A294860 Solution of the equation a(n) = a(n-2) + b(n-2), where a( ) and b( ) are increasing sequences of positive integers such that every positive integer is in one of them and only one term is in both.

1, 2, 4, 6, 9, 13, 17, 23, 28, 35, 42, 50, 58, 68, 77, 88, 98, 110, 122, 135, 148, 162, 177, 192, 208, 224, 241, 258, 277, 295, 315, 334, 355, 375, 398, 419, 443, 465, 490, 513, 539, 564, 591, 617, 645, 672, 701, 729, 760, 789, 821, 851, 884, 915, 949, 981

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The initial values sequences in the following guide are a(0) = 1, a(1) = 2, b(0) = 3.

A294860: a(n) = a(n-2) + b(n-2); not quite complementary

A022939: a(n) = a(n-2) + b(n-2); offset 1, complementary

A294861: a(n) = a(n-2) + b(n-2) + 1

A294862: a(n) = a(n-2) + b(n-2) + 2

A294863: a(n) = a(n-2) + b(n-2) + 3

A294864: a(n) = a(n-2) + b(n-2) + n

A294865: a(n) = a(n-2) + 2*b(n-2)

A294866: a(n) = 2*a(n-1) - a(n-2) + b(n-1)

A294867: a(n) = 2*a(n-1) - a(n-2) + b(n-1) - 1

A294868: a(n) = 2*a(n-1) - a(n-2) + b(n-1) - 2

A294869: a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 1

A294870: a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 2

A294871: a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 3

A294872: a(n) = 2*a(n-1) - a(n-2) + b(n-1) + n

A022942: a(n) = a(n-2) + b(n-1); offset 1

A295998: a(n) = 2*a(n-2) + b(n-2)

Links

Table of n, a(n) for n=0..55.

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

Example

a(0) = 1, a(1) = 2, b(0) = 3, so that a(2) = 4
(b(n)) = (3,4,5,7,8,10,11,12,14,15,...)

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] + 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}]  (* A294860 *)
Table[b[n], {n, 0, 10}]

Crossrefs

Cf. A294861, A294864, A294865.

Sequence in context: A232739 A342712 A006697 * A183920 A079717 A247179

Adjacent sequences: A294857 A294858 A294859 * A294861 A294862 A294863

Keyword

nonn,easy

Author

Clark Kimberling, Nov 16 2017

Extensions

Edited by Clark Kimberling, Dec 02 2017

Status

approved