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A295067
Solution of the complementary equation a(n) = 2*a(n-2) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
3
1, 3, 4, 11, 14, 29, 36, 67, 82, 146, 177, 307, 370, 631, 758, 1281, 1536, 2583, 3094, 5189, 6212, 10403, 12450, 20833, 24928, 41696, 49887, 83424, 99807, 166882, 199649, 333801, 399336, 667641, 798712, 1335323, 1597466, 2670689, 3194976, 5341423, 6389998
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.
The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 1.19..., 1.67...
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 1, a(1) = 3, a(2) = 3, b(0) = 2, b(1) = 5
a(2) = 2*a(0) + b(0) = 4
Complement: (b(n)) = (2, 5, 6, 7, 8, 9, 10, 12, 13, 15, ... )
MATHEMATICA
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1]=5;
a[n_] := a[n] = 2 a[n - 2] + b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}] (* A295067 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Sequence in context: A133621 A216558 A049619 * A263262 A249650 A114951
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 19 2017
STATUS
approved