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A296220 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2), 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 (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 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), 1-13.

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

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Last modified September 20 10:17 EDT 2020. Contains 337264 sequences. (Running on oeis4.)