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
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).
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.
Comments