%I A000982 M1348 N0517
%S A000982 0,1,2,5,8,13,18,25,32,41,50,61,72,85,98,113,128,145,162,181,200,221,
%T A000982 242,265,288,313,338,365,392,421,450,481,512,545,578,613,648,685,722,
%U A000982 761,800,841,882,925,968,1013,1058,1105,1152,1201,1250,1301,1352,1405
%N A000982 Ceiling(n^2/2).
%C A000982 Floor[ arithmetic mean of next n numbers]. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), 
               Mar 11 2003
%C A000982 Pairwise sums of repeated squares (A008794).
%C A000982 Also, number of topologies on n+1 unlabeled elements with exactly 4 elements 
               in the topology. a(3) gives 4 elements a,b,c,d; the valid topologies 
               are (0,a,ab,abcd), (0,a,abc,abcd), (0,ab,abc,abcd), (0,a,bcd,abcd) 
               and (0,ab,cd,abcd), with a count of 5. - Jon Perry (perry(AT)globalnet.co.uk), 
               Mar 05 2004
%C A000982 Euler transform of a(n+1) is length 4 sequence [2,2,0,-1].
%C A000982 Partition n in two parts, say r and s so that r^2 + s^2 is minimal, then 
               a(n) = r^2 +s^2. Geometrical significance: folding a rod with length 
               n units at right angles in such a way that the end points are at 
               the least distance, which is given by a(n)^(1/2) as the hypotenus 
               of a right angled triangle with the sum of the base and height = 
               n units. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 18 
               2004
%C A000982 Convolution of A002061(n)-0^n and (-1)^n. Convolution of n (A001477) 
               with {1,0,2,0,2,0,2...}. Partial sums of repeated odd numbers {0,
               1,1,3,3,5,5,...}. - Paul Barry (pbarry(AT)wit.ie), Jul 22 2004
%C A000982 The ratio of the sum of terms over the total number of terms in an n 
               X n spiral. The sum of terms of an n X n spiral is A037270, or Sum{k=0..n^2,
               k} = (n^4 + n^2)/2 and the total number of terms is n^2. - William 
               A. Tedeschi (fynmun(AT)hotmail.com), Feb 27 2008
%C A000982 Starting with offset 1 = row sums of triangle A158946. [From Gary W. 
               Adamson (qntmpkt(AT)yahoo.com), Mar 31 2009]
%C A000982 Partial sums of A109613. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Dec 05 2009]
%D A000982 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000982 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000982 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques 
               Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000982 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000982 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A000982 S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta and K.M. Tamizhmani, 
               <a href="http://arXiv.org/abs/nlin.SI/0104020">Blending two discrete 
               integrability criteria: ...</a>
%H A000982 MathWorld, <a href="http://mathworld.wolfram.com/Topology.html">Topology</
               a>
%F A000982 a(2n) = 2n^2, a(2n+1) = 2n^2 + 2n + 1.
%F A000982 a(n) = (2n^2 + 1 - (-1)^n) / 4. a(0)=0, a(1)=1, a(n+1)=n+1+max(2*floor(a(n)/
               2); 3*floor(a(n)/3)). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Nov 06 2002
%F A000982 G.f.: (x+x^2+x^3+x^4)/((1-x)*(1-x^2)^2) - Len Smiley (smiley(AT)math.uaa.alaska.edu).
%F A000982 a(n)=a(n-2)+2n-2. - Paul Barry (pbarry(AT)wit.ie), Jul 17 2004
%F A000982 G.f.: x(1+x^2)/((1-x^2)(1-x)^2)=x(1+x^2)/((1+x)(1-x)^3); a(n)=sum{k=0..n, 
               (k^2-k+1-0^k)(-1)^(n-k) }; a(n)=sum{k=0..n, ((1+(-1)^(n-k))-0^(n-k))k 
               }. - Paul Barry (pbarry(AT)wit.ie), Jul 22 2004
%F A000982 a(0) = 0, a(n+1) = a(n) + 2*floor(n/2) + 1. a(n) = A116940(n) - A005843(n). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 27 2006
%F A000982 Starting with offset 1, = row sums of triangle A134444. Also, with offset 
               1, = binomial transform of [1, 1, 2, -2, 4, -8, 16, -32,...]. - Gary 
               W. Adamson (qntmpkt(AT)yahoo.com), Oct 25 2007
%F A000982 a(n) = floor((n^2+1)/2) - William A. Tedeschi (fynmun(AT)hotmail.com), 
               Feb 27 2008
%F A000982 Contribution from Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Sep 12 
               2008: (Start)
%F A000982 a(n) = A004526(n+2) + A000217 (n)
%F A000982 (End)
%F A000982 Contribution from Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 
               05 2008: (Start)
%F A000982 a(n)=a(n-1)+a(n-2)-a(n-3)+2
%F A000982 a(n)=2a(n-1)-2a(n-3)+a(n-4) (End)
%F A000982 a(n)=A004526(n)^2 + A110654(n)^2. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Mar 12 2009]
%p A000982 A000982:=-(1+z**2)/(z+1)/(z-1)**3; [Conjectured by S. Plouffe in his 
               1992 dissertation.]
%p A000982 with (combinat):seq(count(Partition((n^2+1)), size=2), n=0..53); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Mar 28 2008
%Y A000982 Cf. A000096.
%Y A000982 Cf. A134444.
%Y A000982 Cf. A037270.
%Y A000982 A158946 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 31 2009]
%Y A000982 Sequence in context: A076145 A049617 A054074 this_sequence A122221 A083704 
               A111097
%Y A000982 Adjacent sequences: A000979 A000980 A000981 this_sequence A000983 A000984 
               A000985
%K A000982 nonn,easy,new
%O A000982 0,3
%A A000982 N. J. A. Sloane (njas(AT)research.att.com).