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 A294565 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2) - 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 2
 1, 2, 6, 12, 25, 44, 77, 130, 217, 360, 590, 964, 1569, 2549, 4135, 6702, 10856, 17578, 28455, 46055, 74533, 120614, 195173, 315814, 511015, 826858, 1337903, 2164792, 3502727, 5667552, 9170313, 14837900, 24008249, 38846186, 62854473, 101700698, 164555211 (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) + 2*b(1) - b(0) - 2 = 6 Complement: (b(n)) = (3, 4, 5, 7, 8, 10, 11, 13, 14, 15, 16, ...) 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] + 2 b[n - 1] - b[n - 2] - 2; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; Table[a[n], {n, 0, 40}]  (* A294565 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A001622, A294532. Sequence in context: A034882 A175943 A228816 * A210068 A210633 A304710 Adjacent sequences:  A294562 A294563 A294564 * A294566 A294567 A294568 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 15 2017 STATUS approved

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Last modified January 26 05:10 EST 2020. Contains 331273 sequences. (Running on oeis4.)