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A294867
Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1) -1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
2
1, 2, 6, 14, 28, 49, 78, 116, 164, 223, 294, 379, 479, 595, 728, 879, 1049, 1239, 1450, 1683, 1939, 2219, 2524, 2855, 3214, 3602, 4020, 4469, 4950, 5464, 6012, 6595, 7214, 7870, 8564, 9297, 10070, 10884, 11740, 12639, 13582, 14570, 15604, 16685, 17815, 18995
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) - 1 = 6
Complement: (b(n)) = (3, 4, 5, 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] - 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}] (* A294867 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Cf. A294860.
Sequence in context: A305329 A161212 A256058 * A033547 A050531 A290699
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 16 2017
STATUS
approved