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

0,4

Comments

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

All the terms are even numbers (A005843).

Links

Table of n, a(n) for n=0..54.

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

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.

Example

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

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}]]

Prog

(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]];
(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)))

Crossrefs

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

Sequence in context: A024904 A172445 A274221 * A139056 A152519 A240507

Adjacent sequences: A327476 A327477 A327478 * A327480 A327481 A327482

Keyword

nonn,easy

Author

Stefano Spezia, Sep 16 2019

Status

approved