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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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified December 5 06:51 EST 2020. Contains 338944 sequences. (Running on oeis4.)