A297011 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) - b(n), where a(0) = 3, a(1) = 5, b(0) = 1, b(1) = 2, b(2) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
3, 5, 9, 17, 36, 81, 188, 446, 1068, 2569, 6192, 14938, 36052, 87024, 210081, 507166, 1224392, 2955928, 7136225, 17228354, 41592908, 100414144, 242421169, 585256454, 1412934048, 3411124520, 8235183057, 19881490602, 47998164228, 115877819024, 279753802241
Offset: 0
Examples
a(0) = 3, a(1) = 5, b(0) = 1, b(1) = 2, b(2) = 4 a(2) = 2*a(1) + a(0) - b(2) = 9 Complement: (b(n)) = (1, 2, 4, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 18, 19, ...)
Links
- Clark Kimberling, Table of n, a(n) for n = 0..1000
- Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
Programs
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Mathematica
a[0] = 3; a[1] = 5; b[0] = 1; b[1] = 2; b[2] = 4; a[n_] := a[n] = 2 a[n - 1] + a[n - 2] - b[n]; j = 1; While[j < 9, k = a[j] - j - 1; While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; u = Table[a[n], {n, 0, k}]; (* A297011 *) Table[b[n], {n, 0, 25}] (* complement *) Take[u, 30]
Comments