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A295058
Solution of the complementary equation a(n) = 2*a(n-1) - b(n-1), where a(0) = 3, a(1) = 5, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.
2
3, 5, 8, 12, 18, 29, 49, 88, 165, 317, 620, 1225, 2434, 4851, 9683, 19346, 38671, 77320, 154617, 309210, 618395, 1236764, 2473501, 4946974, 9893918, 19787805, 39575578, 79151123, 158302212, 316604389, 633208742
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) = 3, a(1) = 5, b(0) = 1
b(1) = 2 (least "new number")
a(2) = 2*a(1) - b(1) = 8
Complement: (b(n)) = (1, 2, 4, 6, 7, 9, 10, 11, 13, 14, 15, ...)
MATHEMATICA
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 3; 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}] (* A295058 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Cf. A295053.
Sequence in context: A020745 A232896 A227635 * A004398 A286311 A256057
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 18 2017
STATUS
approved