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A294862 Solution of the complementary equation a(n) = a(n-2) + b(n-2) + 2, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 3
1, 2, 6, 8, 13, 17, 24, 29, 37, 43, 53, 60, 71, 80, 92, 102, 115, 126, 140, 153, 168, 182, 198, 214, 231, 248, 266, 284, 303, 322, 343, 363, 385, 406, 429, 452, 476, 500, 525, 550, 576, 602, 629, 656, 685, 713, 743, 772, 803, 833, 866, 897, 931, 963, 998 (list; graph; refs; listen; history; text; internal format)
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..54.

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) + b(0) + 2 = 6

Complement: (b(n)) = (3, 4, 5, 7, 9, 10, 11, 12, 14, 15, 16, ...)

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] + 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}]  (* A294862 *)

Table[b[n], {n, 0, 10}]

CROSSREFS

Cf. A294860, A294863.

Sequence in context: A186703 A054248 A038108 * A087327 A266627 A289753

Adjacent sequences:  A294859 A294860 A294861 * A294863 A294864 A294865

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 16 2017

STATUS

approved

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Last modified May 13 11:25 EDT 2021. Contains 343839 sequences. (Running on oeis4.)