cp's OEIS Frontend

A309805 Maximum number of nonattacking kings placeable on a hexagonal board with edge-length n in Glinski's hexagonal chess.

%I A309805 #27 Apr 04 2024 15:39:55
%S A309805 1,2,7,10,19,24,37,44,61,70,91,102,127,140,169,184,217,234,271,290,
%T A309805 331,352,397,420,469,494,547,574,631,660,721,752,817,850,919,954,1027,
%U A309805 1064,1141,1180,1261,1302,1387,1430,1519,1564,1657,1704,1801,1850,1951,2002
%N A309805 Maximum number of nonattacking kings placeable on a hexagonal board with edge-length n in Glinski's hexagonal chess.
%H A309805 Chess variants, <a href="https://www.chessvariants.com/hexagonal.dir/hexagonal.html">Glinski's Hexagonal Chess</a>
%H A309805 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hexagonal_chess#Gli%C5%84ski&#39;s_hexagonal_chess">Hexagonal chess - Gliński's hexagonal chess</a>
%H A309805 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A309805 a(n) = n^2 - floor(n/2) - floor(n/2)^2.
%F A309805 From _Stefano Spezia_, Aug 18 2019 (Start)
%F A309805 G.f.: - (1 + x + 3*x^2 + x^3)/((- 1 + x)^3*(1 + x)^2).
%F A309805 E.g.f.: (1/8)*exp(-x)*(-1 + 2*x + exp(2*x)*(1 + 4*x + 6*x^2)).
%F A309805 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5.
%F A309805 a(n) = (1/16)*(3 + (-1)^(1+2*n) - 4*n + 12*n^2 - 2*(-1)^n*(1 + 2*n)).
%F A309805 a(2*n-1) = A003215(n).
%F A309805 a(2*n) = A049450(n).
%F A309805 (End)
%e A309805 a(1) = 1
%e A309805 .
%e A309805   o
%e A309805 .
%e A309805 a(2) = 2
%e A309805 .
%e A309805    . .
%e A309805   o . o
%e A309805    . .
%e A309805 .
%e A309805 a(3) = 7
%e A309805 .
%e A309805     o . o
%e A309805    . . . .
%e A309805   o . o . o
%e A309805    . . . .
%e A309805     o . o
%e A309805 .
%e A309805 a(4) = 10
%e A309805 .
%e A309805      . . . .
%e A309805     o . o . o
%e A309805    . . . . . .
%e A309805   o . o . o . o
%e A309805    . . . . . .
%e A309805     o . o . o
%e A309805      . . . .
%e A309805 .
%t A309805 nn:=51; CoefficientList[Series[- (1 + x + 3*x^2 + x^3)/((- 1 + x)^3*(1 + x)^2),{x, 0, nn}], x] (* _Georg Fischer_, May 10 2020 *)
%o A309805 (PARI) a(n) = n^2 - (n\2) - (n\2)^2; \\ _Andrew Howroyd_, Aug 17 2019
%o A309805 (Python)
%o A309805 def A309805(n): return n**2-(m:=n>>1)*(m+1) # _Chai Wah Wu_, Apr 04 2024
%Y A309805 Cf. A003215, A049450.
%Y A309805 Partial sums of A133090.
%K A309805 nonn,easy
%O A309805 1,2
%A A309805 _Sangeet Paul_, Aug 17 2019