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A006209 Generalized Fibonacci numbers A_{n,4}.

%I A006209 M0027 #30 Jan 05 2025 19:51:33
%S A006209 1,1,0,1,0,2,0,3,1,6,2,9,4,18,8,30,16,56,32,101,64,191,128,351,256,
%T A006209 668,512,1257,1026,2402,2056,4592,4122,8854,8272,17092,16608,33212,
%U A006209 33364,64674,67072,126490,134912,248038,271528,487986,546818,962350
%N A006209 Generalized Fibonacci numbers A_{n,4}.
%D A006209 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006209 Bau-Sen Du, <a href="https://doi.org/10.1017/S0004972700002306">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 A006209 Bau-Sen Du, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/27-2/du.pdf">A Simple Method Which Generates Infinitely Many Congruence Identities</a>, Fib. Quart. 27 (1989), 116-124.
%H A006209 Bau-Sen Du, <a href="https://arxiv.org/abs/0706.2421">A Simple Method Which Generates Infinitely Many Congruence Identities</a>, arXiv:0706.2421 [math.NT], 2007.
%H A006209 Bau-Sen Du, <a href="https://arxiv.org/abs/0706.2297">The Minimal Number of Periodic Orbits of Periods Guaranteed in Sharkovskii's Theorem</a>, arXiv:0706.2297 [math.DS], 2007.
%t A006209 max = 100; Clear[b1, b2];
%t A006209 For[n = 1, n <= max, n++,
%t A006209 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];
%t A006209 b2[1][n, n] = b2[2][n, n] = 1];
%t A006209 For[k = 3, k <= max, k++,
%t A006209 For[n = 1, n <= max, n++,
%t A006209 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]];
%t A006209 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 A006209 ]];
%t A006209 phin[n_] := Table[b2[m][n, n] + 2 Sum[If[m + 2 - 2 j > 0, b1[m + 2 - 2j][j, n], 0], {j, 1, n}], {m, 1, max}];
%t A006209 MT[s_List] := Table[DivisorSum[n, MoebiusMu[#] s[[n/#]]&]/n, {n, 1, Length[s]}];
%t A006209 MT[phin[4]] (* _Jean-François Alcover_, Nov 05 2018, adapted from _Max Alekseyev_'s PARI script *)
%o A006209 (PARI) \\ implementation of MT() and phin() is given in A006207
%o A006209 MT(phin(4)) \\ sequence A_{n,4} \\ _Max Alekseyev_, Feb 23 2012
%Y A006209 Cf. A006206 (A_{n,1}), A006207 (A_{n,2}), A006208 (A_{n,3}), A130628 (A_{n,5}), A208092 (A_{n,6}), A006210 (D_{n,2}), A006211 (D_{n,3}), A094392.
%K A006209 nonn
%O A006209 1,6
%A A006209 _N. J. A. Sloane_
%E A006209 Terms a(32) onward from _Max Alekseyev_, Feb 23 2012