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A298111 Solution b( ) of the complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 2*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (b(n)) is the increasing sequence of positive integers not in (a(n)).  See Comments. 2
3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 25, 26, 28, 30, 31, 32, 33, 35, 37, 38, 39, 40, 42, 44, 45, 46, 47, 49, 51, 52, 53, 54, 56, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 75, 76, 78, 79, 81, 82, 83, 84, 86, 87, 88, 90, 92, 93 (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 A298000.  See A297830 for a guide to related sequences.

Conjecture:  1/5 < a(n) - n*sqrt(2) < 3 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; b[2] = 5;

a[n_] := a[n] = a[1]*b[n] - a[0]*b[n - 1] + 2 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}]; (* A298000 *)

v = Table[b[n], {n, 0, k}]; (* A298111 *)

Take[u, 50]

Take[v, 50]

CROSSREFS

Cf. A297830, A298000.

Sequence in context: A258187 A039237 A039181 * A299534 A026454 A026458

Adjacent sequences:  A298108 A298109 A298110 * A298112 A298113 A298114

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Feb 09 2018

STATUS

approved

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Last modified December 5 06:44 EST 2020. Contains 338944 sequences. (Running on oeis4.)