A309805 - OEIS

%I #27 Apr 04 2024 15:39:55

%S 1,2,7,10,19,24,37,44,61,70,91,102,127,140,169,184,217,234,271,290,

%T 331,352,397,420,469,494,547,574,631,660,721,752,817,850,919,954,1027,

%U 1064,1141,1180,1261,1302,1387,1430,1519,1564,1657,1704,1801,1850,1951,2002

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

%H Chess variants, <a href="https://www.chessvariants.com/hexagonal.dir/hexagonal.html">Glinski's Hexagonal Chess</a>

%H 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 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F a(n) = n^2 - floor(n/2) - floor(n/2)^2.

%F From _Stefano Spezia_, Aug 18 2019 (Start)

%F G.f.: - (1 + x + 3*x^2 + x^3)/((- 1 + x)^3*(1 + x)^2).

%F E.g.f.: (1/8)*exp(-x)*(-1 + 2*x + exp(2*x)*(1 + 4*x + 6*x^2)).

%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5.

%F a(n) = (1/16)*(3 + (-1)^(1+2*n) - 4*n + 12*n^2 - 2*(-1)^n*(1 + 2*n)).

%F a(2*n-1) = A003215(n).

%F a(2*n) = A049450(n).

%F (End)

%e a(1) = 1

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%e a(2) = 2

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%e a(3) = 7

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%e a(4) = 10

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%t 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 (PARI) a(n) = n^2 - (n\2) - (n\2)^2; \\ _Andrew Howroyd_, Aug 17 2019

%o (Python)

%o def A309805(n): return n**2-(m:=n>>1)*(m+1) # _Chai Wah Wu_, Apr 04 2024

%Y Cf. A003215, A049450.

%Y Partial sums of A133090.

%K nonn,easy

%O 1,2

%A _Sangeet Paul_, Aug 17 2019