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 A135414 a(1)=a(2)=1 and for n>=3, a(n)=n-a(a(n-2)). 4
 1, 1, 2, 3, 4, 4, 4, 5, 6, 6, 7, 8, 9, 9, 9, 10, 11, 12, 12, 12, 13, 14, 14, 15, 16, 17, 17, 17, 18, 19, 19, 20, 21, 22, 22, 22, 23, 24, 25, 25, 25, 26, 27, 27, 28, 29, 30, 30, 30, 31, 32, 33, 33, 33, 34, 35, 35, 36, 37, 38, 38, 38, 39, 40, 40, 41, 42, 43, 43, 43, 44, 45, 46, 46 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS A generalization of Hofstadter's G-sequence. Contribution from Daniel Platt (d.platt(AT)web.de), Jul 27 2009: (Start) Conjecture: A recursively built tree structure can be obtained from the sequence: .29.30.31.32.33.34.35.36.37.38.39.40.41.42.43.44.45.. ..|..\./...|..|...\.|./...|..|...\.|./...|..\./...|.. .18..19...20.21....22....23.24....25....26..27...28.. ..\...|.../...|.....\..../...|.....|.....\...|.../... ...\..|../....|......\../....|.....|......\..|../.... .....12......13.......14....15....16........17....... ......|........\......|...../......|.........|....... ......|..........\....|.../........|.........|....... ......8...............9...........10........11....... ......|.................\......./............|....... ......|...................\.../..............|....... ......5.....................6................7....... .........\..................|............./.......... ..............\.............|........../............. ....................\.......|....../................. ............................4........................ .........................../......................... ..........................3.......................... ........................./........................... ........................2............................ ......................./............................. ......................1.............................. When constructing the tree node n is connected to node a(n) below: ..n.. ..|.. .a(n) Same procedure as for A005206. Reading the nodes bottom-to-top, left-to-right provides the natural numbers. The tree has a recursive structure: The following construct will give - added on top of its own ends - the above tree: .............. ... . ............./.../.. ............/.../... . ... .....X...X.... ..\...\.../.../..... ...\...\./.../...... ....X...X...X....... .....\..|../........ ......\.|./......... ........X........... (End) LINKS D. Platt, Table of n, a(n) for n=1..1999 [From Daniel Platt (d.platt(AT)web.de), Jul 27 2009] FORMULA a(n)=2+floor(n*phi)+floor((n+1)*phi)-floor((n+3)*phi) where phi=(sqrt(5)-1)/2 n = a(n) + a(a(n-2)) unless n = 2 or n = -3. - Michael Somos, Jun 30 2011 EXAMPLE x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 4*x^6 + 4*x^7 + 5*x^8 + 6*x^9 + 6*x^10 + ... MATHEMATICA a[ n_] := 2 - Boole[ n==0] + Quotient[ n, GoldenRatio] + Quotient[ n + 1, GoldenRatio] - Quotient[ n + 3, GoldenRatio] (* Michael Somos, Jun 30 2011 *) PROG (PARI) a(n)=2+floor(n*(sqrt(5)-1)/2)+floor((n+1)*(sqrt(5)-1)/2)-floor((n+3)*(sqrt(5)-1)/2) (PARI) {a(n) = local(g = (1 + sqrt(5)) / 2); 2 - (n==0) + n\g + (n + 1)\g - (n + 3)\g} /* Michael Somos, Jun 30 2011 */ (Haskell) a135414 n = a135414_list !! (n-1) a135414_list = 1 : 1 : zipWith (-) [3..] (map a135414 a135414_list) -- Reinhard Zumkeller, Nov 12 2011 CROSSREFS Cf. A005206. Sequence in context: A087876 A006158 A340203 * A326821 A099479 A120508 Adjacent sequences:  A135411 A135412 A135413 * A135415 A135416 A135417 KEYWORD nonn AUTHOR Benoit Cloitre, Feb 17 2008, Feb 19 2008 STATUS approved

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Last modified January 22 01:28 EST 2022. Contains 350481 sequences. (Running on oeis4.)