

A295616


Solution of the complementary equation a(n) = 2*a(n1)  a(n3) + b(n2), where a(0) = 1, a(1) = 2, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6, and (a(n)) and (b(n)) are increasing complementary sequences.


2



1, 2, 3, 10, 24, 52, 102, 189, 337, 584, 992, 1661, 2753, 4530, 7416, 12097, 19683, 31970, 51864, 84067, 136187, 220535, 357029, 577898, 935289, 1513578, 2449288, 3963318, 6413090, 10376925, 16790566, 27168077, 43959265, 71128001, 115087963, 186216700
(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 A295613 for a guide to related sequences.
a(n)/a(n1) > (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growthrate of the Fibonacci numbers (A000045).


LINKS

Table of n, a(n) for n=0..35.
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 113.


EXAMPLE

a(0) = 1, a(1) = 2, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6, so that
b(3) = 7 (least "new number")
a(3) = 2*a(2)  a(0) + b(1) = 10
Complement: (b(n)) = (4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, ...)


MATHEMATICA

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; a[2] = 3; b[0] = 4; b[1] = 5; b[2] = 6;
a[n_] := a[n] = 2 a[n  1]  a[n  3] + 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, 30}] (* A295616 *)
Table[b[n], {n, 0, 20}] (* complement *)


CROSSREFS

Cf. A001622, A000045, A295613.
Sequence in context: A320812 A162034 A105286 * A059929 A123029 A103018
Adjacent sequences: A295613 A295614 A295615 * A295617 A295618 A295619


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Nov 25 2017


STATUS

approved



