A295364 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n-2), where a(0) = 1, a(1) = 3, a[2] = 5, b(0) = 2, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
1, 3, 5, 32, 79, 167, 318, 575, 1003, 1710, 2869, 4761, 7840, 12841, 20953, 34100, 55395, 89875, 145690, 236027, 382223, 618802, 1001625, 1621077, 2623404, 4245237, 6869453, 11115560, 17985943, 29102526, 47089591
Offset: 0
Examples
a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, so that b(2) = 6 (least "new number") a(3) = a(2) + a(1) + b(2)*b(1) = 32 Complement: (b(n)) = (2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...)
Links
- Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
Programs
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Mathematica
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; a[0] = 1; a[1] = 3; a[2] = 5; b[0] = 2; b[1] = 4; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1]*b[n - 2]; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; z = 32; u = Table[a[n], {n, 0, z}] (* A295364 *) v = Table[b[n], {n, 0, 10}] (* complement *)
Formula
a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).
Comments