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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
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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)) = A298209 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 March 18 22:11 EDT 2019. Contains 321305 sequences. (Running on oeis4.)