A294412 Solution of the complementary equation a(n) = a(n-1) + 2*b(n-2) + 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 8, 17, 28, 41, 56, 75, 96, 119, 144, 171, 200, 231, 264, 301, 340, 381, 424, 469, 516, 565, 616, 669, 724, 783, 844, 907, 972, 1039, 1108, 1179, 1252, 1327, 1404, 1483, 1564, 1649, 1736, 1825, 1916, 2009, 2104, 2201, 2300, 2401, 2504, 2609, 2716, 2825

Offset

0,2

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.

Links

Table of n, a(n) for n=0..49.

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(1) + 2*b(0) + 1 = 8
Complement: (b(n)) = (2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 15, 16, ...)

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] + 2b[n - 2] + 1;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294412 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A293076, A293765, A022940.

Sequence in context: A368273 A335047 A048230 * A227017 A073433 A059518

Adjacent sequences: A294409 A294410 A294411 * A294413 A294414 A294415

Keyword

nonn,easy

Author

Clark Kimberling, Oct 31 2017

Status

approved