A294864 Solution of the complementary equation a(n) = a(n-2) + b(n-2) + n, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 6, 9, 15, 21, 29, 38, 48, 59, 71, 84, 99, 114, 131, 148, 167, 187, 208, 230, 253, 277, 302, 328, 356, 384, 414, 444, 476, 508, 542, 576, 613, 649, 688, 726, 767, 807, 850, 892, 937, 982, 1029, 1076, 1125, 1174, 1225, 1276, 1329, 1382, 1437, 1493, 1550

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

Table of n, a(n) for n=0..52.

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) = a(0) + b(0) + 2 = 6
Complement: (b(n)) = (3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 16, ...)

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[n - 2] + b[n - 2] + 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}]  (* A294864 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A294860.

Sequence in context: A260699 A084265 A084140 * A103139 A181025 A345051

Adjacent sequences: A294861 A294862 A294863 * A294865 A294866 A294867

Keyword

nonn,easy

Author

Clark Kimberling, Nov 16 2017

Status

approved