A060868 -id:A060868 - cp's OEIS frontend

A238976 a(n) = ((3^(n-1)-1)^2)/4.

0, 1, 16, 169, 1600, 14641, 132496, 1194649, 10758400, 96845281, 871666576, 7845176329, 70607118400, 635465659921, 5719195722256, 51472775849209, 463255025689600, 4169295360346561, 37523658630539536, 337712928837117289, 3039416363020840000, 27354747277647913201, 246192725530212278416

Offset: 1

Kival Ngaokrajang, Mar 07 2014

If the Cantor square fractal is modified as shown in the illustration (see Links), then 4*a(n) is the total number of holes in the modified Cantor square fractal after n iterations. The total number of sides (outside) is 4*A171498(n-1). The total length of the sides (outside) converges to 20 when the initial total side length is 12 (starting with 5 unit squares).
For the Cantor square fractal, the total number of sides (outside) is 4*A168616(n+2). The total number of holes is 4*A060867(n-1) for n > 1. The total length of the sides (outside) converges to 12 with the same initial condition (i.e., 5 unit square); its maximum is 17.333... and is reached at n = 2, 3. The Cantor square fractal and modified one are not true fractals.
See illustrations in links.

Kival Ngaokrajang, Illustrations of initial terms
Eric Weisstein's World of Mathematics, Cantor Square Fractal
Index entries for linear recurrences with constant coefficients, signature (13,-39,27)

Cf. A024023, A060867, A060868, A168616, A171498.

a(n) = ((3^(n-1)-1)^2)/4; \\ Joerg Arndt, Mar 08 2014

a(n) = (A024023(n-1))^2/4.

G.f.: x*(3*x + 1)/((1-x)*(1-3*x)*(1-9*x)). - Ralf Stephan, Mar 14 2014

A379587 Array read by ascending antidiagonals: A(n, k) = (k^n - 1)^2/(k - 1), with k >= 2.

Original entry on oeis.org

0, 1, 0, 9, 2, 0, 49, 32, 3, 0, 225, 338, 75, 4, 0, 961, 3200, 1323, 144, 5, 0, 3969, 29282, 21675, 3844, 245, 6, 0, 16129, 264992, 348843, 97344, 9245, 384, 7, 0, 65025, 2389298, 5589675, 2439844, 335405, 19494, 567, 8, 0, 261121, 21516800, 89467563, 61027344, 12090125, 960000, 37303, 800, 9, 0

Offset: 0

Stefano Spezia, Dec 26 2024

			The array begins as:
    0,     0,      0,       0,        0,        0, ...
    1,     2,      3,       4,        5,        6, ...
    9,    32,     75,     144,      245,      384, ...
   49,   338,   1323,    3844,     9245,    19494, ...
  225,  3200,  21675,   97344,   335405,   960000, ...
  961, 29282, 348843, 2439844, 12090125, 47073606, ...
  ...

Cf. A027620, A060867 (k=2), A060868 (k=3), A060869 (k=4), A060870 (k=5), A060871 (k=7), A361475, A379588 (antidiagonal sums).

A[n_,k_]:=(k^n-1)^2/(k-1); Table[A[n-k+2,k],{n,0,9},{k,2,n+2}]//Flatten

G.f. of column k: (1 - k)*x*(1 + k*x)/((1 - x)*(1 - k*x)*(1 - k^2*x)).
E.g.f. of column k: exp(x)*(1 - 2*exp((k-1)*x) + exp((k^2-1)*x))/(k - 1).
A(2, n) = A027620(n-2) for n > 1.

A086752 Number of n X n matrices over GF(3) with rank n-1.

Original entry on oeis.org

1, 32, 8112, 17971200, 355207057920, 63010655570903040, 100505356319291594711040, 1442361950110091891786121216000, 186276322602412236974585775503690956800, 216505458700483736766078241517019274701019545600, 2264736353104098912130003755084217747715114856943819161600

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 31 2003

Cf. A086699, A053290, A060868.

Table[3^(n^2) * (1 - 1/3^n) * QPochhammer[1/3^n, 3, n-1]/2, {n, 1, 10}] (* Vaclav Kotesovec, Apr 14 2024 *)

For n>=2: a(n) = Product_{j=0..n-2} (3^n - 3^j)^2 / (3^(n-1)- 3^j).

a(n) = ((3^n-1)/2)*Product_{j=0..n-2} (3^n-3^j). - David Wasserman, Mar 28 2005

More terms from David Wasserman, Mar 28 2005

cp's OEIS Frontend

A238976 a(n) = ((3^(n-1)-1)^2)/4.

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A379587 Array read by ascending antidiagonals: A(n, k) = (k^n - 1)^2/(k - 1), with k >= 2.

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A086752 Number of n X n matrices over GF(3) with rank n-1.

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