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 A294542 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 2
 1, 2, 8, 16, 31, 55, 96, 162, 270, 445, 729, 1189, 1934, 3141, 5094, 8255, 13370, 21647, 35040, 56711, 91776, 148513, 240316, 388857, 629202, 1018089, 1647322, 2665444, 4312800, 6978279, 11291115, 18269431, 29560584, 47830054, 77390678, 125220773 (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 A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622). LINKS Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. EXAMPLE a(0) = 1, a(1) = 2, b(0) = 3, so that b(1) = 4 (least "new number"); a(2) = a(1) + a(0) + b(1) + 1 = 8. Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, ...). MATHEMATICA mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; a = 1; a = 3; b = 2; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + 1; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; Table[a[n], {n, 0, 40}]  (* A294542 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A001622, A294532. Sequence in context: A187216 A210729 A294534 * A294553 A295949 A077666 Adjacent sequences:  A294539 A294540 A294541 * A294543 A294544 A294545 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 04 2017 STATUS approved

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Last modified May 15 06:18 EDT 2021. Contains 343909 sequences. (Running on oeis4.)