A294480 Solution of the complementary equation a(n) = a(n-2) + b(n-1) + 2n, where a(0) = 1, a(1) = 3, b(0) = 2.

1, 3, 9, 14, 23, 31, 43, 55, 70, 85, 103, 122, 143, 165, 189, 214, 241, 269, 299, 331, 364, 399, 435, 473, 512, 553, 596, 640, 686, 733, 782, 832, 884, 937, 992, 1048, 1106, 1166, 1227, 1290, 1354, 1420, 1487, 1556, 1626, 1698, 1771, 1846, 1923, 2001, 2081

Offset

0,2

Comments

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294476 for a guide to related sequences.

Links

Clark Kimberling, Table of n, a(n) for n = 0..1000

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

Example

a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
a(2)  = a(0) + b(1) = 9
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 12, 13, 15, ...)

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 2] + b[n - 1] + 2n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294480 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A293076, A293765, A294476.

Sequence in context: A305373 A111907 A309149 * A195972 A274435 A136562

Adjacent sequences: A294477 A294478 A294479 * A294481 A294482 A294483

Keyword

nonn,easy

Author

Clark Kimberling, Nov 01 2017

Status

approved