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 A295141 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences. 5
 1, 2, 8, 22, 57, 142, 348, 847, 2052, 4962, 11988, 28951, 69904, 168774, 407468, 983727, 2374940, 5733626, 13842212, 33418071, 80678377, 194774849, 470228100 (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. LINKS Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. FORMULA a(n+1)/a(n) -> 1 + sqrt(2). EXAMPLE a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4 a(2) =2*a(1) + a(0) + b(0) = 8 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 = 1; a = 2; b = 3; b = 4; a[n_] := a[n] = 2 a[ n - 1] + a[n - 2] + b[n - 2]; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; Table[a[n], {n, 0, 18}]  (* A295141 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A295053, A295142, A295143, A295144. Sequence in context: A005803 A145654 A221880 * A074352 A301555 A261561 Adjacent sequences:  A295138 A295139 A295140 * A295142 A295143 A295144 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 19 2017 STATUS approved

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Last modified November 20 20:28 EST 2019. Contains 329347 sequences. (Running on oeis4.)