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%I #58 Mar 22 2023 13:11:05
%S 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 23,17,34,68,76,63,50,37,28,20,55,110,123,102,81,60,45,32,22,89,178,
%U 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 Stolarsky array read by antidiagonals.
%C Inverse of sequence A064357 considered as a permutation of the positive integers. - _Howard A. Landman_, Sep 25 2001
%C 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 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 See A098861 for the row in which is a given number. - _M. F. Hasler_, Nov 05 2014
%C Named after the American mathematician Kenneth Barry Stolarsky. - _Amiram Eldar_, Jun 11 2021
%D C. Kimberling, "Stolarsky interspersions," Ars Combinatoria 39 (1995) 129-138.
%H Alois P. Heinz, <a href="/A035506/b035506.txt">Antidiagonals n = 0..140, flattened</a>
%H Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/intersp.html">Interspersions</a>.
%H Clark Kimberling, <a href="http://dx.doi.org/10.1090/S0002-9939-1993-1111434-0">Interspersions and dispersions</a>, Proceedings of the American Mathematical Society, Vol. 117 (1993), pp. 313-321.
%H David R. Morrison, <a href="https://web.math.ucsb.edu/~drm/papers/stolarsky.pdf">A Stolarsky array of Wythoff pairs</a>, A collection of manuscripts related to the Fibonacci sequence, Santa Clara, CA: Fibonacci Association, 1980, pp. 134-136.
%H N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StolarskyArray.html">Stolarsky arrays</a>.
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F T(1,k) = 2*T(0,k+1); T(3,k) = 3*T(0,k+2). - _M. F. Hasler_, Nov 05 2014
%e Top left corner of the array is:
%e 1 2 3 5 8 13 21 34 55
%e 4 6 10 16 26 42 68 110 178
%e 7 11 18 29 47 76 123 119 322
%e 9 15 24 39 63 102 165 267 432
%e 12 19 31 50 81 131 212 343 555
%e 14 23 37 60 97 157 254 411 665
%p 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 (* program generates the dispersion array T of the complement of increasing sequence f[n] *)
%t r = 40; r1 = 12; (* r=# rows of T, r1=# rows to show *)
%t c = 40; c1 = 12; (* c=# cols of T, c1=# cols to show *)
%t x = GoldenRatio; f[n_] := Floor[n*x + 1/2]
%t (* f(n) is complement of column 1 *)
%t mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t rows = {NestList[f, 1, c]};
%t Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t t[i_, j_] := rows[[i, j]];
%t TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t (* t=Stolarsky array, A035506 *)
%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
%t (* Stolarsky array as a sequence *)
%t (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%t (* Second program: *)
%t 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 Table[A[n, d-n], {d, 0, 11}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Mar 22 2023, after _Alois P. Heinz_ *)
%o (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 Cf. A035513 (Wythoff array), A035507 (inverse Stolarsky array), A191426.
%Y Main diagonal gives A035489.
%K nonn,tabl,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E More terms from Larry Reeves (larryr(AT)acm.org), Sep 27 2000
%E Extended (terms, Mathematica, example) by _Clark Kimberling_, Jun 03 2011
%E Example corrected by _M. F. Hasler_, Nov 05 2014
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