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A260302
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Maximum water retention of a number octagon of order n.
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2
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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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)
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FORMULA
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a(n) = (1/2)*(7*n^2 - 18*n + 12) (7*n^2 - 18*n + 13) for n > 2.
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)
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EXAMPLE
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(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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A261347 (water retention on a number square).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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