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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.
28
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
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)):
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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;
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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
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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 04 2018
STATUS
approved