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 A295957 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. 2
 1, 4, 11, 22, 41, 72, 123, 206, 342, 562, 919, 1497, 2433, 3948, 6400, 10368, 16789, 27179, 43992, 71196, 115214, 186437, 301679, 488145, 789854, 1278030, 2067916, 3345979, 5413929, 8759943, 14173908, 22933888, 37107834, 60041761, 97149635, 157191437 (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 A295862 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) = 4, b(0) = 2, b(1) = 3, b(2) = 5 b(3) = 6 (least "new number") a(2) = a(1) + a(0) + b(2) + 1 = 11 Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, ...) MATHEMATICA a = 1; a = 4; b = 2; b = 3; b = 5; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 1; j = 1; While[j < 6, 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}];  (* A295957 *) CROSSREFS Cf. A001622, A000045, A295862. Sequence in context: A049836 A177853 A008016 * A259574 A008249 A174405 Adjacent sequences:  A295954 A295955 A295956 * A295958 A295959 A295960 KEYWORD nonn,easy AUTHOR Clark Kimberling, Dec 08 2017 STATUS approved

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Last modified August 5 01:47 EDT 2021. Contains 346456 sequences. (Running on oeis4.)