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A295057
Solution of the complementary equation a(n) = 2*a(n-1) + b(n-1), where a(0) = 2, a(1) = 5, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.
2
2, 5, 13, 30, 66, 139, 286, 581, 1172, 2355, 4722, 9458, 18931, 37878, 75773, 151564, 303147, 606314, 1212649, 2425320, 4850663, 9701350, 19402725, 38805476, 77610979, 155221986, 310444001, 620888033
OFFSET
0,1
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 for a guide to related sequences.
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 2, a(1) = 5, b(0) = 1
b(1) = 3 (least "new number")
a(2) = 2*a(1) + b(1) = 13
Complement: (b(n)) = (1, 3, 4, 6, 7, 8, 9, 10, 11, 12, 14, ...)
MATHEMATICA
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 2; a[1] = 5; b[0] = 1;
a[n_] := a[n] = 2 a[n - 1] + b[n - 1];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}] (* A295057 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Cf. A295053.
Sequence in context: A282153 A054127 A184052 * A309535 A018012 A359673
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 18 2017
STATUS
approved