A327479 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.

0, 0, 0, 4, 6, 12, 16, 28, 32, 44, 52, 68, 76, 92, 104, 124, 136, 156, 172, 196, 212, 236, 256, 284, 304, 332, 356, 388, 412, 444, 472, 508, 536, 572, 604, 644, 676, 716, 752, 796, 832, 876, 916, 964, 1004, 1052, 1096, 1148, 1192, 1244, 1292, 1348, 1396, 1452, 1504

Offset: 0

Author

Stefano Spezia, Sep 16 2019

Keywords

• nonn
• easy

Comments

a(n) is equal to h_1(n) as defined in A309038.

All the terms are even numbers (A005843).

Examples

			Illustrations for n = 3..8:
      __    __               __    __.__             __    __.__.__
     |__|__|__|             |__|__|__.__|           |__|__|__.__.__|
      __|__|__               __|__|__.__             __|__|__    __
     |__|  |__|             |  |  |     |           |  |  |__|__|__|
                            |__|  |__.__|           |  |   __|__|__
                                                    |__|  |__|  |__|
      a(3) = 4                a(4) = 6                  a(5) = 12
   __    __    __.__     __    __    __    __     __    __    __    __.__
  |__|__|__|  |__   |   |__|__|__|  |__|__|__|   |__|__|__|  |__|__|__   |
   __|__|__    __|  |    __|__|__    __|__|__     __|__|__    __|  |  |__|
  |__|  |__|__|__.__|   |__|  |__|__|__|  |__|   |__|  |__|__|__.__|   __
   __    __|__|__.__     __    __|__|__    __     __    __|__|__    __|  |
  |  |__|  |  |     |   |__|__|__|  |__|__|__|   |__|__|  |  |__|__|__.__|
  |__.__.__|  |__.__|    __|__|__    __|__|__     __|__.__|   __|__|__.__
                        |__|  |__|  |__|  |__|   |  |__    __|  |  |     |
                                                 |__.__|  |__.__|  |__.__|
     a(6) = 16                a(7) = 28                 a(8) = 32
		

Links

• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Crossrefs

Cf. A000290, A005843, A309038, A326118, A327480.

Programs

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]];

Maple

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)):
ser := series(gf, x, 55): seq(factorial(n)*coeff(ser, x, n), n = 0..54);

Mathematica

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}]]

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)))

Formula

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)).

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)).

a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n > 10.

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.

Lim_{n->inf} a(n)/A000290(n) = 1/2.