A013824 -id:A013824 - cp's OEIS frontend

A225910 Square array read by antidiagonals: a(m,n) is the number of binary pattern classes in the (m,n)-rectangular grid, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 6, 7, 6, 1, 1, 10, 24, 24, 10, 1, 1, 20, 76, 168, 76, 20, 1, 1, 36, 288, 1120, 1120, 288, 36, 1, 1, 72, 1072, 8640, 16576, 8640, 1072, 72, 1, 1, 136, 4224, 66816, 263680, 263680, 66816, 4224, 136, 1, 1, 272, 16576, 529920, 4197376, 8407040, 4197376, 529920, 16576, 272, 1

Offset: 0

Yosu Yurramendi, May 20 2013

In the square table A000012, A005418, and A225826 to A225834 are the first 11 rows (see example).
In the square table, m odd (see formula). The order of the recurrence equations is 4. Let it be (a1(m),a2(m),a3(m),a4(m)) the characterizing 4-plet of a(m). The sequence a1(m) belongs to A028403 (2^m+2^((m+1)/2)), -a2(m) to A147538 (2^m*(2^((m+1)/2)-1)) and a4(m) to A013824 (2^(2m)*2^((m+1)/2)). -a3(m) sequence formula is 2^m*(2^m+2^((m+1)/2)).
All the coefficients of x in generating functions from A225826 to A225834 belong to A113979.

			Array begins:
  1   1      1         1            1               1                  1 ...
  1   2      3         6           10              20                 36 ...
  1   3      7        24           76             288               1072 ...
  1   6     24       168         1120            8640              66816 ...
  1  10     76      1120        16576          263680            4197376 ...
  1  20    288      8640       263680         8407040          268517376 ...
  1  36   1072     66816      4197376       268517376        17180065792 ...
  1  72   4224    529920     67133440      8590786560      1099516870656 ...
  1 136  16576   4212736   1073790976    274882625536     70368756760576 ...
  1 272  66048  33632256  17180262400   8796137062400   4503599962914816 ...
  1 528 262912 268713984 274878693376 281475261923328 288230376957018112 ...
  ...

Alois P. Heinz, Antidiagonals n = 0..65, flattened
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023.

Cf. A005418, A225826-A225834.

m even and n even:
a(m,n) = 2^(m*n/2-2)*(2^(m*n/2) + 3);
m even and n odd:
a(m,n) = 2^(m*n/2-1)*(2^(m*n/2-1) + 2^(m/2-1) + 1);
m odd and n even:
a(m,n) = 2^(m*n/2-1)*(2^(m*n/2-1) + 2^(n/2-1) + 1);
m odd and n odd:
a(m,n) = 2^((m*n-1)/2-1)*(2^((m*n-1)/2) + 2^((m-1)/2) + 2^((n-1)/2) + 1).
m even:
a(m,n) = 2^m*a(m,n-1) + 2^m*a(m,n-2) - (2^m)^2*a(m,n-3) with n>2, a(m,0)=1, a(m,1)=a(1,m), a(m,2)=a(2,m).
m odd:
a(m,n) = 2^m*a(m,n-1) + 2^m*a(m,n-2) - (2^m)^2*a(m,n-3) - 2^(((m+1)/2)*n-3)*(2^((m-1)/2)-1) with n>2, a(m,0)=1, a(m,1)=a(1,m), a(m,2)=a(2,m).
Only a(1,n) and a(2,n) (A005418 and A225826) sequences are needed to define the others.

A013823 a(n) = 2^(5*n + 2).

Original entry on oeis.org

4, 128, 4096, 131072, 4194304, 134217728, 4294967296, 137438953472, 4398046511104, 140737488355328, 4503599627370496, 144115188075855872, 4611686018427387904, 147573952589676412928, 4722366482869645213696, 151115727451828646838272, 4835703278458516698824704

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (32).

Cf. A000079, A009976, A013822, A013824, A016873.

