The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A295952 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n), where a(0) = 1, a(1) = 5, b(0) = 2, b(1) = 3, b(2) = 4, and (a(n)) and (b(n)) are increasing complementary sequences. 5
 1, 5, 10, 21, 38, 67, 114, 192, 318, 523, 855, 1393, 2264, 3674, 5956, 9649, 15625, 25296, 40944, 66264, 107233, 173523, 280783, 454334, 735146, 1189510, 1924687, 3114229, 5038949, 8153212, 13192196, 21345444, 34537677, 55883160, 90420877, 146304078 (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. 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) = 5, b(0) = 2, b(1) = 3, b(2) = 4; b(3) = 6 (least "new number"); a(2) = a(1) + a(0) + b(2) = 10; Complement: (b(n)) = (2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, ...) MATHEMATICA a = 1; a = 5; b = 2; b = 3; b = 4; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n]; j = 1; While[j < 5, 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}]   (* A295952 *) Table[b[n], {n, 0, 20}]  (* complement *) CROSSREFS Cf. A001622, A000045, A295862. Sequence in context: A242644 A002800 A280077 * A132174 A297301 A132461 Adjacent sequences:  A295949 A295950 A295951 * A295953 A295954 A295955 KEYWORD nonn,easy AUTHOR Clark Kimberling, Dec 08 2017 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified July 26 10:21 EDT 2021. Contains 346294 sequences. (Running on oeis4.)