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A294861 Solution of the complementary equation a(n) = a(n-2) + b(n-2) + 1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 2
1, 2, 5, 7, 12, 16, 22, 27, 34, 41, 49, 57, 67, 76, 87, 97, 109, 121, 134, 147, 161, 176, 191, 207, 223, 240, 257, 276, 294, 314, 333, 354, 374, 397, 418, 442, 464, 489, 512, 538, 563, 590, 616, 644, 671, 700, 728, 759, 788, 820, 850, 883, 914, 948, 980 (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) + 1 = 5

Complement: (b(n)) = (3, 4, 6, 8, 9, 10, 11, 12, 14, 15, 17, ...)

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] + 1;

b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];

Table[a[n], {n, 0, 18}]  (* A294861 *)

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

CROSSREFS

Cf. A294860, A294862.

Sequence in context: A001318 A024702 A226084 * A161664 A080547 A080555

Adjacent sequences:  A294858 A294859 A294860 * A294862 A294863 A294864

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Nov 16 2017

STATUS

approved

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Last modified September 20 12:42 EDT 2018. Contains 315239 sequences. (Running on oeis4.)