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%I #37 Aug 06 2024 04:37:58
%S 1,21,131,471,1251,2751,5321,9381,15421,24001,35751,51371,71631,97371,
%T 129501,169001,216921,274381,342571,422751,516251,624471,748881,
%U 891021,1052501,1235001,1440271
%N Crystal ball sequence for A_4 lattice.
%C Partial sums of A008383.
%H T. D. Noe, <a href="/A008384/b008384.txt">Table of n, a(n) for n = 0..1000</a>
%H J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).
%H H. D. Nguyen, D. Taggart, <a href="https://citeseerx.ist.psu.edu/pdf/8f2f36f22878c984775ed04368b8893879b99458">Mining the OEIS: Ten Experimental Conjectures</a>, 2013; Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
%H <a href="/index/Cor#crystal_ball">Index entries for crystal ball sequences</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = 1 +5*n*(n+1)*(7*n^2+7*n+10)/12. - _T. D. Noe_, Apr 29 2007
%F G.f.: (-1-x^4-16*x^3-36*x^2-16*x)/(x-1)^5. [Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009]
%F a(n) = 5*a(n-1)-10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5), n> 4. - _Harvey P. Dale_, Aug 22 2011
%t Table[1/12 (12-50 n+85 n^2-70 n^3+35 n^4),{n,30}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,21,131,471,1251},30] (* _Harvey P. Dale_, Aug 22 2011 *)
%o (PARI) a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,-5,10,-10,5]^n*[1;21;131;471;1251])[1,1] \\ _Charles R Greathouse IV_, Jun 15 2015
%Y Cf. A008383.
%K nonn,nice,easy
%O 0,2
%A _N. J. A. Sloane_ and _J. H. Conway_
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