A297830 - OEIS

A297830

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

1, 2, 9, 12, 15, 18, 21, 26, 28, 33, 35, 40, 42, 47, 49, 54, 56, 59, 62, 67, 71, 73, 76, 79, 84, 88, 90, 93, 96, 101, 105, 107, 110, 113, 118, 122, 124, 127, 130, 135, 139, 141, 146, 148, 153, 155, 158, 161, 166, 168, 171, 176, 180, 182, 187, 189, 194, 196

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

Guide to related sequences having initial values a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, where (b(n)) is the increasing sequence of positive integers not in (a(n)):

***

a(n) = a(1)*b(n-1) - a(0)*b(n-2) + n (a(n)) = A297826; (b(n)) = A297997

a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n (a(n)) = A297830; (b(n)) = A298003

a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 3*n (a(n)) = A297836; (b(n)) = A298004

a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 4*n (a(n)) = A297837; (b(n)) = A298005

a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 1 (a(n)) = A297831; (b(n)) = A298006

a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 2 (a(n)) = A297832; (b(n)) = A298007

a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 3 (a(n)) = A297833; (b(n)) = A298108

a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 4 (a(n)) = A297834; (b(n)) = A298109

a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n + 1 (a(n)) = A297835;

a(n) = a(1)*b(n-1) - a(0)*b(n-2)+floor(5*n/2) (a(n)) = A297998;

***

For sequences (a(n)) and (b(n)) associated with equations of the form a(n) = a(1)*b(n) - a(0)*b(n-1), see the guide at A297800.

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..10000

EXAMPLE

a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 9.

Complement: (b(n)) = (3,4,5,6,8,10,11,13,14,16,17,19,...)

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;

j = 1; While[j < 100, k = a[j] - j - 1;

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

Table[a[n], {n, 0, k}] (* A297830 *)

CROSSREFS

Cf. A297826, A297836, A297837.

Sequence in context: A032372 A207967 A265413 * A103008 A124742 A152003

Adjacent sequences: A297827 A297828 A297829 * A297831 A297832 A297833

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Feb 04 2018

STATUS

approved