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A296219
Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.
1
1, 3, 10, 17, 21, 25, 29, 33, 38, 45, 49, 53, 57, 61, 66, 73, 77, 82, 89, 93, 98, 105, 109, 114, 121, 125, 130, 137, 141, 145, 150, 157, 161, 165, 169, 173, 178, 185, 189, 194, 201, 205, 210, 217, 221, 226, 233, 237, 242, 249, 253, 257, 262, 269, 273, 277
OFFSET
0,2
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
a(2) = a(0)*b(1) + a(1)*b(0) = 10
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 18, ...)
MATHEMATICA
mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 3; b[0] = 2;
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}]; (* A296219 *)
Table[b[n], {n, 0, 20}]
CROSSREFS
Cf. A296000.
Sequence in context: A356090 A190763 A043405 * A063293 A270997 A356058
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 08 2017
EXTENSIONS
Conjectured g.f. removed by Alois P. Heinz, Jun 25 2018
STATUS
approved