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A295056
Solution of the complementary equation a(n) = 2*a(n-1) + b(n-1), where a(0) = 1, a(1) = 4, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.
2
1, 4, 11, 27, 60, 127, 262, 533, 1076, 2164, 4341, 8696, 17407, 34830, 69677, 139372, 278763, 557546, 1115113, 2230248, 4460519, 8921062, 17842149, 35684324, 71368676, 142737381, 285474792, 570949615
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 (least "new number")
a(2) = 2*a(1) + b(1) = 11
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, ...)
MATHEMATICA
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 4; b[0] = 2;
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}] (* A295056 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Cf. A295053.
Sequence in context: A192965 A305119 A035593 * A160399 A119706 A034345
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 18 2017
STATUS
approved