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%I M0286 #33 Nov 18 2018 07:39:07
%S 1,1,0,1,1,2,2,3,4,6,8,11,16,23,32,46,66,94,136,195,282,408,592,856,
%T 1248,1814,2646,3858,5644,8246,12088,17706,25992,38155,56102,82490,
%U 121474,178902,263776,389033,574304,848069,1253344,1852926,2741164
%N Generalized Fibonacci numbers A_{n,2}.
%C Bau-Sen Du (1985)'s Table 1, p. 6, has this sequence as the third column. - _Jonathan Vos Post_, Jun 18 2007
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Bau-Sen Du, <a href="http://arXiv.org/abs/0706.2297">The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem</a>. Bull. Austral. Math. Soc. 31(1985), 89-103. Corrigendum: 32 (1985), 159.
%H Bau-Sen Du, <a href="http://arXiv.org/abs/0706.2421">A Simple Method Which Generates Infinitely Many Congruence Identities</a>, Fib. Quart., 27 (1989), 116-124.
%t 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]]];
%t 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}];
%t MT[s_List] := Table[ DivisorSum[n, MoebiusMu[#]*s[[n/#]]&]/n, {n, 1, Length[s]}];
%t MT[phin[2]] (* _Jean-François Alcover_, Dec 07 2015, adapted from _Max Alekseyev_'s PARI script *)
%o (PARI)
%o 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)
%o { 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
%o { MT(s) = vector(#s,n,sumdiv(n,d,moebius(d)*s[n/d])/n) } \\ Moebius transform
%o MT( phin(2) ) \\ sequence A_{n,2}
%o \\ _Max Alekseyev_, Feb 23 2012
%Y 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.
%K nonn
%O 1,6
%A _N. J. A. Sloane_
%E arxiv URL replaced with non-cached version by _R. J. Mathar_, Oct 30 2009
%E Terms a(32) onward from _Max Alekseyev_, Feb 23 2012
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