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A298109 Solution b( ) of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 4, 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, 6, 7, 9, 10, 11, 13, 14, 15, 16, 18, 20, 21, 23, 24, 25, 26, 28, 30, 31, 33, 34, 36, 37, 38, 39, 41, 42, 43, 45, 47, 48, 49, 50, 52, 54, 55, 57, 58, 60, 61, 62, 63, 65, 66, 67, 69, 71, 72, 73, 74, 76, 78, 79, 80, 81, 83, 85, 86, 88, 89, 91, 92, 93, 94 (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 A297832.  See A297830 for a guide to related sequences.

Conjecture:  3/2 < a(n) - n*sqrt(2) < 4 for n >= 1.

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] + 2 n - 4;

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}]; (* A297834 *)

v = Table[b[n], {n, 0, k}]; (* A298109 *)

Take[u, 50]

Take[v, 50]

CROSSREFS

Cf. A297830, A297834.

Sequence in context: A061208 A325439 A183860 * A184429 A248185 A130269

Adjacent sequences:  A298106 A298107 A298108 * A298110 A298111 A298112

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Feb 09 2018

STATUS

approved

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Last modified September 20 08:53 EDT 2021. Contains 347579 sequences. (Running on oeis4.)