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%I #38 Oct 05 2022 15:39:42
%S 28,35,56,91,140,203,280,371,476,595,728,875,1036,1211,1400,1603,1820,
%T 2051,2296,2555,2828,3115,3416,3731,4060,4403,4760,5131,5516,5915,
%U 6328,6755,7196,7651,8120,8603,9100,9611,10136,10675,11228,11795,12376,12971
%N Sums of seven consecutive squares: a(n) = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2 + (n+5)^2 + (n+6)^2.
%C a(n) is defined for any n in Z and a(-n) = a(n-6).
%C There are no primes or squares in the sequence because a(n) is a multiple of 7 and 7 is with multiplicity 1: a(n) = 7*((n+3)^2 + 4), and the factor (n+3)^2 + 4 is not a multiple of 7 for any n. A001032 gives the integers k such that the sum of k consecutive squares is a square.
%H Jean-Christophe Hervé, <a href="/A260637/b260637.txt">Table of n, a(n) for n = -3..1000</a>
%H Patrick De Geest, <a href="http://www.worldofnumbers.com/index.html">World!Of Numbers</a>.
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 7*n^2 + 42*n + 91 = 7*(n^2 + 6*n + 13) = 7*((n+3)^2 + 4).
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = a(n-1) + 7*(2*n+7).
%F G.f.: 7*(4 - 7*x + 5*x^2) / (x^3*(1-x)^3). - _Colin Barker_, Nov 12 2015
%F From _Amiram Eldar_, Sep 15 2022: (Start)
%F Sum_{n>=-3} 1/a(n) = coth(2*Pi)*Pi/28 + 1/56.
%F Sum_{n>=-3} (-1)^(n+1)/a(n) = cosech(2*Pi)*Pi/28 + 1/56. (End)
%p A260637:=n->7*((n+3)^2 + 4): seq(A260637(n), n=-3..50); # _Wesley Ivan Hurt_, Nov 17 2015
%t Table[Plus@@(Range[n, n + 6]^2), {n, -3, 96}]
%t Total/@Partition[Range[-3,50]^2,7,1] (* or *) LinearRecurrence[{3,-3,1},{28,35,56},50] (* _Harvey P. Dale_, Oct 05 2022 *)
%o (PARI) vector(100, n, n--; n^2+(n+1)^2+(n+2)^2+(n+3)^2+(n+4)^2+(n+5)^2+(n+6)^2).
%o (PARI) a(n) = 7*n^2 + 42*n + 91;
%o vector(50, n, a(n-4)) \\ _Altug Alkan_, Nov 11 2015
%o (PARI) Vec(-7*(5*x^2-7*x+4)/(x^3*(x-1)^3) + O(x^100)) \\ _Colin Barker_, Nov 12 2015
%o (Magma) [7*((n+3)^2 + 4) : n in [-3..50]]; // _Wesley Ivan Hurt_, Nov 17 2015
%o (SageMath) [7*((n+3)^2 +4) for n in (-3..50)] # _G. C. Greubel_, Aug 24 2022
%Y Cf. A000290, A001844, A027575, A027578, A027865, A120328, A276026.
%Y Cf. A001032.
%K nonn,easy
%O -3,1
%A _Jean-Christophe Hervé_, Nov 11 2015
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