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A295059
Solution of the complementary equation a(n) = 2*a(n-1) + b(n-2), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
2
1, 4, 10, 23, 51, 108, 223, 454, 917, 1845, 3702, 7417, 14848, 29711, 59438, 118893, 237804, 475627, 951274, 1902569, 3805160, 7610344, 15220713, 30441452, 60882931, 121765890, 243531809, 487063648
OFFSET
0,2
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) = 1, a(1) = 4, b(0) = 2, b(1) = 3
a(2) = 2*a(1) + b(0) = 10
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 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}] (* A295059 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Cf. A295053.
Sequence in context: A001980 A266376 A057750 * A118645 A200759 A159347
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 18 2017
STATUS
approved