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 A297835 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n + 1, 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. 2
 1, 2, 10, 13, 16, 19, 22, 25, 30, 32, 37, 39, 44, 46, 51, 53, 58, 60, 65, 67, 70, 73, 78, 82, 84, 87, 90, 95, 99, 101, 104, 107, 112, 116, 118, 121, 124, 129, 133, 135, 138, 141, 146, 150, 152, 155, 158, 163, 167, 169, 174, 176, 181, 183, 186, 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. See A297830 for a guide to related sequences. Conjecture:  a(n) - (2 +sqrt(2))*n < 7 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) = 10. Complement: (b(n)) = (3,4,6,7,8,9,11,12,14,15,17,18,20,...) MATHEMATICA a = 1; a = 2; b = 3; b = 4; a[n_] := a[n] = a*b[n - 1] - a*b[n - 2] + 2 n + 1; 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}]  (* A297835 *) CROSSREFS Cf. A297826, A297830. Sequence in context: A343476 A343477 A296220 * A298000 A058216 A297998 Adjacent sequences:  A297832 A297833 A297834 * A297836 A297837 A297838 KEYWORD nonn,easy AUTHOR Clark Kimberling, Feb 04 2018 STATUS approved

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Last modified September 24 06:13 EDT 2021. Contains 347623 sequences. (Running on oeis4.)