cp's OEIS Frontend

A035288 Number of ways to place a non-attacking white and black bishop on n X n chessboard.

%I A035288 #38 Aug 21 2025 11:16:11
%S A035288 0,8,52,184,480,1040,1988,3472,5664,8760,12980,18568,25792,34944,
%T A035288 46340,60320,77248,97512,121524,149720,182560,220528,264132,313904,
%U A035288 370400,434200,505908,586152,675584,774880,884740,1005888,1139072,1285064,1444660,1618680,1807968
%N A035288 Number of ways to place a non-attacking white and black bishop on n X n chessboard.
%H A035288 Vincenzo Librandi, <a href="/A035288/b035288.txt">Table of n, a(n) for n = 1..1000</a>
%H A035288 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A035288 a(n) = (3*n^4 - 4*n^3 + 3*n^2 - 2*n)/3.
%F A035288 a(n) = 2 * A172123(n). - _Vaclav Kotesovec_, Nov 28 2011
%F A035288 a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5). - _Harvey P. Dale_, Nov 19 2011
%F A035288 G.f.: -4*x^2*(x+1)*(x+2)/(x-1)^5. - _Colin Barker_, Jan 09 2013
%F A035288 E.g.f.: exp(x)*x^2*(12 + 14*x + 3*x^2)/3. - _Stefano Spezia_, Aug 21 2025
%e A035288 There are 52 ways of putting 2 distinct bishops on 3 X 3 so that neither can capture the other.
%t A035288 Table[(3n^4-4n^3+3n^2-2n)/3, {n,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1}, {0,8,52,184,480}, 40] (* _Harvey P. Dale_, Nov 19 2011 *)
%o A035288 (Magma) [(3*n^4-4*n^3+3*n^2-2*n)/3: n in [1..35]]; // _Vincenzo Librandi_, May 04 2013
%o A035288 (PARI) a(n)=(3*n^4-4*n^3+3*n^2-2*n)/3; \\ _Joerg Arndt_, May 04 2013
%Y A035288 Cf. A172123.
%K A035288 nonn,easy
%O A035288 1,2
%A A035288 _Erich Friedman_