cp's OEIS Frontend

A143166 a(n) = n*(8*n^2 + 1)/3.

%I A143166 #38 Aug 07 2025 14:47:52
%S A143166 0,3,22,73,172,335,578,917,1368,1947,2670,3553,4612,5863,7322,9005,
%T A143166 10928,13107,15558,18297,21340,24703,28402,32453,36872,41675,46878,
%U A143166 52497,58548,65047,72010,79453,87392,95843,104822,114345,124428,135087,146338,158197
%N A143166 a(n) = n*(8*n^2 + 1)/3.
%C A143166 One fourth of the sum of p^2 + q^2 over the discrete square frame of length 2*n centered around the origin (called the 2n-frame). See the Wolfdieter Lang link below.
%C A143166 Because the summation over p*q becomes zero due to symmetry, this is also the sum over, e.g., (p+q)^2.
%C A143166 The total number of sites (vertices) s(n) of a square around (0,0) with length 2*n, is (2*n+1)^2. The 2n-frame borders 8*n = s(n) - s(n-1) sites, for n>=1. For n=0 only the site (0,0) is considered.
%C A143166 The author was led to consider such sums by a (much more difficult) question asked by R. Thomale.
%C A143166 Convolution of 4*j-1 with 4*j-3, j=1..n. For n=4: [1,5,9,13] convolved with [3,7,11,15] gives a(4) = 1*(15) + 5*(11) + 9*(7) + 13*(3) = 172. - _J. M. Bergot_, May 27 2017
%H A143166 Vincenzo Librandi, <a href="/A143166/b143166.txt">Table of n, a(n) for n = 0..1000</a>
%H A143166 Wolfdieter Lang, <a href="/A143166/a143166.txt">The squares for n=0..4</a>.
%H A143166 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A143166 a(n) = (1/4)*(S(n) - S(n-1)), with a(0)=0 and S(n):=sum(sum(p^2+q^2,p=-n..+n),q=-n..+n) = 2*sum(sum((p^2,p=-n..+n), q=-n..+n) = 2*sum(p^2,p=-n..+n)*sum(1,q=-n..n) = 2*2*(n*(n+1)*(2*n+1))/6)*(2*n+1) = (2/3)*n*(n+1)*(2*n+1)^2.
%F A143166 a(n) = n*(8*n^2 + 1)/3.
%F A143166 G.f.: x*(3 + 10*x + 3*x^2)/(1-x)^4. - _Vincenzo Librandi_, Feb 05 2014
%F A143166 a(n) = T(n, 0) + 2*(Sum_{k=1..n-1}T(n,k)) + T(n,k), with the triangle T(n, k) = A069011(n, k) = n^2 + k^2. - _Wolfdieter Lang_, Aug 15 2019
%F A143166 From _Elmo R. Oliveira_, Aug 07 2025: (Start)
%F A143166 E.g.f.: exp(x)*x*(9 + 24*x + 8*x^2)/3.
%F A143166 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
%F A143166 a(n) = A118465(n)/3. (End)
%e A143166 The total sums S(n) are [0, 12, 100, 392, 1080, 2420, 4732, 8400, 13872, 21660, 32340, ...].
%e A143166 The 2n-frame sums are 4*a(n) = [0, 12, 88, 292, 688, 1340, 2312, 3668, 5472, 7788, 10680, 14212, 18448, 23452, 29288, 36020, 43712, 52428, 62232, 73188, 85360]. The sum is over 8*n numbers.
%e A143166 For n=1 the 8 numbers of the 2-frame are 2,1,2; 1,1; 2,1,2, summing to 4*a(1)=12.
%p A143166 A143166:=n->n*(8*n^2+1)/3; seq(A143166(n), n=0..50); # _Wesley Ivan Hurt_, Feb 03 2014
%t A143166 Table[n (8 n^2 + 1)/3, {n, 0, 50}] (* _Wesley Ivan Hurt_, Feb 03 2014 *)
%t A143166 CoefficientList[Series[x (3 + 10 x + 3 x^2)/(1 - x)^4,{x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 05 2014 *)
%o A143166 (Magma) [n*(8*n^2+1)/3: n in [0..40]]; // _Vincenzo Librandi_, Feb 05 2014
%Y A143166   Cf. A069011, A118465.
%K A143166 nonn,easy
%O A143166 0,2
%A A143166 _Wolfdieter Lang_, Sep 15 2008