A092438 -id:A092438 - cp's OEIS frontend

A092440 a(n) = 2^(2n+1) - 2^(n+1) + 1.

1, 5, 25, 113, 481, 1985, 8065, 32513, 130561, 523265, 2095105, 8384513, 33546241, 134201345, 536838145, 2147418113, 8589803521, 34359476225, 137438429185, 549754765313, 2199021158401, 8796088827905, 35184363700225

Offset: 0

Christopher Hanusa (chanusa(AT)math.washington.edu), Mar 24 2004

Arises from enumeration of domino tilings of Aztec Pillow-like regions.
Each beginning with 1, the subsequences of A046899 are 1; 1,2; 1,3,6; 1,4,10,20 and so forth.  Create triangles with the sides being equal to each of these subsequences; the interior members T(i,j)=T(i-1,j-1) + T(i-1,j).  The sum of all members for each triangle will reproduce the terms of this sequence.  Example using the fourth subsequence 1,4,10,20 will give row(1)=1; row(2)=4,4; row(3)=10,8,10; row(4)=20,18,18,20 giving a sum for all members of 113, the fourth term in the sequence. - J. M. Bergot, Oct 17 2012
Also, the number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 510", based on the 5-celled von Neumann neighborhood. - Robert Price, May 04 2016
Let M be some square matrix of rank 2^n, containing the positive real value X everywhere except on the diagonal; let Y be some complex value with phase 3*Pi/4 everywhere else (thus all coefficients on the diagonal). Then, for M to be a unitary matrix, X must be 1/sqrt(a(n)). - Thomas Baruchel, Aug 10 2020

J. Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 13).
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..500
J. Propp, Publications and Preprints
J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A092437, A092438, A092439, A092441, A092442, A092443.

Table[2^(2n + 1) - 2^(n + 1) + 1, {n, 0, 200}] (* Robert Price, May 04 2016 *)

a(n)=2^(2*n+1)-2^(n+1)+1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 2^(2n+1) - 2^(n+1) + 1.
From Colin Barker, Nov 22 2012: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3).
G.f.: -(4*x^2-2*x+1)/((x-1)*(2*x-1)*(4*x-1)). (End)
a(n) = A000225(n)^2 + (A000225(n) + 1)^2. - César Aguilera, May 28 2023

A092443 Sequence arising from enumeration of domino tilings of Aztec Pillow-like regions.

Original entry on oeis.org

3, 12, 50, 210, 882, 3696, 15444, 64350, 267410, 1108536, 4585308, 18929092, 78004500, 320932800, 1318498920, 5409723510, 22169259090, 90751353000, 371125269900, 1516311817020, 6189965556060, 25249187564640, 102917884095000, 419218847880300, 1706543186909652

Offset: 1

Christopher Hanusa (chanusa(AT)math.washington.edu), Mar 24 2004

The sequence 1, 3, 12, 50, ... is ((n+2)/2)*C(2n,n) with g.f. F(1/2,3;2;4x). - Paul Barry, Sep 18 2008

			a(3) = 5!/2!2! + 6!/3!3! = 50.

James Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 13).

Michael De Vlieger, Table of n, a(n) for n = 1..1659
Sanjay Moudgalya, Abhinav Prem, Rahul Nandkishore, Nicolas Regnault, and B. Andrei Bernevig, Thermalization and its absence within Krylov subspaces of a constrained Hamiltonian, arXiv:1910.14048 [cond-mat.str-el], 2019.
James Propp, Publications and Preprints.
James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics, Cambridge University Press, Cambridge, 1999, pp. 255-291.
Eric Rowland and Jason Wu, The entries of the Sinkhorn limit of an m X n matrix, arXiv:2409.02789 [math.NT], 2024. See p. 11.

Cf. A000984, A001622, A001700, A002457.

Cf. A092437, A092438, A092439, A092440, A092441, A092442.

Array[Binomial[2 # + 1, # + 1] &[# - 1]*(# + 2) &, 22] (* Michael De Vlieger, Dec 17 2017 *)

combinat::catalan(n) *binomial(n+2,2) $ n = 1..22 // Zerinvary Lajos, Feb 15 2007

a(n) = (n+2)*binomial(2*n-1, n); \\ Altug Alkan, Dec 17 2017

a(n) = (2*n-1)!/((n-1)!)^2+(2*n)!/(n!)^2 = A002457(n-1) + A000984(n).
a(n) = (n+2)*A001700(n-1). - Vladeta Jovovic, Jul 12 2004
n*a(n) + (-7*n+4)*a(n-1) + 6*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 30 2012
From Amiram Eldar, Jan 27 2024: (Start)
Sum_{n>=1} 1/a(n) = 4*Pi*(11*sqrt(3)-3*Pi)/9 - 13.
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(phi)*(13*sqrt(5)-30*log(phi))/5 - 11, where phi is the golden ratio (A001622). (End)
From Peter Bala, Aug 02 2024: (Start)
a(n) = 1/(n + 1)^2 * Sum_{k = 1..n+1} (k^3)*binomial(n+1, k)^2 = hypergeom([2, -n, -n], [1, 1], 1).
a(n) = 2*(n + 2)*(2*n - 1)/(n*(n + 1)) * a(n-1) with a(1) = 3. (End)

A092437 Triangle read by rows, arising from enumeration of domino tilings of Aztec Pillow-like regions.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 5, 6, 6, 1, 1, 5, 13, 26, 30, 20, 1, 1, 5, 13, 41, 90, 140, 140, 70, 1, 1, 5, 13, 41, 121, 302, 560, 742, 630, 252

Offset: 0

Christopher Hanusa (chanusa(AT)math.washington.edu), Mar 24 2004

The rows are of lengths 1, 3, 5, 7, ...
Call the first row row 0 and entries starting from 0. Then entries i=0 through k in row k are A046717(i).
In row k, entry k+1 is sequence A092438 and entry k+2 is sequence A092439.
In row k, entry 2k-1 is A002457(k-1) and entry 2k is A000984(k).

			Triangle begins:
  1;
  1, 1, 2;
  1, 1, 5, 6, 6;
  1, 1, 5, 13, 26, 30, 20;
  ...

James Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 13).

James Propp, Publications and Preprints.
James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics.

Cf. A092438-A092443.

cp's OEIS Frontend

A092440 a(n) = 2^(2n+1) - 2^(n+1) + 1.

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A092443 Sequence arising from enumeration of domino tilings of Aztec Pillow-like regions.

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A092437 Triangle read by rows, arising from enumeration of domino tilings of Aztec Pillow-like regions.

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