A047838 a(n) = floor(n^2/2) - 1.

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

Offset: 2

Author

Michael Somos, May 07 1999

Keywords

• nonn
• easy

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:

1 0 1 1 1

1 0 1 0 1

1 0 1 0 1

1 0 0 0 1

1 1 1 1 1 (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 also represents 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 overlaping two or more segments. In this case, a(0)=0, a(1)=1, a(2)=3, and so on. - Marco Ripà, Jan 28 2014

Also the longest path length in the n X n white bishop graph. - Eric W. Weisstein, Mar 27 2018

a(n) is the number of right triangles with sides n*(h-floor(h)), floor(h) and h, where h is the hypotenuse. - Andrzej Kukla, Apr 14 2021

Examples

			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 + ...
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 2..10000
• Laurent Bulteau, Samuele Giraudo and Stéphane Vialette, Disorders and permutations , 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Article No. 18; pp. 18:1-18:14.
• Graham Cormode, Notes on the organization number of a permutation.
• Eric Weisstein's World of Mathematics, Longest Path Problem.
• Eric Weisstein's World of Mathematics, White Bishop Graph.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Complement of A047839. First difference is A052928.

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

Partial sums: A213759(n-1) for n > 1. - Guenther Schrack, May 12 2018

Programs

Magma

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

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 *)
Floor[Range[2, 20]^2/2] - 1 (* Eric W. Weisstein, Mar 27 2018 *)
Table[((-1)^n + 2 n^2 - 5)/4, {n, 2, 20}] (* Eric W. Weisstein, Mar 27 2018 *)
CoefficientList[Series[(-1 - x - x^2 + x^3)/((-1 + x)^3 (1 + x)), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 27 2018 *)

PARI

a(n) = n^2\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

a(n) = 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

a(n) = A000217(n-1) + A004526(n-2), for n > 1. - J. Stauduhar, Oct 20 2017

From Guenther Schrack, May 12 2018: (Start)

Set a(0) = a(1) = -1, a(n) = a(n-2) + 2*n - 2 for n > 1.

a(n) = A000982(n-1) + n - 2 for n > 1.

a(n) = 2*A033683(n) - 3 for n > 1.

a(n) = A061925(n-1) + n - 3 for n > 1.

a(n) = A074148(n) - n - 1 for n > 1.

a(n) = A105343(n-1) + n - 4 for n > 1.

a(n) = A116940(n-1) - n for n > 1.

a(n) = A179207(n) - n + 1 for n > 1.

a(n) = A183575(n-2) + 1 for n > 2.

a(n) = A265284(n-1) - 2*n + 1 for n > 1.

a(n) = 2*A290743(n) - 5 for n > 1. (End)

E.g.f.: 1 + x + ((x^2 + x - 2)*cosh(x) + (x^2 + x - 3)*sinh(x))/2. - Stefano Spezia, May 06 2021

Sum_{n>=2} 1/a(n) = 3/2 + tan(sqrt(3)*Pi/2)*Pi/(2*sqrt(3)) - cot(Pi/sqrt(2))*Pi/(2*sqrt(2)). - Amiram Eldar, Sep 15 2022

Extensions

Edited by Charles R Greathouse IV, Apr 23 2010