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 A130168 a(n) = (b(n) + b(n+1))/3, where b(n) = A000366(n). 3
 1, 3, 15, 111, 1131, 15123, 256335, 5364471, 135751731, 4084163643, 144039790455, 5884504366431, 275643776229531, 14673941326078563, 880908054392169375, 59226468571935857991, 4432461082611507366531, 367227420727722013775883, 33514867695588319595233095 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 COMMENTS As remarked by Gessel, A000366 has a combinatorial interpretation via a certain 2n X n array; this sequence is for a similar array of size (2n-1) X (n-1). In effect, Dellac gives a combinatorial reason why the elements of A000366 are alternately -1 and +1 modulo 3. Dellac also shows that all the terms of this sequence are odd. LINKS Hippolyte Dellac, Note sur l'élimination, méthode de parallélogramme, Annales de la Faculté des Sciences de Marseille, XI (1901), 141-164. [Warning 76 Mb; go to p. 81 in the pdf file] FORMULA G.f.: 2*(1+x)/(3*x^3)*Q(0) - 2/(3*x) - 1/x^2 - 2/(3*x^3), where Q(k) = 1 - x*(k+1)^2/( x*(k+1)^2 - 2/(1 - x*(k+1)^2/( x*(k+1)^2 - 2/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2013 MATHEMATICA b[n_] := (-2^(-1))^(n-2)*Sum[Binomial[n, k]*(1-2^(n+k+1))* BernoulliB[n+k+1], {k, 0, n}]; a[n_] := (b[n] + b[n+1])/3; a /@ Range[2, 20] (* Jean-François Alcover, Apr 08 2021 *) CROSSREFS Cf. A000366, A130169. Sequence in context: A254789 A112936 A001063 * A267083 A089945 A135083 Adjacent sequences:  A130165 A130166 A130167 * A130169 A130170 A130171 KEYWORD nonn AUTHOR Don Knuth, Aug 02 2007 STATUS approved

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Last modified July 26 22:49 EDT 2021. Contains 346300 sequences. (Running on oeis4.)