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A059255
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Both sum of n+1 consecutive squares and sum of the immediately following n consecutive squares.
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10
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0, 25, 365, 2030, 7230, 19855, 45955, 94220, 176460, 308085, 508585, 802010, 1217450, 1789515, 2558815, 3572440, 4884440, 6556305, 8657445, 11265670, 14467670, 18359495, 23047035, 28646500, 35284900, 43100525, 52243425, 62875890, 75172930, 89322755, 105527255, 124002480
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OFFSET
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0,2
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COMMENTS
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In 1879, Dostor gave formulas for all solutions -- see the Dickson link. - Jonathan Sondow, Jun 21 2014
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REFERENCES
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Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 68 at p. 152.
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LINKS
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FORMULA
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a(n) = n*(n + 1)*(2n + 1)*(12n^2 + 12n + 1)/6.
G.f.: (5x(1+x)(5+x(38+5x)))/(x-1)^6.
a(0)=0, a(1)=25, a(2)=365, a(3)=2030, a(4)=7230, a(5)=19855, a(n) = 6a(n-1)-15a(n-2)+20a(n-3)-15a(n-4)+6a(n-5)-a(n-6). (End)
a(n) = (4*T(n)-n)^2+(4*T(n)-n+1)^2+...+(4*T(n))^2 = (4*T(n)+1)^2+(4*T(n)+2)^2+...+(4*T(n)+n)^2, where T = A000217. See Boardman (2000). - Jonathan Sondow, Mar 07 2013
a(0)=0, a(n) = 25 + 340*C(n-1,1) + 1325*C(n-1,2) + 2210*C(n-1,3) + 1680*C(n-1,4) + 480*C(n-1,5) for n >= 1, where C(a,b) are binomial coefficients. - Kieren MacMillan, Sep 16 2014
E.g.f.: exp(x)*x*(150 + 945*x + 1010*x^2 + 300*x^3 + 24*x^4)/6. - Stefano Spezia, Aug 05 2024
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EXAMPLE
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a(3) = 2030 = 21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2.
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MAPLE
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MATHEMATICA
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Table[1/6(-1+n)(-n+14n^2-36n^3+24n^4), {n, 40}] (* or *) LinearRecurrence[ {6, -15, 20, -15, 6, -1}, {0, 25, 365, 2030, 7230, 19855}, 40] (* Harvey P. Dale, May 09 2011 *)
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PROG
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(Magma) [n*(n+1)*(2*n+1)*(12*n^2+12*n+1)/6 : n in [0..50]]; // Wesley Ivan Hurt, Jun 21 2014
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CROSSREFS
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The n+1 consecutive squares start with the square of A014105, while the n consecutive squares start with the square of A001844.
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KEYWORD
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nice,nonn,easy,changed
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AUTHOR
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STATUS
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approved
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