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%I #121 Sep 15 2022 02:23:20
%S 1,3,7,11,17,23,31,39,49,59,71,83,97,111,127,143,161,179,199,219,241,
%T 263,287,311,337,363,391,419,449,479,511,543,577,611,647,683,721,759,
%U 799,839,881,923,967,1011,1057,1103,1151,1199,1249,1299,1351,1403
%N a(n) = floor(n^2/2) - 1.
%C 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
%C From _Dmitry Kamenetsky_, Nov 29 2006: (Start)
%C 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:
%C 1 0 1 1 1
%C 1 0 1 0 1
%C 1 0 1 0 1
%C 1 0 0 0 1
%C 1 1 1 1 1 (End)
%C 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
%C a(n) = number of (w,x,y) with all terms in {0,...,n} and w = |x+y-w|. - _Clark Kimberling_, Jun 11 2012
%C 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
%C Also the longest path length in the n X n white bishop graph. - _Eric W. Weisstein_, Mar 27 2018
%C 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
%H Reinhard Zumkeller, <a href="/A047838/b047838.txt">Table of n, a(n) for n = 2..10000</a>
%H Laurent Bulteau, Samuele Giraudo and Stéphane Vialette, <a href="http://igm.univ-mlv.fr/~giraudo/Data/Papers/Disorders%20and%20permutations.pdf">Disorders and permutations </a>, 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Article No. 18; pp. 18:1-18:14.
%H Graham Cormode, <a href="/A047838/a047838.txt">Notes on the organization number of a permutation</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LongestPathProblem.html">Longest Path Problem</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WhiteBishopGraph.html">White Bishop Graph</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F a(2)=1; for n > 2, a(n) = a(n-1) + n - 1 + (n-1 mod 2). - _Benoit Cloitre_, Jan 12 2003
%F 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
%F 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
%F G.f.: x^2*(1+x+x^2-x^3)/((1-x)^3*(1+x)). - _R. J. Mathar_, Sep 09 2008
%F a(n) = floor((n^2 + 4*n + 2)/2). - _Gary Detlefs_, Feb 10 2010
%F a(n) = abs(A188653(n)). - _Reinhard Zumkeller_, Apr 13 2011
%F a(n) = (2*n^2 + (-1)^n - 5)/4. - _Bruno Berselli_, Sep 14 2011
%F a(n) = a(-n) = A007590(n) - 1.
%F a(n) = A080827(n) - 2. - _Kevin Ryde_, Aug 24 2013
%F a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n > 4. - _Wesley Ivan Hurt_, Aug 06 2015
%F a(n) = A000217(n-1) + A004526(n-2), for n > 1. - _J. Stauduhar_, Oct 20 2017
%F From _Guenther Schrack_, May 12 2018: (Start)
%F Set a(0) = a(1) = -1, a(n) = a(n-2) + 2*n - 2 for n > 1.
%F a(n) = A000982(n-1) + n - 2 for n > 1.
%F a(n) = 2*A033683(n) - 3 for n > 1.
%F a(n) = A061925(n-1) + n - 3 for n > 1.
%F a(n) = A074148(n) - n - 1 for n > 1.
%F a(n) = A105343(n-1) + n - 4 for n > 1.
%F a(n) = A116940(n-1) - n for n > 1.
%F a(n) = A179207(n) - n + 1 for n > 1.
%F a(n) = A183575(n-2) + 1 for n > 2.
%F a(n) = A265284(n-1) - 2*n + 1 for n > 1.
%F a(n) = 2*A290743(n) - 5 for n > 1. (End)
%F E.g.f.: 1 + x + ((x^2 + x - 2)*cosh(x) + (x^2 + x - 3)*sinh(x))/2. - _Stefano Spezia_, May 06 2021
%F 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
%e 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 + ...
%p seq(floor((n^2+4*n+2)/2), n=0..20) # _Gary Detlefs_, Feb 10 2010
%t Table[Floor[n^2/2] - 1, {n, 2, 60}] (* _Robert G. Wilson v_, Aug 31 2006 *)
%t LinearRecurrence[{2, 0, -2, 1}, {1, 3, 7, 11}, 60] (* _Harvey P. Dale_, Jan 16 2015 *)
%t Floor[Range[2, 20]^2/2] - 1 (* _Eric W. Weisstein_, Mar 27 2018 *)
%t Table[((-1)^n + 2 n^2 - 5)/4, {n, 2, 20}] (* _Eric W. Weisstein_, Mar 27 2018 *)
%t CoefficientList[Series[(-1 - x - x^2 + x^3)/((-1 + x)^3 (1 + x)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Mar 27 2018 *)
%o (PARI) a(n) = n^2\2 - 1
%o (Magma) [Floor(n^2/2)-1 : n in [2..100]]; // _Wesley Ivan Hurt_, Aug 06 2015
%Y Complement of A047839. First difference is A052928.
%Y Cf. A000217, A007590, A080827, A134151, A135151, A135152, A188653.
%Y Partial sums: A213759(n-1) for n > 1. - _Guenther Schrack_, May 12 2018
%K nonn,easy
%O 2,2
%A _Michael Somos_, May 07 1999
%E Edited by _Charles R Greathouse IV_, Apr 23 2010
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