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 A295138 Solution of the complementary equation a(n) = 3*a(n-2) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 2
 1, 2, 7, 11, 27, 41, 90, 133, 282, 412, 860, 1251, 2596, 3770, 7806, 11329, 23438, 34008, 70336, 102047, 211032, 306166, 633122, 918526, 1899395, 2755608, 5698216, 8266856, 17094681, 24800602, 51284078, 74401842 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 for a guide to related sequences. The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 1.45..., 2.06... LINKS Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. EXAMPLE a(0) = 1, a(1) = 2, b(0) = 3 a(2) =3*a(0) + b(1) = 7 Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, ... ) MATHEMATICA mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; a[0] = 1; a[1] = 2; b[0] = 3; a[n_] := a[n] = 3 a[n - 2] + b[n - 1]; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; Table[a[n], {n, 0, 18}]  (* A295138 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A295053. Sequence in context: A201630 A023862 A024479 * A023864 A024857 A024481 Adjacent sequences:  A295135 A295136 A295137 * A295139 A295140 A295141 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 19 2017 STATUS approved

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Last modified September 18 07:09 EDT 2021. Contains 347510 sequences. (Running on oeis4.)