A006209 Generalized Fibonacci numbers A_{n,4}. (Formerly M0027)

1, 1, 0, 1, 0, 2, 0, 3, 1, 6, 2, 9, 4, 18, 8, 30, 16, 56, 32, 101, 64, 191, 128, 351, 256, 668, 512, 1257, 1026, 2402, 2056, 4592, 4122, 8854, 8272, 17092, 16608, 33212, 33364, 64674, 67072, 126490, 134912, 248038, 271528, 487986, 546818, 962350

Offset

1,6

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..48.

Bau-Sen Du, The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem. Bull. Austral. Math. Soc. 31(1985), 89-103. Corrigendum: 32 (1985), 159.

Bau-Sen Du, A Simple Method Which Generates Infinitely Many Congruence Identities, Fib. Quart. 27 (1989), 116-124.

Bau-Sen Du, A Simple Method Which Generates Infinitely Many Congruence Identities, arXiv:0706.2421 [math.NT], 2007.

Bau-Sen Du, The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem, arXiv:0706.2297 [math.DS], 2007.

Mathematica

max = 100; 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 - 2 j > 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[4]] (* Jean-François Alcover, Nov 05 2018, adapted from Max Alekseyev's PARI script *)

Prog

(PARI) \\ implementation of MT() and phin() is given in A006207
MT(phin(4)) \\ sequence A_{n, 4} \\ Max Alekseyev, Feb 23 2012

Crossrefs

Cf. A006206 (A_{n,1}), A006207 (A_{n,2}), A006208 (A_{n,3}), A130628 (A_{n,5}), A208092 (A_{n,6}), A006210 (D_{n,2}), A006211 (D_{n,3}), A094392.

Sequence in context: A166117 A078051 A130627 * A005307 A340385 A143351

Adjacent sequences: A006206 A006207 A006208 * A006210 A006211 A006212

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved