A027575 - OEIS

%I #82 Sep 15 2022 02:24:24

%S 14,30,54,86,126,174,230,294,366,446,534,630,734,846,966,1094,1230,

%T 1374,1526,1686,1854,2030,2214,2406,2606,2814,3030,3254,3486,3726,

%U 3974,4230,4494,4766,5046,5334,5630,5934,6246,6566,6894,7230,7574,7926,8286,8654,9030

%N a(n) = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2.

%C Summation of n^2 taken 4 at a time. - Al Hakanson (hawkuu(AT)gmail.com), May 20 2009

%C Terms are congruent to (2,0,0) mod 6. - _Ezhilarasu Velayutham_, Apr 04 2019

%H G. C. Greubel, <a href="/A027575/b027575.txt">Table of n, a(n) for n = 0..1000</a>

%H Patrick De Geest, <a href="http://www.worldofnumbers.com/sumsquare.htm">Palindromic Sums of Squares of Consecutive Integers</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>.

%F a(n) = 4*n^2 + 12*n + 14. - Al Hakanson (hawkuu(AT)gmail.com), May 20 2009

%F a(n) = a(n-1) + 8*(n+1) for n>0, a(0)=14. - _Vincenzo Librandi_, Nov 19 2010

%F G.f.: 2*(7-6*x+3*x^2)/(1-x)^3. - _Colin Barker_, Feb 17 2012

%F From _Jean-Christophe Hervé_, Nov 11 2015: (Start)

%F a(n) = (2*n+3)^2 + 5 = A016754(n+1) + 5, hence a(n) is never square.

%F The last formula defines a(n) for n < 0; then we have a(-n) = a(n-3) for all n. (End)

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Wesley Ivan Hurt_, Apr 16 2021

%F E.g.f.: 2*(7 + 8*x + 2*x^2)*exp(x). - _G. C. Greubel_, Aug 25 2022

%F Sum_{n>=0} 1/a(n) = tanh(sqrt(5)*Pi/2)*Pi/(4*sqrt(5)) - 1/6. - _Amiram Eldar_, Sep 15 2022

%t Table[n^2 + (n + 1)^2 + (n + 2)^2 + (n + 3)^2, {n, 0, 42}] (* _Alonso del Arte_, Feb 17 2012 *)

%t Table[Total[Range[n,n+3]^2],{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{14,30,54},50] (* _Harvey P. Dale_, Jan 23 2017 *)

%t Total/@Partition[Range[0,50]^2,4,1] (* _Harvey P. Dale_, Feb 08 2020 *)

%o (Sage) [i^2+(i+1)^2+(i+2)^2+(i+3)^2 for i in range(0,50)] # _Zerinvary Lajos_, Jul 03 2008

%o (PARI) vector(100, n, n--; n^2+(n+1)^2+(n+2)^2+(n+3)^2) \\ _Altug Alkan_, Nov 11 2015

%o (Magma) [2*(2*n^2 +6*n +7): n in [0..50]]; // _G. C. Greubel_, Aug 25 2022

%Y Cf. A000290, A001844, A027578, A027865, A120328, A260637, A276026.

%Y Cf. A016754, A027577.

%K nonn,easy

%O 0,1

%A _Patrick De Geest_