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A294559
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Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 2*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
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2
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1, 2, 13, 28, 57, 104, 183, 312, 523, 866, 1423, 2327, 3793, 6166, 10008, 16226, 26289, 42573, 68923, 111560, 180550, 292180, 472803, 765059, 1237941, 2003083, 3241112, 5244286, 8485492, 13729875, 22215467, 35945445, 58161018, 94106572, 152267702, 246374389
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OFFSET
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0,2
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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) + 2*b(0) = 13
Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, ...)
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MATHEMATICA
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mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + 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, 40}] (* A294559 *)
Table[b[n], {n, 0, 10}]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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