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

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, 256, 1177, 512, 2220, 1024, 4176, 2048, 7930, 4098, 15044, 8200, 28738, 16410, 54937, 32848, 105474, 65760, 202845, 131668, 391316, 263680, 756223, 528128

Offset: 1

Author

Jonathan Vos Post, Jun 18 2007

Keywords

• nonn

Comments

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

Links

• Bau-Sen Du, The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem, arXiv:0706.2297 [math.DS], 2007; Bull. Austral. Math. Soc. 31(1985), 89-103. Corrigendum: 32 (1985), 159.

Crossrefs

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.

Programs

Mathematica

max = 50; Clear[b1, b2];
For[n = 1, n <= max, n++,
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];
For[k = 3, k <= max, k++,
For[n = 1, n <= max, n++,
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]]];
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}];
MT[s_List] := Table[DivisorSum[n, MoebiusMu[#] s[[n/#]] &]/n, {n, 1, Length[s]}];
MT[phin[5]] (* Jean-François Alcover, Nov 06 2018, adapted from Max Alekseyev's PARI script *)

PARI

\\ implementation of MT() and phin() is given in A006207
MT(phin(5)) \\ sequence A_{n,5} \\ Max Alekseyev

Extensions

Terms a(32) onward from Max Alekseyev, Feb 23 2012