The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A295359 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - 2*b(n-3), where a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6, and (a(n)) and (b(n)) are increasing complementary sequences. 2
 1, 3, 5, 14, 24, 41, 68, 112, 183, 298, 484, 786, 1275, 2064, 3342, 5409, 8754, 14166, 22923, 37092, 60019, 97116, 157138, 254257, 411398, 665658, 1077059, 1742720, 2819782, 4562505, 7382290, 11944798, 19327091, 31271892, 50598986 (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 A295357 for a guide to related sequences. LINKS Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. FORMULA a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622). EXAMPLE a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6, so that b(3) = 7 (least "new number") a(3) = a(1) + a(0) + b(2) + b(1) - 2* b(0) = 14 Complement: (b(n)) = (2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 15, ...) MATHEMATICA mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; a = 1; a = 3; a = 5; b = 2; b = 4; b = 6; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] - 2*b[n - 3]; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; z = 32; u = Table[a[n], {n, 0, z}]   (* A295359 *) v = Table[b[n], {n, 0, 10}]  (* complement *) CROSSREFS Cf. A001622, A295357. Sequence in context: A179304 A026645 A026667 * A104208 A026777 A007136 Adjacent sequences:  A295356 A295357 A295358 * A295360 A295361 A295362 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 21 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 | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified July 2 09:32 EDT 2022. Contains 354999 sequences. (Running on oeis4.)