%I A001033 M4999 N2152
%S A001033 1,16,25,33,49,52,64,73,97,100,121,148,169,177,193,196,241,244,249,256,
%T A001033 276,289,292,297,313,337,361,388,393,400,409,457,481,484,528,529,537,
%U A001033 577,592,625,628,649,673,676,708,724,753,772,784,793,832,841,852,897,913,
               961,964,976,996
%N A001033 Numbers n such that the sum of the squares of n consecutive odd numbers 
               x^2 + (x+2)^2 + ... +(x+2n-2)^2 = k^2 for some integer k. The least 
               values of x and k for each n are in A056131 and A056132, respectively.
%C A001033 Papers by Sollfrey, Hunter and Makowski correct and extend the work of 
               Alfred. However, they do not consider n = 97, 241, 244, 276, 528 
               and 832, which are in this sequence. I have verified that there are 
               no other n<1000. - T. D. Noe (noe(AT)sspectra.com), Oct 24 2007
%C A001033 A134419 shows how A001032 and this sequence are related. - T. D. Noe, 
               Nov 04 2007
%D A001033 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001033 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001033 U. Alfred, Sums of squares of consecutive odd integers, Math. Mag., 40 
               (1967), 194-199.
%D A001033 J. A. H. Hunter, A note on sums of squares of consecutive odd numbers, 
               Math. Mag. 42 (1969), 145.
%D A001033 Andrzej Makowski, Remark on the paper "Sums of squares of consecutive 
               odd numbers", Math. Mag. 43 (1970), 212-213.
%D A001033 William Sollfrey, Note on sums of squares of consecutive odd numbers, 
               Math. Mag. 41 (1968), 255-258.
%H A001033 T. D. Noe, <a href="b001033.txt">Table of n, a(n) for n=1..100</a>
%F A001033 We must solve m*(3*x^2+6*m*x-6*x+4*m^2-6*m+2)/3 = k^2 in integers (x, 
               m, k). - N. J. A. Sloane (njas(AT)research.att.com).
%F A001033 For a given n, we must determine whether the generalized Pell equation 
               4n*y^2 + 4y*n^2 + n(4n^2-1)/3 = k^2 has any integer solutions with 
               y>=0. Note that x=2y+1 will be the first odd number being squared. 
               If there are solutions then n is in this sequence. - T. D. Noe (noe(AT)sspectra.com), 
               Oct 24 2007
%e A001033 a(1) = 1 from 1^2. a(2) = 16 from 27^2 + 29^2 + ... + 55^2 + 57^2 = 172^2. 
               a(4) = 33 from 91^2 + 93^2 + ... + 153^2 + 155^2 = 715^2.
%Y A001033 Cf. A056131, A056132.
%Y A001033 Sequence in context: A095409 A111026 A124186 this_sequence A100647 A030666 
               A030676
%Y A001033 Adjacent sequences: A001030 A001031 A001032 this_sequence A001034 A001035 
               A001036
%K A001033 nonn,nice,easy
%O A001033 1,2
%A A001033 N. J. A. Sloane (njas(AT)research.att.com).
%E A001033 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com)
%E A001033 Corrected and extended by T. D. Noe (noe(AT)sspectra.com), Oct 24 2007