This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A084569 #17 Mar 19 2025 05:57:54
%S A084569 1,3,9,21,44,82,142,230,355,525,751,1043,1414,1876,2444,3132,3957,
%T A084569 4935,6085,7425,8976,10758,12794,15106,17719,20657,23947,27615,31690,
%U A084569 36200,41176,46648,52649,59211,66369,74157,82612,91770,101670,112350,123851
%N A084569 Partial sums of A084570.
%C A084569 Conjecture: a(n) is the number of perimeter-magic (hollow) squares of order 3 with magic sum n+3. Order 3 means each of the 4 edges has 3 elements >=1; the square has 8 elements. The elements do not need to be distinct, and squares obtained by rotations are counted only once. The square (read ccw) for magic sum 3 has elements 1 1 1 1 1 1 1 1. The 3 squares with magic sum 4 are 1 1 2 1 1 1 2 1, 1 1 2 1 1 2 1 2 and 1 2 1 2 1 2 1 2. - _R. J. Mathar_, Mar 08 2025
%H A084569 R. J. Mathar, <a href="https://zenodo.org/records/15001365">Generating perimeter-magic polygons</a>, C++ (2025)
%H A084569 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,0,5,-4,1).
%F A084569 a(n) = (-1)^n/8 + (n^4 + 6*n^3 + 17*n^2 + 30*n + 21)/24.
%F A084569 a(n) = Sum_{k=0..n} Sum_{j=0..k} Sum_{i=0..j} (i + (-1)^i).
%F A084569 G.f.: ( -1+x-2*x^2 ) / ( (1+x)*(x-1)^5 ). - _R. J. Mathar_, Mar 08 2025
%F A084569 a(n)+a(n+1) = A116701(n+3)-1. - _R. J. Mathar_, Mar 08 2025
%t A084569 Accumulate[LinearRecurrence[{3,-2,-2,3,-1},{1,2,6,12,23},50]] (* or *) LinearRecurrence[{4,-5,0,5,-4,1},{1,3,9,21,44,82},50] (* _Harvey P. Dale_, Nov 12 2014 *)
%Y A084569 Cf. A116701.
%K A084569 easy,nonn
%O A084569 0,2
%A A084569 _Paul Barry_, May 31 2003