A006207 Generalized Fibonacci numbers A_{n,2}. (Formerly M0286)

1, 1, 0, 1, 1, 2, 2, 3, 4, 6, 8, 11, 16, 23, 32, 46, 66, 94, 136, 195, 282, 408, 592, 856, 1248, 1814, 2646, 3858, 5644, 8246, 12088, 17706, 25992, 38155, 56102, 82490, 121474, 178902, 263776, 389033, 574304, 848069, 1253344, 1852926, 2741164

Offset

1,6

Comments

Bau-Sen Du (1985)'s Table 1, p. 6, has this sequence as the third 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

Table of n, a(n) for n=1..45.

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.

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 - 2*j][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[2]] (* Jean-François Alcover, Dec 07 2015, adapted from Max Alekseyev's PARI script *)

Prog

(PARI)
b1 = vector(100, k, matrix(100, 100)); b2 = vector(100, k, matrix(100, 100)); for(n=1, 100, for(j=1, n, 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, 100, for(n=1, 100, for(j=1, n-1, 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]; )); \\ Computing arrays b(k, 1, j, n) and b(k, 2, j, n)
{ phin(n) = vector(100, m, b2[m][n, n] + 2*sum(j=1, n, if(m+2-2*j>0, b1[m+2-2*j][j, n]))) } \\ sequence phi_n
{ MT(s) = vector(#s, n, sumdiv(n, d, moebius(d)*s[n/d])/n) } \\ Moebius transform
MT( phin(2) ) \\ sequence A_{n, 2}
\\ Max Alekseyev, Feb 23 2012

Crossrefs

Cf. A006206 (A_{n,1}), A006208 (A_{n,3}), A006209 (A_{n,4}), A130628 (A_{n,5}), A208092 (A_{n,6}), A006210 (D_{n,2}), A006211 (D_{n,3}), A094392.

Sequence in context: A103632 A214076 A067859 * A318403 A362048 A334626

Adjacent sequences: A006204 A006205 A006206 * A006208 A006209 A006210

Keyword

nonn

Author

N. J. A. Sloane

Extensions

arxiv URL replaced with non-cached version by R. J. Mathar, Oct 30 2009

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

Status

approved