Search: id:A002414
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%I A002414 M4609 N1966
%S A002414 1,9,30,70,135,231,364,540,765,1045,1386,1794,2275,2835,3480,4216,5049,
%T A002414 5985,7030,8190,9471,10879,12420,14100,15925,17901,20034,22330,24795,
%U A002414 27435,30256,33264,36465,39865,43470,47286,51319,55575,60060,64780
%N A002414 Octagonal pyramidal numbers: n(n+1)(2n-1)/2.
%C A002414 Number of ways of covering 2n x 2n lattice with 2n^2 dominoes with exactly 
               2 horizontal dominoes.
%C A002414 Equals binomial transform of [1, 8, 13, 6, 0, 0, 0,...]. - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Jun 14 2008
%C A002414 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Mar 07 
               2009: (Start)
%C A002414 Sequence of the absolute values of the z^1 coefficients of the polynomials 
               in the GF3 denominators of A156927. See A157704 for background information.
%C A002414 (End)
%D A002414 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002414 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002414 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.
%D A002414 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, 
               p. 194.
%D A002414 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 
               256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see 
               vol. 2, p. 2.
%D A002414 M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Physical 
               Review, 124 (1961), 1664-1672.
%D A002414 P.W. Kasteleyn, The statistics of dimers on a lattice, Physica, 27(1961), 
               1209-1225.
%H A002414 T. D. Noe, <a href="b002414.txt">Table of n, a(n) for n=1..1000</a>
%H A002414 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 A002414 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 A002414 <a href="Sindx_Do.html#domino">Index entries for sequences related to 
               dominoes</a>
%F A002414 a(n) = odd numbers * triangular numbers - Xavier Acloque Oct 27 2003
%F A002414 a(n)= n*(n+1)*(2*n-1)/2, n>=1. G.f.: x*(1+5*x)/(1-x)^4.
%F A002414 (2*n+1)*binomial(2+n,2). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               May 12 2006
%F A002414 a(n) = A000578(n) + A000217(n-1) - Kieren MacMillan (kieren(AT)alumni.rice.edu), 
               Mar 19 2007
%e A002414 a(2) = 9 since there are 9 ways to cover a 4 X 4 lattice with 8 dominoes, 
               2 of which is horizontal and the other 6 are vertical.
%p A002414 A002414 := n-> 1/2*n*(n+1)*(2*n-1);
%p A002414 A002414:=(1+5*z)/(z-1)**4; [Conjectured by S. Plouffe in his 1992 dissertation.]
%p A002414 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Mar 07 
               2009: (Start)
%p A002414 nmax:=38; for n from 0 to nmax do fz(n):=product((1-(k+1)*z)^(1+3*k), 
               k=0..n); c(n):= abs(coeff(fz(n),z,1)); end do: a:=n-> c(n): seq(a(n), 
               n=0..nmax);
%p A002414 (End)
%p A002414 a:=n->sum (j*(n+1)+n*(j-1),j=0..n): seq(a(n),n=1..40);# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 18 2009]
%t A002414 f[n_]:=6*n+1; s1=s2=0;lst={};Do[a=f[n];s1+=a;s2+=s1;AppendTo[lst,s2],
               {n,0,6!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 
               25 2009]
%t A002414 Table[Sum[(n^2 - i), {i, 0, n}], {n, 1, 40}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jul 11 2009]
%Y A002414 Cf. A004003.
%Y A002414 Cf. A002411.
%Y A002414 Cf. A093563 (( 6, 1) Pascal, column m=3). A000567 (differences).
%Y A002414 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Mar 07 
               2009: (Start)
%Y A002414 Cf. A156927, A157704.
%Y A002414 (End)
%Y A002414 Sequence in context: A005919 A084370 A000439 this_sequence A000440 A161684 
               A054310
%Y A002414 Adjacent sequences: A002411 A002412 A002413 this_sequence A002415 A002416 
               A002417
%K A002414 nonn,easy,nice
%O A002414 1,2
%A A002414 N. J. A. Sloane (njas(AT)research.att.com).
%E A002414 Additional comments from Yong Kong (ykong(AT)curagen.com), May 06 2000
%E A002414 More terms from Larry Reeves (larryr(AT)acm.org), May 09 2000

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