cp's OEIS Frontend

A000413 Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)*Pi*n^(3/2), P(n) = A(n) - V(n); A000092 gives values of n where |P(n)| sets a new record; sequence gives A(A000092(n)).

%I A000413 M4367 N1833 #39 Jul 02 2022 14:53:27
%S A000413 1,7,19,57,81,251,437,691,739,1743,3695,6619,8217,9771,14771,15155,
%T A000413 16831,18805,26745,30551,41755,46297,54339,72359,86407,96969,131059,
%U A000413 344859,395231,519963,607141,677397,741509,893019,917217,1288415,1406811,1789599,1827927,3085785,3216051,3444439,3524869
%N A000413 Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)*Pi*n^(3/2), P(n) = A(n) - V(n); A000092 gives values of n where |P(n)| sets a new record; sequence gives A(A000092(n)).
%C A000413 The initial value a(0) = 1 corresponds to an initial A000092(0) = 0 which is the index of a record in the sense that the value P(0) = 0 is larger than all preceding values, because there are none. - _M. F. Hasler_, May 04 2022
%D A000413 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000413 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000413 W. C. Mitchell, <a href="http://dx.doi.org/10.1090/S0025-5718-1966-0195834-3">The number of lattice points in a k-dimensional hypersphere</a>, Math. Comp., 20 (1966), 300-310.
%F A000413 a(n) = A117609(A000092(n)), considering A000092(0) = 0. - _M. F. Hasler_, May 04 2022
%t A000413 P[n_] := (s = Sum[SquaresR[3, k], {k, 0, n}]) - Round[(4/3)*Pi*n^(3/2)]; record = 0; A000092 = Reap[For[n = 0, n <= 10^4, n++, If[(p = Abs[P[n]]) > record, record = p; Print[s]; Sow[s]]]][[2, 1]] (* _Jean-François Alcover_, Feb 08 2016, after _M. F. Hasler_ in A000092 *)
%Y A000413 Cf. A000323, A000036, A000092, A000099, A000223.
%Y A000413 Cf. A117609 (A(n) in name).
%K A000413 nonn
%O A000413 0,2
%A A000413 _N. J. A. Sloane_
%E A000413 Revised Jun 28 2005
%E A000413 a(37)-a(42) from _Vincenzo Librandi_, Aug 21 2016