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A262063
The first of seven consecutive positive integers the sum of the squares of which is equal to the sum of the squares of six consecutive positive integers.
2
78, 2100, 54594, 1417416, 36798294, 955338300, 24801997578, 643896598800, 16716509571294, 433985352254916, 11266902649056594, 292505483523216600, 7593875668954575078, 197148261909295735500, 5118260933972734547994, 132877636021381802512416
OFFSET
1,1
COMMENTS
For the first of the corresponding six consecutive positive integers, see A262062.
FORMULA
a(n) = 27*a(n-1)-27*a(n-2)+a(n-3) for n>3.
G.f.: 6*x*(x-13) / ((x-1)*(x^2-26*x+1)).
a(n) = (13+2*sqrt(42))^(-n)*(6-sqrt(42)+(6+sqrt(42))*(13+2*sqrt(42))^(2*n))/4-3. - Colin Barker, Mar 03 2016
EXAMPLE
78 is in the sequence because 78^2 + ... + 84^2 = 45955 = 85^2 + ... + 90^2.
MATHEMATICA
CoefficientList[Series[6 (x - 13)/((x - 1) (x^2 - 26 x + 1)), {x, 0, 20}], x] (* Vincenzo Librandi, Sep 10 2015 *)
LinearRecurrence[{27, -27, 1}, {78, 2100, 54594}, 30] (* Harvey P. Dale, May 17 2018 *)
PROG
(PARI) Vec(6*x*(x-13)/((x-1)*(x^2-26*x+1)) + O(x^20))
(Magma) I:=[78, 2100, 54594]; [n le 3 select I[n] else 27*Self(n-1)-27*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Sep 10 2015
CROSSREFS
Cf. A262062.
Sequence in context: A187590 A172217 A036524 * A140934 A133239 A210407
KEYWORD
nonn,easy
AUTHOR
Colin Barker, Sep 09 2015
STATUS
approved