A005338 Number of ways in which n identical balls can be distributed among 5 boxes in a row such that each pair of adjacent boxes contains at least 4 balls.

1, 8, 31, 85, 190, 360, 610, 956, 1415, 2005, 2745, 3655, 4756, 6070, 7620, 9430, 11525, 13931, 16675, 19785, 23290, 27220, 31606, 36480, 41875, 47825, 54365, 61531, 69360, 77890, 87160, 97210, 108081, 119815, 132455, 146045, 160630

Offset: 8

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 8..1000
• D. R. Breach, Letter to N. J. A. Sloane, Jun 1980
• Philippe Flajolet, Balls and Urns, etc., A problem in submarine detection (solution to problem 68-16).
• M. Hayes (proposer) and D. R. Breach (solver), A combinatorial problem, Problem 68-16, SIAM Rev. 12 (1970), 294-297.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A005337, A005339, A005340.

Programs

Magma

I:=[1, 8, 31, 85, 190, 360, 610]; [n le 7 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, May 11 2012

Mathematica

f[x_] := x^8*(1 + 3*x + x^2 - 11*x^5 + 7*x^6)/(1 - x)^5; Drop[ CoefficientList[ Series[f[x], {x, 0, 44}], x], 8] (* Jean-François Alcover, Oct 05 2011, after Vladeta Jovovic *)
LinearRecurrence[{5,-10,10,-5,1},{1,8,31,85,190,360,610},40] (* Harvey P. Dale, Aug 26 2019 *)

Formula

G.f.: x^8*(1 + 3*x + x^2 - 11*x^5 + 7*x^6)/(1 - x)^5. - Vladeta Jovovic, Apr 13 2008

a(n) = (n^4 + 10*n^3 - 445*n^2 + 2690*n - 1656)/24 for n > 9. - Colin Barker, May 10 2012

Extensions

Corrected and extended by Vladeta Jovovic, Apr 13 2008

Name clarified by Alois P. Heinz, Oct 02 2017