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 A297833 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 3, 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. 3
 1, 2, 6, 9, 14, 16, 21, 23, 26, 29, 34, 38, 40, 43, 46, 51, 55, 57, 62, 64, 69, 71, 74, 77, 82, 84, 87, 92, 96, 98, 103, 105, 110, 112, 115, 118, 123, 125, 128, 133, 137, 139, 142, 145, 150, 154, 156, 159, 162, 167, 171, 173, 178, 180, 185, 187, 190, 193 (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. See A297830 for a guide to related sequences. Conjecture:  -2 < a(n) - (2 +sqrt(2))*n <= 1 for n >= 1. 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) = 6. Complement: (b(n)) = (3,4,5,7,8,10,12,13,15,17,18,19,...) MATHEMATICA a = 1; a = 2; b = 3; b = 4; a[n_] := a[n] = a*b[n - 1] - a*b[n - 2] + 2 n - 3; 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}]  (* A297833 *) CROSSREFS Cf. A297826, A297830. Sequence in context: A048162 A226823 A171866 * A049508 A184824 A184836 Adjacent sequences:  A297830 A297831 A297832 * A297834 A297835 A297836 KEYWORD nonn,easy AUTHOR Clark Kimberling, Feb 04 2018 STATUS approved

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Last modified September 16 15:56 EDT 2021. Contains 347473 sequences. (Running on oeis4.)