A260302 Maximum water retention of a number octagon of order n.

0, 0, 231, 1378, 4753, 12246, 26335, 50086, 87153, 141778, 218791, 323610, 462241, 641278, 867903, 1149886, 1495585, 1913946, 2414503, 3007378, 3703281, 4513510, 5449951, 6525078, 7751953, 9144226, 10716135, 12482506, 14458753, 16660878, 19105471, 21809710

Offset: 1

Author

Craig Knecht, Nov 10 2015

Keywords

• nonn
• easy

Comments

A number octagon fills an octagon on a square grid with the smallest unique natural numbers.

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.

Examples

			      (22 23 24)
   (37  1  2  3 25)
(36  4  5  6  7  8  26)
(35  9 10 11 12 13  27)
(34 14 15 16 17 18  28)
   (33 19 20 21 29)
      (32 31 30)
The largest values (22 - 37) form the dam with the value 22 being the spillway.
		

Links

• G. C. Greubel and Craig Knecht, Table of n, a(n) for n = 1..1000 (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)
• Craig Knecht, Octagon volume calculation.
• Craig Knecht, Volume retention of a number square.
• Wikipedia, Water retention on mathematical surfaces
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A261347 (water retention on a number square).

Programs

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

Mathematica

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 *)

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

Formula

a(n) = (1/2)*(7*n^2 - 18*n + 12) (7*n^2 - 18*n + 13) for n > 2.

From Colin Barker, Nov 11 2015: (Start)

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>7.

G.f.: -x^3*(10*x^4-49*x^3+173*x^2+223*x+231) / (x-1)^5. (End)