OFFSET
0,2
COMMENTS
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
REFERENCES
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.
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 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(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)
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
KEYWORD
easy,nonn
AUTHOR
Christopher Hanusa (chanusa(AT)math.washington.edu), Mar 24 2004
STATUS
approved