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A130628 Related to the minimal number of periodic orbits of periods guaranteed by Sharkovskii's theorem. 6

%I

%S 1,1,0,1,0,2,0,3,0,6,1,9,2,18,4,30,8,56,16,99,32,186,64,337,128,635,

%T 256,1177,512,2220,1024,4176,2048,7930,4098,15044,8200,28738,16410,

%U 54937,32848,105474,65760,202845,131668,391316,263680,756223,528128

%N Related to the minimal number of periodic orbits of periods guaranteed by Sharkovskii's theorem.

%C Bau-Sen Du's [1985/2007] Table 1, p. 6, has this sequence as the 6th column, denoted A_{n,5}.

%H Bau-Sen Du, <a href="https://arxiv.org/abs/0706.2297">The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem</a>, arXiv:0706.2297 [math.DS], 2007; Bull. Austral. Math. Soc. 31(1985), 89-103. Corrigendum: 32 (1985), 159.

%t max = 50; Clear[b1, b2];

%t For[n = 1, n <= max, n++,

%t For[j = 1, j <= n, j++, b1[1][j, n] = 0; b1[2][j, n] = 1; b2[1][j, n] = b2[2][j, n] = 0]; b2[1][n, n] = b2[2][n, n] = 1];

%t For[k = 3, k <= max, k++,

%t For[n = 1, n <= max, n++,

%t For[j = 1, j <= n-1, j++, b1[k][j, n] = b1[k-2][1, n] + b1[k-2][j+1, n]; b2[k][j, n] = b2[k-2][1, n] + b2[k-2][j+1, n]]; b1[k][n, n] = b1[k-2][1, n] + b1[k-1][n, n]; b2[k][n, n] = b2[k-2][1, n] + b2[k-1][n, n]]];

%t phin[n_] := Table[b2[m][n, n] + 2 Sum[If[m + 2 - 2j > 0, b1[m + 2 - 2j][j, n], 0], {j, 1, n}], {m, 1, max}];

%t MT[s_List] := Table[DivisorSum[n, MoebiusMu[#] s[[n/#]] &]/n, {n, 1, Length[s]}];

%t MT[phin[5]] (* _Jean-Fran├žois Alcover_, Nov 06 2018, adapted from _Max Alekseyev_'s PARI script *)

%o (PARI) \\ implementation of MT() and phin() is given in A006207

%o MT(phin(5)) \\ sequence A_{n,5} \\ _Max Alekseyev_

%Y Cf. A006206 (A_{n,1}), A006207 (A_{n,2}), A006208 (A_{n,3}), A006209 (A_{n,4}), A208092 (A_{n,6}), A006210 (D_{n,2}), A006211 (D_{n,3}), A094392.

%K nonn

%O 1,6

%A _Jonathan Vos Post_, Jun 18 2007

%E Terms a(32) onward from _Max Alekseyev_, Feb 23 2012

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Last modified June 5 23:10 EDT 2020. Contains 334858 sequences. (Running on oeis4.)