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A295962
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Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
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2
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3, 4, 11, 20, 37, 64, 109, 182, 302, 496, 811, 1321, 2147, 3484, 5648, 9150, 14818, 23989, 38829, 62841, 101694, 164560, 266280, 430867, 697175, 1128071, 1825276, 2953378, 4778686, 7732097, 12510817, 20242949, 32753803, 52996790, 85750632, 138747462
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OFFSET
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0,1
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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. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).
See A295862 for a guide to related sequences.
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LINKS
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Clark Kimberling, Table of n, a(n) for n = 0..2000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
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EXAMPLE
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a(0) = 3, a(1) = 4, b(0) = 1, b(1) = 2, b(2) = 5
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) - 1 = 11
Complement: (b(n)) = (1, 2, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21, ...)
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MATHEMATICA
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a[0] = 3; a[1] = 4; b[0] = 1; b[1] = 2; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - 1;
j = 1; While[j < 6, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, k}]; (* A295962 *)
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CROSSREFS
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Cf. A001622, A000045, A295862.
Sequence in context: A025079 A222770 A036652 * A097072 A049977 A000677
Adjacent sequences: A295959 A295960 A295961 * A295963 A295964 A295965
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KEYWORD
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nonn,easy
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AUTHOR
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Clark Kimberling, Dec 08 2017
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STATUS
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approved
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