cp's OEIS Frontend

A027578 Sums of five consecutive squares: a(n) = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2.

%I A027578 #73 Aug 16 2023 07:43:33
%S A027578 30,55,90,135,190,255,330,415,510,615,730,855,990,1135,1290,1455,1630,
%T A027578 1815,2010,2215,2430,2655,2890,3135,3390,3655,3930,4215,4510,4815,
%U A027578 5130,5455,5790,6135,6490,6855,7230,7615,8010,8415,8830,9255,9690,10135,10590,11055
%N A027578 Sums of five consecutive squares: a(n) = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2 + (n+4)^2.
%C A027578 a(n) is defined for n < 0 and a(-n) = a(n-4) for any n; a(-3) = a(-1) = 15, a(-2) = 10. - _Jean-Christophe Hervé_, Nov 11 2015
%H A027578 Vincenzo Librandi, <a href="/A027578/b027578.txt">Table of n, a(n) for n = 0..1000</a>
%H A027578 Patrick De Geest, <a href="http://www.worldofnumbers.com/sumsquare.htm">Palindromic Sums of Squares of Consecutive Integers</a>.
%H A027578 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%H A027578 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>.
%F A027578 a(n) = 5*A059100(n+2).
%F A027578 From _Colin Barker_, Mar 29 2012: (Start)
%F A027578 G.f.: 5*(6-7*x+3*x^2)/(1-x)^3.
%F A027578 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. (End)
%F A027578 a(n) = 5*(n + 2)^2 + 10. a(n) is never square. - _Bruno Berselli_, Jul 29 2015
%F A027578 E.g.f.: 5*(6 + 5*x + x^2)*exp(x). - _G. C. Greubel_, Aug 24 2022
%F A027578 From _Amiram Eldar_, Sep 15 2022: (Start)
%F A027578 Sum_{n>=0} 1/a(n) = coth(sqrt(2)*Pi)*Pi/(10*sqrt(2)) - 7/60.
%F A027578 Sum_{n>=0} (-1)^n/a(n) = cosech(sqrt(2)*Pi)*Pi/(10*sqrt(2)) + 1/60. (End)
%p A027578 A027578:=n->5*(n+2)^2+10: seq(A027578(n), n=0..50); # _Wesley Ivan Hurt_, Nov 12 2015
%t A027578 Table[5 (n + 2)^2 + 10, {n, 0, 50}] (* _Bruno Berselli_, Jul 29 2015 *)
%t A027578 Total/@Partition[Range[0,50]^2,5,1] (* or *) LinearRecurrence[{3,-3,1},{30,55,90},50] (* _Harvey P. Dale_, Mar 06 2018 *)
%o A027578 (Sage) [i^2+(i+1)^2+(i+2)^2+(i+3)^2+(i+4)^2 for i in range(0,50)] # _Zerinvary Lajos_, Jul 03 2008
%o A027578 (Magma) [n^2+(n+1)^2+(n+2)^2+(n+3)^2+(n+4)^2: n in [0..50] ]; // _Vincenzo Librandi_, Jun 17 2011
%o A027578 (PARI) vector(100, n, n--; n^2+(n+1)^2+(n+2)^2+(n+3)^2+(n+4)^2) \\ _Altug Alkan_, Nov 11 2015
%Y A027578 Cf. A000290, A001844, A027575, A027865, A120328, A260637, A276026.
%Y A027578 Cf. A027580.
%Y A027578 Subsequence of A174069, A174070, A174071.
%K A027578 nonn,easy
%O A027578 0,1
%A A027578 _Patrick De Geest_