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 A295862 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. 32
 1, 3, 9, 18, 34, 60, 104, 175, 291, 479, 784, 1278, 2078, 3373, 5470, 8863, 14354, 23239, 37616, 60879, 98520, 159425, 257972, 417425, 675426, 1092881, 1768338, 2861251, 4629622, 7490908, 12120566, 19611511, 31732115, 51343665, 83075820, 134419526, 217495388 (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).  Following is a guide to related sequences: ***** Complementary equation: a(n) = a(n-1) + a(n-2) + b(n); initial values (a(0), a(1); b(0), b(1), b(2)): A295862:  (1,3; 2,4,5) A295947:  (2,4; 1,3,5) A295948:  (3,4; 1,2,5) A295949:  (1,2; 3,4,5) A295950:  (1,4; 2,3,5) A295951:  (2,3; 1,4,5) A295952:  (1,5; 2,3,4) Complementary equation: a(n) = a(n-1) + a(n-2) + b(n) + 1; initial values (a(0), a(1); b(0), b(1), b(2)): A295953:  (1,3; 2,4,5) A295954:  (2,4; 1,3,5) A295955:  (3,4; 1,2,5) A295956:  (1,2; 3,4,5) A295957:  (1,4; 2,3,5) A295958:  (2,3; 1,4,5) A295959:  (1,5; 2,3,4) Complementary equation: a(n) = a(n-1) + a(n-2) + b(n) - 1; initial values (a(0), a(1); b(0), b(1), b(2)): A295860:  (1,3; 2,4,5) A295961:  (2,4; 1,3,5) A295962:  (3,4; 1,2,5) A295963:  (1,2; 3,4,5) A295964:  (1,4; 2,3,5) A295965:  (2,3; 1,4,5) A295966:  (1,5; 2,3,4) LINKS Clark Kimberling, Table of n, a(n) for n = 0..3000 Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. FORMULA a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(2) + f(n-2)*b(3) + ... + f(2)*b(n-1) + f(1)*b(n), where f(n) = A000045(n), the n-th Fibonacci number. EXAMPLE a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, so that b(3) = 6 (least "new number"); a(2) = a(1) + a(0) + b(2) = 9; Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, ...) MATHEMATICA a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n]; 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}]  (*A295862*) Table[b[n], {n, 0, 20}] (*complement*) CROSSREFS Cf. A001622, A000045, A295947. Sequence in context: A256524 A210970 A293406 * A246695 A132920 A127645 Adjacent sequences:  A295859 A295860 A295861 * A295863 A295864 A295865 KEYWORD nonn,easy AUTHOR Clark Kimberling, Dec 08 2017 STATUS approved

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Last modified January 24 19:12 EST 2021. Contains 340411 sequences. (Running on oeis4.)