A294400 Solution of the complementary equation a(n) = a(n-1) + b(n-2) + n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 7, 14, 23, 34, 48, 64, 82, 102, 124, 148, 175, 204, 235, 268, 303, 340, 379, 420, 464, 510, 558, 608, 660, 714, 770, 828, 888, 950, 1015, 1082, 1151, 1222, 1295, 1370, 1447, 1526, 1607, 1690, 1775, 1862, 1951, 2043, 2137, 2233, 2331, 2431, 2533, 2637

Offset: 0

Author

Clark Kimberling, Oct 31 2017

Keywords

• nonn
• easy

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A022940 for a guide to related sequences.

Examples

			a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2)  = a(1) + b(0) + 2 = 7
Complement: (b(n)) = (2, 4, 5, 6, 8, 10, 11, 12, 13, 15, 16, ...)
		

Links

• Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Crossrefs

Cf. A293076, A293765, A022940.

Programs

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + 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, 40}]  (* A294400 *)
Table[b[n], {n, 0, 10}]