as a sequence *) %t A035506 (* Program by _Peter J. C. Moses_, Jun 01 2011 *) %t A035506 (* Second program: *) %t A035506 A[n_, k_] := Module[{t, a, b}, t = (1+Sqrt[5])/2; a = Floor[n*(t+1)+1+t/2]; b = Round[a*t]; ({b, a}.MatrixPower[{{1, 1}, {1, 0}}, k])[[2]]]; %t A035506 Table[A[n, d-n], {d, 0, 11}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Mar 22 2023, after _Alois P. Heinz_ *) %o A035506 (PARI) {Stolarsky(r,c)= tau=(1+sqrt(5))/2; a=floor(r*(1+tau)-tau/2); b=round(a*tau); if(c==1,a, if(c==2,b, for(i=1,c-2,d=a+b; a=b; b=d; ); d))} \\ _Randall L Rathbun_, Jan 25 2002 %Y A035506 Cf. A035513 (Wythoff array), A035507 (inverse Stolarsky array), A191426. %Y A035506 Main diagonal gives A035489. %K A035506 nonn,tabl,easy,nice %O A035506 0,2 %A A035506 _N. J. A. Sloane_ %E A035506 More terms from Larry Reeves (larryr(AT)acm.org), Sep 27 2000 %E A035506 Extended (terms, Mathematica, example) by _Clark Kimberling_, Jun 03 2011 %E A035506 Example corrected by _M. F. Hasler_, Nov 05 2014 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE