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A296223
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Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.
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2
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1, 3, 9, 34, 124, 453, 1654, 6040, 22055, 80532, 294058, 1073735, 3920679, 14316124, 52274468, 190877084, 696976221, 2544966858, 9292793804, 33932079081, 123900951107, 452416889887, 1651973131976, 6032080786047, 22025781112962, 80425818360771
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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 A295862 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
a(2) = a(0)*b(1) + a(1)*b(0) - 1 = 9
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, ...)
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MATHEMATICA
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mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 1}] - 1;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
u = Table[a[n], {n, 0, 200}] (* A296223 *)
Table[b[n], {n, 0, 20}]
N[Table[a[n]/a[n - 1], {n, 1, 200, 10}], 200];
RealDigits[Last[t], 10][[1]] (* A296224 *)
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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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