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A006208 Generalized Fibonacci numbers A_{n,3}.

%I A006208 M0148 #31 Oct 01 2018 08:20:00
%S A006208 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,
%T A006208 850,1050,1673,2128,3328,4320,6649,8788,13366,17920,26957,36610,54634,
%U A006208 74932,111057,153656,226514,315616,463243,649334,949823,1337984,1951760
%N A006208 Generalized Fibonacci numbers A_{n,3}.
%C A006208 Bau-Sen Du's [1985/2007] Table 1, p. 6, has this sequence as the 4th column. - _Jonathan Vos Post_, Jun 18 2007
%D A006208 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006208 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 A006208 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 A006208 max = 50;
%t A006208 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}];
%t A006208 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}];
%t A006208 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}];
%t A006208 MT[s_] := Table[DivisorSum[n, MoebiusMu[#] s[[n/#]]&]/n, {n, Length[s]}];
%t A006208 MT[phin[3]] (* _Jean-François Alcover_, Oct 01 2018, after _Max Alekseyev_ in A006207 *)
%o A006208 (PARI) \\ implementation of MT() and phin() is given in A006207
%o A006208 MT(phin(3)) \\ sequence A_{n,3} \\ _Max Alekseyev_, Feb 23 2012
%Y A006208 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.
%K A006208 nonn
%O A006208 1,6
%A A006208 _N. J. A. Sloane_
%E A006208 arxiv URL replaced with non-cached version, and duplicate of a reference removed, by _R. J. Mathar_, Oct 30 2009
%E A006208 Terms a(32) onward from _Max Alekseyev_, Feb 23 2012