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 A260302 #59 Sep 08 2022 08:46:13
%S A260302 0,0,231,1378,4753,12246,26335,50086,87153,141778,218791,323610,
%T A260302 462241,641278,867903,1149886,1495585,1913946,2414503,3007378,3703281,
%U A260302 4513510,5449951,6525078,7751953,9144226,10716135,12482506,14458753,16660878,19105471,21809710
%N A260302 Maximum water retention of a number octagon of order n.
%C A260302 A number octagon fills an octagon on a square grid with the smallest unique natural numbers.
%C A260302 The sum of the interior values for a number hexagon on a circular lattice is A079903. There are nice illustrations for this by Mathar at A257594.
%H A260302 G. C. Greubel and Craig Knecht, <a href="/A260302/b260302.txt">Table of n, a(n) for n = 1..1000</a> (Terms 1 through 32 were computed by Craig Knecht; terms 33 through 1000 by G. C. Greubel, Nov 13 2015; term 451 = 1002105368551 corrected by _Georg Fischer_, May 24 2019)
%H A260302 Craig Knecht, <a href="/A260302/a260302.jpg">Octagon volume calculation.</a>
%H A260302 Craig Knecht, <a href="/A260302/a260302_1.jpg">Volume retention of a number square.</a>
%H A260302 Wikipedia, <a href="http://en.wikipedia.org/wiki/Water_retention_on_mathematical_surfaces">Water retention on mathematical surfaces</a>
%H A260302 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A260302 a(n) = (1/2)*(7*n^2 - 18*n + 12) (7*n^2 - 18*n + 13) for n > 2.
%F A260302 From _Colin Barker_, Nov 11 2015: (Start)
%F A260302 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>7.
%F A260302 G.f.: -x^3*(10*x^4-49*x^3+173*x^2+223*x+231) / (x-1)^5. (End)
%e A260302 (22 23 24)
%e A260302 (37 1 2 3 25)
%e A260302 (36 4 5 6 7 8 26)
%e A260302 (35 9 10 11 12 13 27)
%e A260302 (34 14 15 16 17 18 28)
%e A260302 (33 19 20 21 29)
%e A260302 (32 31 30)
%e A260302 The largest values (22 - 37) form the dam with the value 22 being the spillway.
%t A260302 Table[-KroneckerDelta[n,1] - 10*KroneckerDelta[n,2] + (1/2)*((7*n^2-18*n+12)^2+(7*n^2-18*n+12)), {n, 1, 30}] (* _G. C. Greubel_, Nov 13 2015 *)
%o A260302 (PARI) concat(vector(2), Vec(-x^3*(10*x^4-49*x^3+173*x^2+223*x+231)/(x-1)^5 + O(x^100))) \\ _Colin Barker_, Nov 11 2015
%o A260302 (Magma) [0,0] cat [(1/2)*(7*n^2-18*n+12)*(7*n^2-18*n+13): n in [3..60]]; // _Vincenzo Librandi_, Nov 20 2015
%Y A260302 Cf. A261347 (water retention on a number square).
%K A260302 nonn,easy
%O A260302 1,3
%A A260302 _Craig Knecht_, Nov 10 2015