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

1,3

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.

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).

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)

Example

      (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.

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A350368 A183672 A183664 * A211717 A029569 A152102

Adjacent sequences: A260299 A260300 A260301 * A260303 A260304 A260305

Keyword

nonn,easy

Author

Craig Knecht, Nov 10 2015

Status

approved