

A092440


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


28



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

0,2


COMMENTS

Arises from enumeration of domino tilings of Aztec Pillowlike 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(i1,j1) + T(i1,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(10=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^n1 of the twodimensional cellular automaton defined by "Rule 510", based on the 5celled 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


REFERENCES

J. Propp, Enumeration of matchings: problems and progress, pp. 255291 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.


LINKS

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 5Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (7,14,8).


FORMULA

a(n) = 2^(2n+1)2^(n+1)+1.
From Colin Barker, Nov 22 2012: (Start)
a(n) = 7*a(n1)14*a(n2)+8*a(n3).
G.f.: (4*x^22*x+1)/((x1)*(2*x1)*(4*x1)). (End)


MATHEMATICA

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


PROG

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


CROSSREFS

Cf. A092437, A092438, A092439, A092441, A092442, A092443.
Sequence in context: A290920 A267228 A183926 * A196985 A124534 A261383
Adjacent sequences: A092437 A092438 A092439 * A092441 A092442 A092443


KEYWORD

easy,nonn


AUTHOR

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


STATUS

approved



