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A298005 Solution b( ) of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 4*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)).  See Comments. 2
3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 82 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values.  The solution a( ) is given at A297837.  See A297830 for a guide to related sequences.

Conjecture: 7/10 < a(n) - n*L < 3 for n >= 1, where L = -1 + sqrt(5).

LINKS

Table of n, a(n) for n=0..65.

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

MATHEMATICA

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;

a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 4 n;

j = 1; While[j < 80000, k = a[j] - j - 1;

While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k

u = Table[a[n], {n, 0, k}]; (* A297837 *)

v = Table[b[n], {n, 0, k}]; (* A298005 *)

Take[u, 50]

Take[v, 50]

CROSSREFS

Cf. A297830, A297837.

Sequence in context: A130231 A333491 A304816 * A216442 A298296 A099474

Adjacent sequences:  A298002 A298003 A298004 * A298006 A298007 A298008

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Feb 09 2018

STATUS

approved

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Last modified June 23 13:59 EDT 2021. Contains 345402 sequences. (Running on oeis4.)