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 A296225 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0) + n, where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences. 2
 1, 3, 12, 44, 161, 588, 2147, 7839, 28621, 104498, 381533, 1393015, 5086038, 18569636, 67799608, 247543185, 903805055, 3299883119, 12048205018, 43989207775, 160609019998, 586399678681, 2141004179974, 7817021504815, 28540731390577, 104205079621096 (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 A295862 for a guide to related sequences. LINKS Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. EXAMPLE a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4 a(2) = a(0)*b(1) + a(1)*b(0) + 2 = 12 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, ...) MATHEMATICA mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &]; a = 1; a = 3; b = 2; a[n_] := a[n] = n + Sum[a[k]*b[n - k - 1], {k, 0, 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, 200}]  (* A296225 *) Table[b[n], {n, 0, 20}] N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200]; RealDigits[Last[t], 10][] (* A296226 *) CROSSREFS Cf. A296000, A296226. Sequence in context: A167477 A190051 A220633 * A109437 A331473 A005656 Adjacent sequences:  A296222 A296223 A296224 * A296226 A296227 A296228 KEYWORD nonn,easy AUTHOR Clark Kimberling, Dec 10 2017 STATUS approved

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Last modified April 17 08:40 EDT 2021. Contains 343064 sequences. (Running on oeis4.)