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%I #31 Sep 08 2022 08:46:24
%S 0,0,0,4,6,12,16,28,32,44,52,68,76,92,104,124,136,156,172,196,212,236,
%T 256,284,304,332,356,388,412,444,472,508,536,572,604,644,676,716,752,
%U 796,832,876,916,964,1004,1052,1096,1148,1192,1244,1292,1348,1396,1452,1504
%N a(n) is the minimum number of squares of unit area that must be removed from an n X n square to obtain a connected figure without holes and of the longest perimeter.
%C a(n) is equal to h_1(n) as defined in A309038.
%C All the terms are even numbers (A005843).
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-2,1).
%F O.g.f.: 2*x^3*(-2 + x - 2*x^2 + x^3 - 2*x^4 + 3*x^5 - 2*x^6 + x^7)/((-1 + x)^3*(1 + x + x^2 + x^3)).
%F E.g.f.: 8 + 4*x + 2*x^2 + x^4/12 + (1/4)*(-7*exp(-x) + exp(x)*(-25 + 6*x + 2*x^2) - 4*sin(x)).
%F a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n > 10.
%F a(n) = (1/4)*(- 25 + 2*n*(2 + n) - 7*cos(n*Pi) - 4*sin(n*Pi/2)) for n > 4, a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 4, a(4) = 6.
%F Lim_{n->inf} a(n)/A000290(n) = 1/2.
%e Illustrations for n = 3..8:
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%e __|__|__ __|__|__.__ __|__|__ __
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%e |__| |__.__| | | __|__|__
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%e a(3) = 4 a(4) = 6 a(5) = 12
%e __ __ __.__ __ __ __ __ __ __ __ __.__
%e |__|__|__| |__ | |__|__|__| |__|__|__| |__|__|__| |__|__|__ |
%e __|__|__ __| | __|__|__ __|__|__ __|__|__ __| | |__|
%e |__| |__|__|__.__| |__| |__|__|__| |__| |__| |__|__|__.__| __
%e __ __|__|__.__ __ __|__|__ __ __ __|__|__ __| |
%e | |__| | | | |__|__|__| |__|__|__| |__|__| | |__|__|__.__|
%e |__.__.__| |__.__| __|__|__ __|__|__ __|__.__| __|__|__.__
%e |__| |__| |__| |__| | |__ __| | | |
%e |__.__| |__.__| |__.__|
%e a(6) = 16 a(7) = 28 a(8) = 32
%p gf := 8+4*x+2*x^2+(1/12)*x^4+1/4*(-7*exp(-x)+exp(x)*(2*x^2+6*x-25)-4*sin(x)):
%p ser := series(gf, x, 55): seq(factorial(n)*coeff(ser, x, n), n = 0..54);
%t Join[{0,0,0,4,6},Table[(1/4)*(-25+2n*(2+n)-7*Cos[n*Pi]-4*Sin[n*Pi/2]),{n,5,54}]]
%o (Magma) I:=[0, 0, 0, 4, 6, 12, 16, 28, 32, 44, 52]; [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..55]];
%o (PARI) concat([0, 0, 0], Vec(2*x^3*(-2+x-2*x^2+x^3-2*x^4+3*x^5-2*x^6+x^7)/((-1+x)^3*(1+x+x^2+x^3))+O(x^55)))
%Y Cf. A000290, A005843, A309038, A326118, A327480.
%K nonn,easy
%O 0,4
%A _Stefano Spezia_, Sep 16 2019
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