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A295367
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Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*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.
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4
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1, 2, 15, 37, 82, 161, 299, 532, 921, 1563, 2616, 4335, 7133, 11692, 19097, 31095, 50534, 82009, 132963, 215434, 348903, 564889, 914392, 1479931, 2395025, 3875712, 6271549, 10148131, 16420610, 26569733, 42991399, 69562254, 112554843
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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 A295357 for a guide to related sequences.
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LINKS
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FORMULA
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a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).
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EXAMPLE
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a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that
b(2) = 5 (least "new number")
a(2) = a(1) + a(0) + b(1)*b(0) = 15
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 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] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1]*b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
z = 32; u = Table[a[n], {n, 0, z}] (* A295367 *)
v = Table[b[n], {n, 0, 10}] (* complement *)
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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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