A135414 - OEIS

A135414

a(1)=a(2)=1 and for n>=3, a(n)=n-a(a(n-2)).

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 A356992 A099479

Adjacent sequences: A135411 A135412 A135413 * A135415 A135416 A135417

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Feb 17 2008, Feb 19 2008

STATUS

approved