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 A297837 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 4*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. 7
 1, 2, 13, 18, 23, 28, 33, 38, 43, 48, 53, 60, 64, 69, 74, 81, 85, 90, 95, 102, 106, 111, 116, 123, 127, 132, 137, 144, 148, 153, 158, 165, 169, 174, 179, 186, 190, 195, 200, 207, 211, 216, 221, 228, 232, 237, 242, 247, 252, 259, 263, 268, 275, 279, 284, 289 (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. For a guide to related sequences, see A297830. Conjecture:  a(n) - (3 + sqrt(5))*n < 3 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) = 13. Complement: (b(n)) = (3,4,5,6,7,8,9,10,11,12,14,15,16,17,19,20,...) MATHEMATICA a = 1; a = 2; b = 3; b = 4; a[n_] := a[n] = a*b[n - 1] - a*b[n - 2] + 4 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}]  (* A297836 *) CROSSREFS Cf. A297826, A297830, A297836. Sequence in context: A128852 A191765 A063615 * A246358 A262688 A020585 Adjacent sequences:  A297834 A297835 A297836 * A297838 A297839 A297840 KEYWORD nonn,easy AUTHOR Clark Kimberling, Feb 04 2018 STATUS approved

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Last modified August 6 19:52 EDT 2020. Contains 336256 sequences. (Running on oeis4.)