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A180274 Integers whose squares are the sums of 24 consecutive squares.

%I A180274 #33 Aug 12 2024 01:38:30
%S A180274 70,106,158,182,274,430,650,1022,1546,1786,2702,4250,6430,10114,15302,
%T A180274 17678,26746,42070,63650,100118,151474,174994,264758,416450,630070,
%U A180274 991066,1499438,1732262,2620834,4122430,6237050,9810542,14842906,17147626,25943582
%N A180274 Integers whose squares are the sums of 24 consecutive squares.
%C A180274 The corresponding starts of 24 consecutive squares to be summed are A094196.
%H A180274 Colin Barker, <a href="/A180274/b180274.txt">Table of n, a(n) for n = 1..1000</a>
%H A180274 K. S. Brown, <a href="http://www.mathpages.com/home/kmath147.htm">Sum of Consecutive Nth Powers Equals an Nth Power</a>
%H A180274 Fun With Num3ers, <a href="https://benvitalenum3ers.wordpress.com/2016/07/27/sets-of-24-consecutive-squares-whose-sum-is-a-square/">Sets of 24 consecutive squares whose sum is a square</a>, July 27 2016.
%H A180274 V. Pletser, <a href="http://arxiv.org/abs/1409.7972">Finding all squared integers expressible as the sum of consecutive squared integers using generalized Pell equation solutions with Chebyshev polynomials</a>, arXiv preprint arXiv:1409.7972 [math.NT], 2014. See Table 2 p. 8.
%H A180274 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,10,0,0,0,0,0,-1).
%F A180274 a(n) = +10*a(n-6) -a(n-12). G.f. ( 70+106*x+158*x^2+182*x^3+274*x^4+430*x^5-50*x^6-38*x^7-34*x^8-34*x^9-38*x^10-50*x^11 ) / ( 1-10*x^6+x^12 ). - _Joerg Arndt_, Jan 17 2011
%F A180274 a(n) = sqrt( 24*(A094196(n))^2 +552*A094196(n)+4324) . - _R. J. Mathar_, Jan 20 2011
%p A180274 A094196 := proc(n) if n <= 12 then op(n,[1, 9, 20, 25, 44, 76, 121, 197, 304, 353, 540, 856]) ; else 10*procname(n-6)-procname(n-12)+92 ; end if ; end proc:
%p A180274 A180274 := proc(n) local a96 ; a96 := A094196(n) ; 24*a96^2+552*a96+4324 ; sqrt(%) ; end proc:
%p A180274 seq(A180274(n),n=1..30) ; # _R. J. Mathar_, Jan 20 2011
%t A180274 Select[Sqrt[#]&/@(Total[#]&/@Partition[Range[900000]^2, 24, 1]), IntegerQ] (* _Harvey P. Dale_, Jan 21 2011 *)
%t A180274 t={70, 106, 158, 182, 274, 430, 650, 1022, 1546, 1786, 2702, 4250}; Do[AppendTo[t, 10*t[[-6]] - t[[-12]]], {n, 13, 100}]; t
%o A180274 (PARI) { for(n=1,999999,t=((n+23)*(n+24)*(2*n+47)-n*(n-1)*(2*n-1))/6;if(issquare(t),print1(ceil(sqrt(t)),","))) }
%o A180274 (PARI) Vec(-2*x*(25*x^11+19*x^10+17*x^9+17*x^8+19*x^7+25*x^6-215*x^5-137*x^4-91*x^3-79*x^2-53*x-35) / (x^12-10*x^6+1) + O(x^100)) \\ _Colin Barker_, May 09 2015
%Y A180274 Cf. A094196.
%Y A180274 Cf. A001032 (24 is a term of that sequence).
%K A180274 nonn,easy
%O A180274 1,1
%A A180274 _Zhining Yang_, Jan 17 2011