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 A294557 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n - 1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 4
 1, 2, 11, 24, 49, 90, 159, 272, 457, 759, 1250, 2046, 3336, 5425, 8807, 14281, 23140, 37476, 60674, 98211, 158949, 257228, 416249, 673552, 1089879, 1763512, 2853475, 4617074, 7470639, 12087806, 19558541, 31646446, 51205089, 82851640, 134056837, 216908588 (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) + b(0) + 1 = 11 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 12, 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] + b[n - 1] + b[n - 2] + 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, 40}] (* A294557 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A001622, A294532. Sequence in context: A297545 A256905 A294547 * A009189 A012213 A012251 Adjacent sequences: A294554 A294555 A294556 * A294558 A294559 A294560 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 15 2017 EXTENSIONS Definition corrected by Georg Fischer, Sep 27 2020 STATUS approved

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Last modified January 29 23:01 EST 2023. Contains 359939 sequences. (Running on oeis4.)