A294865 Solution of the complementary equation a(n) = a(n-2) + 2*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 7, 10, 17, 22, 33, 40, 55, 64, 81, 92, 111, 124, 147, 162, 187, 204, 233, 252, 283, 304, 337, 360, 395, 420, 457, 484, 525, 554, 597, 628, 673, 706, 755, 790, 841, 878, 931, 970, 1025, 1066, 1123, 1166, 1225, 1270, 1331, 1378, 1443, 1492, 1559, 1610

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..51.

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) + 2*b(0) = 7
Complement: (b(n)) = (3, 4, 5, 6, 8, 9, 11, 12, 13, 14, 15, ...)

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] + 2*b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}]  (* A294865 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A294860.

Sequence in context: A059316 A295825 A140115 * A105770 A240469 A257335

Adjacent sequences: A294862 A294863 A294864 * A294866 A294867 A294868

Keyword

nonn,easy

Author

Clark Kimberling, Nov 16 2017

Status

approved