The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A294414 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4. 18
 1, 3, 10, 22, 43, 78, 136, 231, 387, 641, 1053, 1721, 2803, 4555, 7391, 11981, 19409, 31429, 50879, 82352, 133278, 215679, 349008, 564740, 913803, 1478600, 2392462, 3871123, 6263648, 10134836, 16398551, 26533456, 42932078, 69465607, 112397760, 181863444 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The initial values of each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4: A294413: a(n) = a(n-1) + a(n-2) - b(n-1) + 6 A294414: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) A294415: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 1 A294416: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n A294417: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - n A294418: a(n) = a(n-1) + a(n-2) + b(n-1) + 2*b(n-2) A294419: a(n) = a(n-1) + a(n-2) + 2*b(n-1) + 2*b(n-2) A294420: a(n) = a(n-1) + a(n-2) + 2*b(n-1) + b(n-2) A294421: a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2) A294422: a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + 1 A294423: a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + n A294424: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - 1 A294425: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - 2 A294426: a(n) = a(n-1) + 2*a(n-2) + b(n-1) + b(n-2) - 3 Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio. LINKS Table of n, a(n) for n=0..35. Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. EXAMPLE a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that a(2) = a(1) + a(0) + b(1) + b(0) = 6 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 11, 12, 13, ...) MATHEMATICA mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2]; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; Table[a[n], {n, 0, 40}] (* A294414 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A293076, A293765, A022940. Sequence in context: A006503 A248851 A023554 * A299336 A222629 A070880 Adjacent sequences: A294411 A294412 A294413 * A294415 A294416 A294417 KEYWORD nonn,easy AUTHOR Clark Kimberling, Oct 31 2017 EXTENSIONS Definition corrected by Georg Fischer, Sep 27 2020 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 June 22 15:32 EDT 2024. Contains 373587 sequences. (Running on oeis4.)