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A297831
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Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 1, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments.
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3
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1, 2, 8, 11, 14, 17, 22, 24, 29, 31, 36, 38, 43, 45, 48, 51, 56, 60, 62, 65, 68, 73, 77, 79, 82, 85, 90, 94, 96, 99, 102, 107, 111, 113, 118, 120, 125, 127, 130, 133, 138, 140, 143, 148, 152, 154, 159, 161, 166, 168, 171, 174, 179, 181, 184, 189, 193, 195
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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 A297830 for a guide to related sequences.
Conjecture: a(n) - (2 +sqrt(2))*n < 5/2 for n >= 1.
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LINKS
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Clark Kimberling, Table of n, a(n) for n = 0..10000
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EXAMPLE
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a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 8.
Complement: (b(n)) = (3,4,5,6,7,9,10,12,13,15,16,18,19,...)
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MATHEMATICA
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a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 2 n - 1;
j = 1; While[j < 100, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k
Table[a[n], {n, 0, k}] (* A297831 *)
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CROSSREFS
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Cf. A297826, A297830.
Sequence in context: A216538 A077820 A360649 * A045086 A064105 A129516
Adjacent sequences: A297828 A297829 A297830 * A297832 A297833 A297834
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KEYWORD
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nonn,easy
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AUTHOR
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Clark Kimberling, Feb 04 2018
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STATUS
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approved
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