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A293765
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Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + 2, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.
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41
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1, 3, 10, 20, 38, 67, 115, 193, 321, 528, 864, 1408, 2289, 3715, 6023, 9758, 15802, 25583, 41409, 67017, 108452, 175496, 283976, 459501, 743507, 1203039, 1946578, 3149650, 5096262, 8245947, 13342245, 21588229, 34930512, 56518780, 91449333, 147968155
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OFFSET
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0,2
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COMMENTS
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The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The initial values of each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4:
Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio. See A293358 for a guide to related sequences.
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LINKS
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EXAMPLE
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a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2) = a(1) + a(0) + b(1) + 2 = 10;
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, ...)
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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; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + 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}] (* A293765 *)
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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