cp's OEIS Frontend

A046895 Sizes of successive clusters in Z^4 lattice.

%I A046895 #37 Jun 21 2024 11:05:39
%S A046895 1,9,33,65,89,137,233,297,321,425,569,665,761,873,1065,1257,1281,1425,
%T A046895 1737,1897,2041,2297,2585,2777,2873,3121,3457,3777,3969,4209,4785,
%U A046895 5041,5065,5449,5881,6265,6577,6881,7361,7809,7953,8289,9057
%N A046895 Sizes of successive clusters in Z^4 lattice.
%C A046895 Number of lattice points inside or on the 4-sphere x^2 + y^2 + z^2 + u^2 = n. - _T. D. Noe_, Mar 14 2009
%H A046895 T. D. Noe and Charles R Greathouse IV, <a href="/A046895/b046895.txt">Table of n, a(n) for n = 0..10000</a> (terms up to 1000 from Noe)
%H A046895 A. Walfisz, <a href="http://dx.doi.org/10.1002/zamm.19640441217">Weylsche Exponentialsummen in der neueren Zahlentheorie</a>, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Volume 44, Issue 12, page 607, 1964.
%F A046895 a(n) = A122510(4,n). a(n^2) = A055410(n). - _R. J. Mathar_, Apr 21 2010
%F A046895 G.f.: T3(q)^4/(1-q) where T3(q) = 1 + 2*Sum_{k>=1} q^(k^2). - _Joerg Arndt_, Apr 08 2013
%F A046895 Pi^2/2 * (sqrt(n)-1)^4 < a(n) < Pi^2/2 * (sqrt(n)+1)^4 for n > 0. - _Charles R Greathouse IV_, Feb 17 2015
%F A046895 a(n) = Pi^2/2 * n^2 + O(n (log n)^(2/3)) using a result of Walfisz. - _Charles R Greathouse IV_, Feb 18 2015
%F A046895 a(n) = 1 + 8*A024916(n) - 32*A024916(floor(n/4)) by Jacobi's four-square theorem. - _Peter J. Taylor_, Jun 03 2020
%t A046895 Accumulate[ Table[ SquaresR[4, n], {n, 0, 42}]] (* _Jean-François Alcover_, May 11 2012 *)
%t A046895 QP = QPochhammer; s = (QP[q^2]^5/(QP[q]^2*QP[q^4]^2))^4/(1-q) + O[q]^50; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 25 2015, after _Joerg Arndt_ *)
%o A046895 (PARI)
%o A046895 q='q+O('q^66);
%o A046895 Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^4/(1-q))
%o A046895 /* _Joerg Arndt_, Apr 08 2013 */
%o A046895 (Python)
%o A046895 from math import isqrt
%o A046895 def A046895(n): return 1+((-(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))&-1)<<2)+(((t:=isqrt(m:=n>>2))**2*(t+1)-sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))&-1)<<4) # _Chai Wah Wu_, Jun 21 2024
%Y A046895 Partial sums of A000118.
%Y A046895 Cf. A117609
%K A046895 nonn,easy,nice
%O A046895 0,2
%A A046895 _N. J. A. Sloane_