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A295070
Solution of the complementary equation a(n) = a(n-2) + b(n-1) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
2
1, 2, 8, 11, 19, 24, 35, 43, 57, 68, 84, 97, 115, 130, 150, 168, 191, 211, 236, 259, 287, 312, 342, 369, 401, 430, 464, 495, 531, 565, 604, 640, 681, 719, 762, 802, 848, 891, 939, 984, 1034, 1081, 1133, 1182, 1236, 1287, 1343, 1396, 1454, 1510, 1571, 1629
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. Conjecture: a(n)/a(n-1) -> 1.
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 1, a(1) = 2, b(0) = 3
a(2) = a(0) + b(1) + b(0) = 8
Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15, ... )
MATHEMATICA
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 2 a[n - 2] + b[n - 1] + 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}] (* A295070 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Cf. A295053.
Sequence in context: A143189 A056550 A074263 * A009420 A189328 A090746
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 19 2017
STATUS
approved