

A296220


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


4



1, 2, 10, 13, 16, 19, 22, 25, 29, 34, 38, 43, 47, 52, 56, 61, 65, 70, 74, 79, 82, 86, 91, 94, 97, 101, 106, 109, 113, 118, 121, 124, 128, 133, 136, 140, 145, 148, 151, 155, 160, 163, 167, 172, 175, 178, 182, 187, 190, 194, 199, 202, 205, 209, 214, 217, 221
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OFFSET

0,2


COMMENTS

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295862 for a guide to related sequences.
P. Majer proved that a(n)/n > 4, that (a(n)  4*n) is unbounded, and that a( ) is not a linear recurrence sequence; see the Math Overflow link and A297964.  Clark Kimberling, Feb 10 2018


LINKS

Peter J. C. Moses, Table of n, a(n) for n = 0..9999
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 113.
Pietro Majer, Limit associated with complementary sequences Math Overflow.


EXAMPLE

a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4;
a(2) = a(0)*b(1) + a(1)*b(0) = 10.
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, ...).


MATHEMATICA

mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[0]*b[n  1] + a[1]*b[n  2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n  1}]]];
u = Table[a[n], {n, 0, 500}]; (* A296220 *)
Table[b[n], {n, 0, 20}]


CROSSREFS

Cf. A296000.
Sequence in context: A189079 A045218 A177856 * A297835 A298000 A058216
Adjacent sequences: A296217 A296218 A296219 * A296221 A296222 A296223


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Dec 08 2017


STATUS

approved



