A296220 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
1, 2, 10, 13, 16, 19, 22, 25, 29, 34, 38, 43, 47, 52, 56, 61, 65, 70, 74, 79, 82, 86, 91, 94, 97, 101, 106, 109, 113, 118, 121, 124, 128, 133, 136, 140, 145, 148, 151, 155, 160, 163, 167, 172, 175, 178, 182, 187, 190, 194, 199, 202, 205, 209, 214, 217, 221
Offset: 0
Examples
a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4; a(2) = a(0)*b(1) + a(1)*b(0) = 10. Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, ...).
Links
- Peter J. C. Moses, Table of n, a(n) for n = 0..9999
- Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
- Pietro Majer, Limit associated with complementary sequences Math Overflow.
Crossrefs
Cf. A296000.
Programs
-
Mathematica
mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &]; a[0] = 1; a[1] = 2; b[0] = 3; a[n_] := a[n] = a[0]*b[n - 1] + a[1]*b[n - 2]; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; u = Table[a[n], {n, 0, 500}]; (* A296220 *) Table[b[n], {n, 0, 20}]
Comments