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!)
 A294545 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) - 1, where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 2
 1, 2, 6, 12, 24, 43, 75, 127, 212, 351, 576, 941, 1532, 2489, 4038, 6545, 10602, 17167, 27790, 44979, 72793, 117797, 190616, 308440, 499084, 807553, 1306667, 2114251, 3420950, 5535234, 8956218, 14491487, 23447741, 37939265, 61387044, 99326348 (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. See A294532 for a guide to related sequences. Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622). LINKS Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. EXAMPLE a(0) = 1, a(1) = 2, b(0) = 3, so that b(1) = 4 (least "new number"); a(2) = a(1) + a(0) + b(1) -1 = 6. Complement: (b(n)) = (3, 4, 5, 7, 8, 9, 11, 13, 14, 15, 16, ...). MATHEMATICA mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; a = 1; a = 3; b = 2; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] - 1; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; Table[a[n], {n, 0, 40}]  (* A294545 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A001622, A294532. Sequence in context: A054061 A294563 A307211 * A305104 A118224 A227068 Adjacent sequences:  A294542 A294543 A294544 * A294546 A294547 A294548 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 04 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 May 16 00:49 EDT 2021. Contains 343937 sequences. (Running on oeis4.)