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A130628 Related to the minimal number of periodic orbits of periods guaranteed by Sharkovskii's theorem. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

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

LINKS

Table of n, a(n) for n=1..49.

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.

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-Fran├žois Alcover, Nov 06 2018, adapted from Max Alekseyev's PARI script *)

PROG

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

MT(phin(5)) \\ sequence A_{n, 5} \\ Max Alekseyev

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.

Sequence in context: A082857 A208092 A081155 * A249431 A331176 A076563

Adjacent sequences:  A130625 A130626 A130627 * A130629 A130630 A130631

KEYWORD

nonn

AUTHOR

Jonathan Vos Post, Jun 18 2007

EXTENSIONS

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

STATUS

approved

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Last modified April 9 20:55 EDT 2020. Contains 333363 sequences. (Running on oeis4.)