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 A296292 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1), where a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 4
 2, 4, 12, 31, 67, 133, 248, 444, 772, 1315, 2217, 3686, 6083, 9977, 16298, 26545, 43147, 70032, 113557, 184007, 298024, 482535, 781109, 1264242, 2045999, 3310941, 5357694, 8669445, 14028035, 22698437, 36727492, 59427014, 96155658, 155583893, 251740843 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 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..1000 Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. EXAMPLE a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5 a(2) = a(0) + a(1) + 2*b(1) = 12 Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, ...) MATHEMATICA a = 2; a = 4; b = 1; b = 3; a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n-1]; j = 1; While[j < 10, 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}]; (* A296292 *) Table[b[n], {n, 0, 20}]    (* complement *) CROSSREFS Cf. A001622, A296245. Sequence in context: A148189 A148190 A151434 * A287966 A148191 A141312 Adjacent sequences:  A296289 A296290 A296291 * A296293 A296294 A296295 KEYWORD nonn,easy AUTHOR Clark Kimberling, Dec 14 2017 STATUS approved

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Last modified September 27 16:20 EDT 2020. Contains 337383 sequences. (Running on oeis4.)