[2^(5*n+2): n in [0..15]]; // Vincenzo Librandi, Jul 07 2011

2^(5*Range[0, 20] + 2) (* Paolo Xausa, Feb 21 2025 *)

makelist(2^(5*n+2),n,0,20); /* Martin Ettl, Oct 21 2012 */

From Philippe Deléham, Nov 24 2008: (Start)
a(n) = 32*a(n-1); a(0)=4.
G.f.: 4/(1-32*x).
a(n) = 4*A009976(n). (End)
From Elmo R. Oliveira, Feb 20 2025: (Start)
E.g.f.: 4*exp(32*x).
a(n) = A000079(A016873(n)). (End)
a(n) = 2*A013822(n) = A013824(n)/2. - Paolo Xausa, Feb 21 2025

A013896 a(n) = 20^(5*n + 3).

Original entry on oeis.org

8000, 25600000000, 81920000000000000, 262144000000000000000000, 838860800000000000000000000000, 2684354560000000000000000000000000000, 8589934592000000000000000000000000000000000, 27487790694400000000000000000000000000000000000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..50
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (3200000).

Subsequence of A009964.

Cf. A013824, A013856, A016885.

[20^(5*n+3): n in [0..10]]; // Vincenzo Librandi, May 27 2011

20^(5Range[0,20]+3) (* or *) NestList[3200000#&,8000,20] (* Harvey P. Dale, Dec 05 2021 *)

a(n) = 3200000*a(n-1), a(0)=8000. - Vincenzo Librandi, May 27 2011
From Elmo R. Oliveira, Jul 11 2025: (Start)
G.f.: 8000/(1-3200000*x).
E.g.f.: 8000*exp(3200000*x).
a(n) = A013824(n)*A013856(n) = A009964(A016885(n)). (End)

A386394 a(n) = (5^(2n) + 32^(5*n-2))/7.

Original entry on oeis.org

7, 199, 5743, 168151, 4990207, 149920999, 4553331343, 139603087351, 4314710904607, 134256051681799, 4200826222176943, 132042253318646551, 4165747461421299007, 131813802646096802599, 4180788781690478542543, 132853918439479834845751, 4228042325697967752173407

Offset: 1

Stefano Spezia, Jul 20 2025

a(n) is integer for n > 0.

Proof: It is sufficient to prove that 5^(2*n) + 3*2^(5*n-2) is divisible by 7. Since from 5 == -2 (mod 7) follows that 5^(2*n) == 2^(2*n) (mod 7) and from 2^5 == 2^2 (mod 7) follows 2^(5*n-2) == 2^(2*(n-1)) (mod 7), one gets that 5^(2*n) + 3*2^(5*n-2) == 2^(2*n) + 3*2^(2*(n-1)) (mod 7) and 2^(2*n) + 3*2^(2*(n-1)) = 2^(2*(n-1))*(4 + 3) == 0 (mod 7). QED

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 5.1.24 on page 158.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (57,-800).

Cf. A009969, A013824.

[(5^(2*n)+3*2^(5*n-2))/7 : n in [1..20]]; // Vincenzo Librandi, Jul 21 2025

a[n_]:=(5^(2n)+3*2^(5n-2))/7; Array[a,17]

a(n) = 57*a(n-1) - 800*a(n-2) for n > 2.
G.f.: x*(7 - 200*x)/((1 - 25*x)*(1 - 32*x)).
E.g.f.: (3*exp(32*x) + 4*exp(25*x) - 7)/28.
a(n) = (A009969(n) + 3*A013824(n-1))/7.

cp's OEIS Frontend

A225910 Square array read by antidiagonals: a(m,n) is the number of binary pattern classes in the (m,n)-rectangular grid, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

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A013823 a(n) = 2^(5*n + 2).

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Formula

A013896 a(n) = 20^(5*n + 3).

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Magma

Mathematica

Formula

A386394 a(n) = (5^(2*n) + 3*2^(5*n-2))/7.

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Author

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Crossrefs

Programs

Magma

Mathematica

Formula

A386394 a(n) = (5^(2n) + 32^(5*n-2))/7.