%I #16 Nov 16 2017 09:39:07

%S 14,29,30,50,54,55,77,86,90,91,110,126,135,139,140,149,174,190,194,

%T 199,203,204,230,245,255,271,280,284,285,294,302,330,355,365,366,371,

%U 380,384,385,415,434,446,451,476,492,501,505,506,509,510,534,559,590,595

%N Numbers that can be written as a sum of at least 3 consecutive squares.

%C Numbers of the form (a(a+1)(2a+1)-b(b+1)(2b+1))/6 where a >= b+3 and b >= 0. - _Robert Israel_, Jul 18 2017

%H Robert Israel, <a href="/A174070/b174070.txt">Table of n, a(n) for n = 1..10000</a>

%e 14 = 1^2 + 2^2 + 3^2, 29 = 2^2 + 3^2 + 4^2.

%e 30 = 1^2 + 2^2 + 3^2 + 4^2, 50 = 3^2 + 4^2 + 5^2.

%p N:= 1000: # to get all terms <= N

%p R:= [seq(b*(b+1)*(2*b+1)/6, b=0..ceil(sqrt(N/3)))]:

%p sort(convert(select(`<=`, {seq(seq(R[i]-R[j],j=1..i-3),i=1..nops(R))},N),list)); # _Robert Israel_, Jul 18 2017

%t max=50^2;lst={};Do[z=n^2+(n+1)^2;Do[z+=(n+x)^2;If[z>max,Break[]];AppendTo[lst,z],{x,2,max/2}],{n,max/2}];Union[lst]

%t (* Second program: *)

%t Function[s, Function[t, Union@ Flatten@ Map[TakeWhile[#, # < t[[1, -1]] &] &, t]]@ Map[Total /@ Partition[s, #, 1] &, Range[3, Length@ s]]][Range[16]^2] (* _Michael De Vlieger_, Jul 18 2017 *)

%t Module[{nn=30,sq},sq=Range[nn]^2;Take[Union[Flatten[Table[Total/@ Partition[ sq,n,1],{n,3,nn-2}]]],2nn]] (* _Harvey P. Dale_, Nov 16 2017 *)

%Y Cf. A111774, A138591, A174069.

%K nonn

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, Mar 06 2010