A006208 Generalized Fibonacci numbers A_{n,3}.

1, 1, 0, 1, 0, 2, 1, 3, 2, 6, 4, 9, 8, 18, 16, 32, 32, 61, 64, 115, 128, 224, 258, 431, 520, 850, 1050, 1673, 2128, 3328, 4320, 6649, 8788, 13366, 17920, 26957, 36610, 54634, 74932, 111057, 153656, 226514, 315616, 463243, 649334, 949823, 1337984, 1951760

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Bau-Sen Du's [1985/2007] Table 1, p. 6, has this sequence as the 4th column. - Jonathan Vos Post, Jun 18 2007

References

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

Links

• 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.

Crossrefs

Cf. A006206 (A_{n,1}), A006207 (A_{n,2}), A006209 (A_{n,4}), A130628 (A_{n,5}), A208092 (A_{n,6}), A006210 (D_{n,2}), A006211 (D_{n,3}), A094392.

Programs

Mathematica

max = 50;
Do[Do[b1[1][j, n] = 0; b1[2][j, n] = 1; b2[1][j, n] = b2[2][j, n] = 0, {j, n}]; b2[1][n, n] = b2[2][n, n] = 1, {n, max}];
Do[Do[Do[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], {j, n-1}]; 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], {n, max}], {k, 3, max}];
phin[n_] := Table[b2[m][n, n] + 2 Sum[If[m+2-2j > 0, b1[m+2-2j][j, n], 0], {j, n}], {m, max}];
MT[s_] := Table[DivisorSum[n, MoebiusMu[#] s[[n/#]]&]/n, {n, Length[s]}];
MT[phin[3]] (* Jean-François Alcover, Oct 01 2018, after Max Alekseyev in A006207 *)

PARI

\\ implementation of MT() and phin() is given in A006207
MT(phin(3)) \\ sequence A_{n,3} \\ Max Alekseyev, Feb 23 2012

Extensions

arxiv URL replaced with non-cached version, and duplicate of a reference removed, by R. J. Mathar, Oct 30 2009

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