login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A047838 a(n) = floor(n^2/2) - 1. 11
1, 3, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337, 363, 391, 419, 449, 479, 511, 543, 577, 611, 647, 683, 721, 759, 799, 839, 881, 923, 967, 1011, 1057, 1103, 1151, 1199, 1249, 1299, 1351, 1403 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

Define the organization number of a permutation pi_1, pi_2, ..., pi_n to be the following. Start at 1, count the steps to reach 2, then the steps to reach 3, etc. Add them up. Then the maximal value of the organization number of any permutation of [1..n] for n = 0, 1, 2, 3, ... is given by 0, 1, 3, 7, 11, 17, 23, ... (this sequence). This was established by Graham Cormode (graham(AT)research.att.com), Aug 17 2006, see link below, answering a question raised by Tom Young (mcgreg265(AT)msn.com) and Barry Cipra, Aug 15 2006

From Dmitry Kamenetsky, Nov 29 2006: (Start)

This is the length of the longest non-self-intersecting spiral drawn on an n X n grid. E.g., for n=5 the spiral has length 17:

10111

10101

10101

10001

11111 (End)

It appears that a(n+1) is the maximum number of consecutive integers (beginning with 1) that can be placed, one after another, on an n-peg Towers of Hanoi, such that the sum of any two consecutive integers on any peg is a square. See the problem: http://online-judge.uva.es/p/v102/10276.html. - Ashutosh Mehra, Dec 06 2008

a(n) = number of (w,x,y) with all terms in {0,...,n} and w=|x+y-w|. - Clark Kimberling, Jun 11 2012

The same sequence represents also the solution to the "pigeons problem": maximal value of the sum of the lengths of n-1 line segments (connected at their end-points) required to pass through n trail dots, with unit distance between adjacent points, visiting all of them without overlap two or more segments. In this case, a(0)=0, a(1)=1, a(2)=3, and so on. - Marco Ripà, Jan 28 2014

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Graham Cormode, Notes on the organization number of a permutation.

Marco Ripà, The rectangular spiral solution for the n_1xn_2x...xn_k points problem, viXra.org.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

FORMULA

a(2)=1; for n>2, a(n) = a(n-1) + n - 1 + (n-1 mod 2). - Benoit Cloitre, Jan 12 2003

T(n-1) + floor(n/2) - 1 = T(n) - floor((n+3)/2), where T(n) is the n-th triangular number (A000217). - Robert G. Wilson v, Aug 31 2006

Equals (n-1)-th row sums of triangles A134151 and A135152. Also, = binomial transform of [1, 2, 2, -2, 4, -8, 16, -32, ...]. - Gary W. Adamson, Nov 21 2007

G.f.: x^2(1+x+x^2-x^3)/((1-x)^3*(1+x)). - R. J. Mathar, Sep 09 2008

a(n) = floor((n^2+4*n+2)/2). - Gary Detlefs, Feb 10 2010

a(n) = abs(A188653(n)). - Reinhard Zumkeller, Apr 13 2011

a(n) = (2*n^2+(-1)^n-5)/4. - Bruno Berselli, Sep 14 2011

a(n) = a(-n) = A007590(n) - 1.

a(n) = A080827(n) - 2. - Kevin Ryde, Aug 24 2013

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4), n>4. - Wesley Ivan Hurt, Aug 06 2015

EXAMPLE

x^2 + 3*x^3 + 7*x^4 + 11*x^5 + 17*x^6 + 23*x^7 + 31*x^8 + 39*x^9 + 49*x^10 + ...

MAPLE

seq(floor((n^2+4*n+2)/2), n=0..20) # Gary Detlefs, Feb 10 2010

MATHEMATICA

Table[ Floor[n^2/2] - 1, {n, 2, 60}] (* Robert G. Wilson v, Aug 31 2006 *)

LinearRecurrence[{2, 0, -2, 1}, {1, 3, 7, 11}, 60] (* Harvey P. Dale, Jan 16 2015 *)

PROG

(PARI) a(n) = n^2\2 - 1

(MAGMA) [Floor(n^2/2)-1 : n in [2..100]]; // Wesley Ivan Hurt, Aug 06 2015

CROSSREFS

Complement of A047839. First difference is A052928.

Cf. A000217, A007590, A052928, A080827, A134151, A135151, A135152, A188653.

Sequence in context: A023234 A237662 A134707 * A188653 A262502 A211076

Adjacent sequences:  A047835 A047836 A047837 * A047839 A047840 A047841

KEYWORD

nonn,easy

AUTHOR

Michael Somos, May 07 1999

EXTENSIONS

Edited by Charles R Greathouse IV, Apr 23 2010

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 29 21:48 EDT 2017. Contains 284288 sequences.