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A295069
Solution of the complementary equation a(n) = 2*a(n-2) - b(n-2) + n, where a(0) = 3, a(1) = 4, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.
3
3, 4, 7, 9, 13, 17, 24, 31, 45, 59, 86, 114, 168, 223, 331, 441, 657, 877, 1309, 1748, 2612, 3490, 5218, 6974, 10430, 13941, 20853, 27875, 41699, 55743, 83391, 111479, 166775, 222951, 333543, 445895, 667079, 891783, 1334150, 1783558, 2668292, 3567108
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.
The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 1.33..., 1.49...
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 3, a(1) = 4, b(0) = 1
a(2) = 2*a(0) - b(0) + 2 = 7
Complement: (b(n)) = (1, 2, 5, 6, 8, 10, 11, 12, 14, 15, 16, 18, ... )
MATHEMATICA
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 3; a[1] = 4; b[0] = 1;
a[n_] := a[n] = 2 a[n - 2] + b[n - 1] + n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}] (* A295069 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Sequence in context: A061568 A146994 A330146 * A349360 A103054 A140208
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 19 2017
STATUS
approved