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 A294170 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 2*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. 4
 1, 2, 12, 26, 53, 97, 171, 292, 490, 813, 1337, 2187, 3564, 5794, 9404, 15247, 24703, 40005, 64766, 104832, 169662, 274561, 444294, 718929, 1163300, 1882309, 3045692, 4928087, 7973868, 12902047, 20876010, 33778155, 54654266, 88432525, 143086898, 231519533 (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. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622). See A296245 for a guide to related sequences. LINKS Clark Kimberling, Table of n, a(n) for n = 0..2000 Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. EXAMPLE a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5 a(2) = a(0) + a(1) + b(2) + 4 = 12 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, ...) MATHEMATICA a = 1; a = 2; b = 3; b = 4; b = 5; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 2 n; j = 1; While[j < 16, k = a[j] - j - 1; While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; Table[a[n], {n, 0, k}];  (* A294170 *) CROSSREFS Cf. A296245, A296491, A296492. Sequence in context: A098707 A152811 A294552 * A102960 A166151 A154149 Adjacent sequences:  A294167 A294168 A294169 * A294171 A294172 A294173 KEYWORD nonn,easy AUTHOR Clark Kimberling, Feb 10 2018 STATUS approved

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Last modified May 31 00:53 EDT 2020. Contains 334747 sequences. (Running on oeis4.)