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A007204 Crystal ball sequence for D_4 lattice.

%I A007204 M5182 #100 Feb 15 2024 04:41:01
%S A007204 1,25,169,625,1681,3721,7225,12769,21025,32761,48841,70225,97969,
%T A007204 133225,177241,231361,297025,375769,469225,579121,707281,855625,
%U A007204 1026169,1221025,1442401,1692601,1974025,2289169,2640625,3031081,3463321,3940225,4464769,5040025
%N A007204 Crystal ball sequence for D_4 lattice.
%C A007204 Equals binomial transform of [1, 24, 120, 192, 96, 0, 0, 0, ...]. - _Gary W. Adamson_, Aug 13 2009
%C A007204 Hypotenuse of Pythagorean triangles with hypotenuse a square: A057769(n)^2 + A069074(n-1)^2 = a(n)^2. - _Martin Renner_, Nov 12 2011
%C A007204 Numbers n such that n*x^4 + x^2 + 1 is reducible. - _Arkadiusz Wesolowski_, Nov 04 2013
%D A007204 Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 106, table 53.
%D A007204 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007204 T. D. Noe, <a href="/A007204/b007204.txt">Table of n, a(n) for n = 0..1000</a>
%H A007204 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 A007204 <a href="/index/Cor#crystal_ball">Index entries for crystal ball sequences</a>
%H A007204 <a href="/index/Da#D4">Index entries for sequences related to D_4 lattice</a>
%H A007204 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A007204 G.f.: (1 + 54*x^2 + 20*x + 20*x^3 + x^4)/(1-x)^5.
%F A007204 a(0)=1, a(1)=25, a(2)=169, a(3)=625, a(4)=1681, a(n)=5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Harvey P. Dale_, Mar 03 2013
%F A007204 Sum_{n>=0} 1/a(n) = Pi*(sinh(Pi) - Pi)/(2*(cosh(Pi) + 1)) = 1.0487582722070177... - _Ilya Gutkovskiy_, Nov 18 2016
%F A007204 a(n) = A016754(n) + A060300(n). - _Bruce J. Nicholson_, Apr 14 2017
%F A007204 a(n) = A001844(n)^2 = (2*n^2+2*n+1)^2. - _Bruce J. Nicholson_, May 15 2017
%F A007204 a(n) = A000583(n+1) + A099761(n) + 2*A005563(n-1)*A000290(n). - _Charlie Marion_, Jan 14 2021
%F A007204 E.g.f.: exp(x)*(1 + 24*x + 60*x^2 + 32*x^3 + 4*x^4). - _Stefano Spezia_, Jun 06 2021
%p A007204 A007204:=n->(2*n^2+2*n+1)^2; seq(A007204(n), n=0..30);
%t A007204 Table[(2n^2+2n+1)^2,{n,0,30}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,25,169,625,1681},40] (* _Harvey P. Dale_, Mar 03 2013 *)
%o A007204 (Magma) [(2*n^2+2*n+1)^2: n in [0..40]]; // _Vincenzo Librandi_, Nov 18 2016
%o A007204 (PARI) a(n)=(2*n^2+2*n+1)^2 \\ _Charles R Greathouse IV_, Feb 08 2017
%Y A007204 Cf. A016754, A057769, A060300, A069074.
%Y A007204 Cf. A000290, A000583, A001844, A005563, A099761.
%K A007204 nonn,easy,nice
%O A007204 0,2
%A A007204 _N. J. A. Sloane_ and _J. H. Conway_, Apr 28 1994
%E A007204 More terms from _Harvey P. Dale_, Mar 03 2013