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%I M4609 N1966 #160 May 27 2023 18:29:11
%S 1,9,30,70,135,231,364,540,765,1045,1386,1794,2275,2835,3480,4216,
%T 5049,5985,7030,8190,9471,10879,12420,14100,15925,17901,20034,22330,
%U 24795,27435,30256,33264,36465,39865,43470,47286,51319,55575,60060,64780
%N Octagonal pyramidal numbers: a(n) = n*(n+1)*(2*n-1)/2.
%C Number of ways of covering a 2n X 2n lattice with 2n^2 dominoes with exactly 2 horizontal dominoes. - Yong Kong (ykong@curagen.com), May 06 2000
%C Equals binomial transform of [0, 1, 7, 6, 0, 0, 0, ...]. - _Gary W. Adamson_, Jun 14 2008, corrected Oct 25 2012
%C Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF3 denominators of A156927. See A157704 for background information. - _Johannes W. Meijer_, Mar 07 2009
%C This sequence is related to A000326 by a(n) = n*A000326(n) - Sum_{i=0..n-1} A000326(i) and this is the case d=3 in the identity n*(n*(d*n-d+2)/2)-Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - _Bruno Berselli_, Apr 21 2010
%C 2*a(n) gives the principal diagonal of the convolution array A213819. - _Clark Kimberling_, Jul 04 2012
%C Partial sums of the figurate octagonal numbers A000567. For each sequence with a linear recurrence with constant coefficients, the values reduced modulo some constant m generate a periodic sequence. For this sequence, these Pisano periods have length 1, 4, 3, 8, 5, 12, 7, 16, 9, 20, 11, 24, 13, 28, 15, 32, 17, ... for m >= 1. - _Ant King_, Oct 26 2012
%C Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 773", based on the 5-celled von Neumann neighborhood. - _Robert Price_, May 23 2016
%C On a square grid of side length n+1, the number of embedded rectangles (where each side is greater than 1). For example, in a 2 X 2 square there is one rectangle, in a 3 X 3 square there are nine rectangles, etc. - _Peter Woodward_, Nov 26 2017
%C a(n) is the sum of the numbers in the n X n square array A204154(n). - _Ali Sada_, Jun 21 2019
%C Sum of all multiples of n and n+1 that are <= n^2. - _Wesley Ivan Hurt_, May 25 2023
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
%D 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 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002414/b002414.txt">Table of n, a(n) for n = 1..1000</a>
%H B. Berselli, A description of the transform in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian).
%H M. E. Fisher, <a href="http://dx.doi.org/10.1103/PhysRev.124.1664">Statistical mechanics of dimers on a plane lattice</a>, Physical Review, 124 (1961), 1664-1672.
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
%H P. W. Kasteleyn, <a href="http://dx.doi.org/10.1016/0031-8914(61)90063-5">The Statistics of Dimers on a Lattice</a>, Physica, 27 (1961), 1209-1225.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>
%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = odd numbers * triangular numbers = (2*n-1)* binomial(n+1,2). - Xavier Acloque, Oct 27 2003
%F G.f.: x*(1+5*x)/(1-x)^4. [Conjectured by _Simon Plouffe_ in his 1992 dissertation.]
%F a(n) = A000578(n) + A000217(n-1). - _Kieren MacMillan_, Mar 19 2007
%F a(-n) = -A160378(n). - _Michael Somos_, Mar 17 2011
%F From _Ant King_, Oct 26 2012: (Start)
%F a(n) = a(n-1) + n*(3*n-2).
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 6.
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
%F a(n) = n*A000326(n) - A002411(n-1), see Berselli's comment.
%F a(n) = (n+1)*(2*A000567(n)+n)/6.
%F a(n) = A000292(n) + 5*A000292(n-1) = binomial(n+2,3)+5*binomial(n+1,3).
%F a(n) = A002413(n) + A000292(n-1).
%F a(n) = A000217(n) + 6*A000292(n-1).
%F Sum_{n>=1} 1/a(n) = 2*(4*log(2)-1)/3 = 1.1817258148265...
%F (End)
%F a(n) = Sum_{i=0..n-1} (n-i)*(6*i+1), with a(0)=0. - _Bruno Berselli_, Feb 10 2014
%F E.g.f.: x*(2 + 7*x + 2*x^2)*exp(x)/2. - _Ilya Gutkovskiy_, May 23 2016
%F a(n) = A080851(6,n-1). - _R. J. Mathar_, Jul 28 2016
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(Pi + 1 - 4*log(2))/3. - _Amiram Eldar_, Jul 02 2020
%e 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. - Yong Kong (ykong@curagen.com), May 06 2000
%e G.f. = x + 9*x^2 + 30*x^3 + 70*x^4 + 135*x^5 + 231*x^6 + 364*x^7 + 540*x^8 + 765*x^9 + ...
%p A002414 := n-> 1/2*n*(n+1)*(2*n-1): seq(A002414(n), n=1..100);
%t LinearRecurrence[{4,-6,4,-1},{1,9,30,70},40] (* _Harvey P. Dale_, Apr 12 2013 *)
%o (PARI) {a(n) = (2*n - 1) * n * (n + 1) / 2} \\ _Michael Somos_, Mar 17 2011
%o (Magma) [n*(n+1)*(2*n-1)/2: n in [1..50]]; // _Vincenzo Librandi_, May 24 2016
%Y Cf. A000578, A004003, A160378.
%Y Cf. A093563 (( 6, 1) Pascal, column m=3). A000567 (differences).
%Y Cf. A156927, A157704. - _Johannes W. Meijer_, Mar 07 2009
%Y Cf. A000326.
%Y Cf. similar sequences listed in A237616.
%Y Cf. A260234 (largest prime factor of a(n+1)).
%K nonn,easy,nice
%O 1,2
%A _N. J. A. Sloane_
%E More terms from Larry Reeves (larryr(AT)acm.org), May 09 2000
%E Incorrect formula deleted by _Ant King_, Oct 04 2012
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