A326118 a(n) is the largest number of squares of unit area connected only at corners and without holes that can be inscribed in an n X n square.
0, 1, 2, 5, 6, 9, 14, 21, 24, 29, 36, 45, 50, 57, 66, 77, 84, 93, 104, 117, 126, 137, 150, 165, 176, 189, 204, 221, 234, 249, 266, 285, 300, 317, 336, 357, 374, 393, 414, 437, 456, 477, 500, 525, 546, 569, 594, 621, 644, 669, 696, 725, 750, 777, 806, 837, 864, 893
Offset: 0
Examples
Illustrations for n = 1..7:
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a(1) = 1 a(2) = 2 a(3) = 5
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a(4) = 6 a(5) = 9
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a(6) = 14 a(7) = 21
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).
Programs
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Magma
I:=[0, 1, 2, 5, 6, 9, 14, 21, 24]; [n le 9 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-4)-2*Self(n-5)+Self(n-6): n in [1..58]];
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Mathematica
Join[{0,1,2},Table[(1/8)*(-29+12*n+2*n^2-3(-1)^n-12*Sin[n*Pi/2]),{n,3,57}]] -
PARI
concat([0], Vec(x*(-1-2*x^2+2*x^3-x^4-2*x^5+2*x^7)/((-1+x)^3*(1+x)*(1+x^2))+O(x^58)))
Formula
O.g.f.: x*(1 + 2*x^2 - 2*x^3 + x^4 + 2*x^5 - 2*x^7)/((1 - x)^3*(1 + x)*(1 + x^2)).
E.g.f.: -3*exp(-x)/8 + (2 + x)^2 + exp(x)/8*(-29 + 2*x*(7 + x)) - 3*sin(x)/2.
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n > 8.
a(n) = (1/8)*(-29 + 12*n + 2*n^2 - 3*(-1)^n - 12*sin(n*Pi/2)) for n > 2, a(0) = 0, a(1) = 1, a(2) = 2.
Limit_{n->oo} a(n)/A000290(n) = 1/4.
Comments