A006207 Generalized Fibonacci numbers A_{n,2}.

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

Author

N. J. A. Sloane

Keywords

• nonn

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

• 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}), 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.

Programs

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 *)

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

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