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A294868
Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1) -2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
2
1, 2, 5, 12, 24, 42, 67, 100, 142, 195, 260, 338, 430, 537, 660, 800, 958, 1135, 1332, 1550, 1791, 2056, 2346, 2662, 3005, 3376, 3776, 4206, 4667, 5160, 5686, 6246, 6841, 7472, 8140, 8846, 9591, 10377, 11205, 12076, 12991, 13951, 14957, 16010, 17111, 18261
OFFSET
0,2
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294860 for a guide to related sequences.
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 1, a(1) = 2, b(0) = 3
b(1) = 4 (least "new number")
a(2) = 2*a(1) - a(0) + b(1) - 2 = 5
Complement: (b(n)) = (3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 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 - 1] - a[n - 2] + b[n - 1] - 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}] (* A294868 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Cf. A294860.
Sequence in context: A127787 A116733 A116721 * A192981 A116713 A261369
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 16 2017
STATUS
approved