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A180274
Integers whose squares are the sums of 24 consecutive squares.
13
70, 106, 158, 182, 274, 430, 650, 1022, 1546, 1786, 2702, 4250, 6430, 10114, 15302, 17678, 26746, 42070, 63650, 100118, 151474, 174994, 264758, 416450, 630070, 991066, 1499438, 1732262, 2620834, 4122430, 6237050, 9810542, 14842906, 17147626, 25943582
OFFSET
1,1
COMMENTS
The corresponding starts of 24 consecutive squares to be summed are A094196.
FORMULA
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
a(n) = sqrt( 24*(A094196(n))^2 +552*A094196(n)+4324) . - R. J. Mathar, Jan 20 2011
MAPLE
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:
A180274 := proc(n) local a96 ; a96 := A094196(n) ; 24*a96^2+552*a96+4324 ; sqrt(%) ; end proc:
seq(A180274(n), n=1..30) ; # R. J. Mathar, Jan 20 2011
MATHEMATICA
Select[Sqrt[#]&/@(Total[#]&/@Partition[Range[900000]^2, 24, 1]), IntegerQ] (* Harvey P. Dale, Jan 21 2011 *)
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
PROG
(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)), ", "))) }
(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
CROSSREFS
Cf. A094196.
Cf. A001032 (24 is a term of that sequence).
Sequence in context: A007621 A353061 A051971 * A075004 A043222 A039399
KEYWORD
nonn,easy
AUTHOR
Zhining Yang, Jan 17 2011
STATUS
approved