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A327480 a(n) is the maximum number of squares of unit area that can be removed from an n X n square while still obtaining a connected figure without holes and of the longest perimeter.

%I A327480 #42 Dec 17 2024 08:44:00
%S A327480 0,0,2,4,8,12,22,28,40,48,64,76,94,108,130,148,172,192,220,244,274,
%T A327480 300,334,364,400,432,472,508,550,588,634,676,724,768,820,868,922,972,
%U A327480 1030,1084,1144,1200,1264,1324,1390,1452,1522,1588,1660,1728,1804,1876,1954
%N A327480 a(n) is the maximum number of squares of unit area that can be removed from an n X n square while still obtaining a connected figure without holes and of the longest perimeter.
%C A327480 a(n) is equal to h_1(n) + h_2(n) as defined in A309038.
%H A327480 Stefano Spezia, <a href="/A327480/b327480.txt">Table of n, a(n) for n = 0..10000</a>
%H A327480 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-2,1).
%F A327480 O.g.f.: 2*x^2*(1 + x^2 + 2*x^4 - 2*x^5 + 2*x^6 - 2*x^7 + x^8)/((1 - x)^3*(1 + x)*(1 + x^2)).
%F A327480 E.g.f.: (1/24)*exp(-x)*(33 + 9*exp(2*x)*(7 - 2*x + 2*x^2) - 2*exp(x)*(48 + 12*x^2 + x^4) - 12*exp(x)*sin(x)).
%F A327480 a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n > 10.
%F A327480 a(n) = (1/8)*(21 - 12*n + 6*n^2 + 11*(-1)^n + 4*A056594(n+1)) for n > 4, a(0) = 0, a(1) = 0, a(2) = 2, a(3) = 4, a(4) = 8. [corrected by _Jason Yuen_, Dec 17 2024]
%F A327480 Limit_{n->oo} a(n)/A000290(n) = 3/4.
%e A327480 Illustrations for n = 2..7:
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%e A327480                              |__|  |__|              |__|  |     |
%e A327480                                                            |__ __|
%e A327480       a(2) = 2                a(3) = 4                  a(4) = 8
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%e A327480                         |__|        |__|     __|__|__    __|__|__
%e A327480                                             |__|  |__|  |__|  |__|
%e A327480       a(5) = 12           a(6) = 22               a(7) = 28
%p A327480 gf := (1/24)*exp(-x)*(33+9*exp(2*x)*(2*x^2-2*x+7)-2*exp(x)*(x^4+12*x^2+48)-12*exp(x)*sin(x)); ser := series(gf, x, 53):
%p A327480 seq(factorial(n)*coeff(ser, x, n), n = 0 .. 52)
%t A327480 Join[{0,0,2,4,8},Table[(1/8)*(21-12n+6n^2+11*(-1)^n-4*Sin[n*Pi/2]),{n,5,52}]]
%o A327480 (Magma) I:=[0, 0, 2, 4, 8, 12, 22, 28, 40, 48, 64]; [n le 11 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-4)-2*Self(n-5)+Self(n-6): n in [1..53]];
%o A327480 (PARI) concat([0, 0], Vec(2*x^2*(1+x^2+2*x^4-2*x^5+2*x^6-2*x^7+x^8)/((1-x)^3*(1+x)*(1+x^2))+O(x^53)))
%Y A327480 Cf. A000290, A056594, A309038, A326118, A327479.
%K A327480 nonn,easy
%O A327480 0,3
%A A327480 _Stefano Spezia_, Sep 16 2019