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.

0, 0, 2, 4, 8, 12, 22, 28, 40, 48, 64, 76, 94, 108, 130, 148, 172, 192, 220, 244, 274, 300, 334, 364, 400, 432, 472, 508, 550, 588, 634, 676, 724, 768, 820, 868, 922, 972, 1030, 1084, 1144, 1200, 1264, 1324, 1390, 1452, 1522, 1588, 1660, 1728, 1804, 1876, 1954

Offset

0,3

Comments

a(n) is equal to h_1(n) + h_2(n) as defined in A309038.

Links

Stefano Spezia, Table of n, a(n) for n = 0..10000

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

Formula

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)).

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)).

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

a(n) = (1/8)*(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.

Limit_{n->oo} a(n)/A000290(n) = 3/4.

Example

Illustrations for n = 2..7:
      __                      __    __                __    __
     |__|__                  |__|__|__|              |__|__|__|
        |__|                  __|__|__                __|__|__ __
                             |__|  |__|              |__|  |     |
                                                           |__ __|
      a(2) = 2                a(3) = 4                  a(4) = 8
   __    __ __ __     __    __    __         __    __    __    __
  |__|__|__ __ __|   |__|__|__|  |__|__     |__|__|__|  |__|__|__|
   __|__|__    __     __|__|__    __|__|     __|__|__    __|__|__
  |  |  |__|__|__|   |__|  |__|__|__|       |__|  |__|__|__|  |__|
  |  |   __|__|__     __    __|__|__         __    __|__|__    __
  |__|  |__|  |__|   |__|__|__|  |__|__     |__|__|__|  |__|__|__|
                        |__|        |__|     __|__|__    __|__|__
                                            |__|  |__|  |__|  |__|
      a(5) = 12           a(6) = 22               a(7) = 28

Maple

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):
seq(factorial(n)*coeff(ser, x, n), n = 0 .. 52)

Mathematica

Join[{0, 0, 2, 4, 8}, Table[(1/8)*(21-12n+6n^2+11*(-1)^n-4*Sin[n*Pi/2]), {n, 5, 52}]]

Prog

(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]];
(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)))

Crossrefs

Cf. A000290, A056594, A309038, A326118, A327479.

Sequence in context: A027677 A103787 A365076 * A330022 A362261 A032473

Adjacent sequences: A327477 A327478 A327479 * A327481 A327482 A327483

Keyword

nonn,easy

Author

Stefano Spezia, Sep 16 2019

Status

approved