A295754 Solution of the complementary equation a(n) = a(n-1) + a(n-3) + a(n-4) + b(n-4), where a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 4, b(0) = 5, b(1) = 6, b(2) = 7, b(3) = 8, and (a(n)) and (b(n)) are increasing complementary sequences.
1, 2, 3, 4, 12, 23, 37, 61, 105, 175, 284, 463, 757, 1231, 1994, 3231, 5237, 8481, 13726, 22215, 35955, 58186, 94152, 152348, 246516, 398882, 645411, 1044305, 1689734, 2734059, 4423808, 7157881, 11581709, 18739612, 30321339, 49060968, 79382329, 128443321
Offset: 0
Examples
a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 4, b(0) = 5, b(1) = 6, b(2) = 7, b(3) = 8, so that b(4) = 9 (least "new number") a(4) = a(3) + a(1) + a(0) + b(0) = 12 Complement: (b(n)) = (5, 6, 7, 8, 9, 10, 11, 13, 14, 15, ...)
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] = 2; a[2] = 3; a[3] = 4; b[0] = 5; b[1] = 6; b[2] = 7; b[3] = 8; a[n_] := a[n] = a[n - 1] + a[n - 3] + a[n - 4] + b[n - 4]; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; z = 36; Table[a[n], {n, 0, z}] (* A295754 *) Table[b[n], {n, 0, 20}] (*complement *)
Comments