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%I #39 Nov 26 2017 11:15:09
%S 1,1,2,4,16,64,512,4096,65536,1048576,33554432,1073741824,68719476736,
%T 4398046511104,562949953421312,72057594037927936,18446744073709551616,
%U 4722366482869645213696,2417851639229258349412352
%N a(n) = 2*a(n-1)*a(n-2)/a(n-3), with a(0)=a(1)=1.
%C a(n+1) is the Hankel transform of A135052. - _Paul Barry_, Nov 15 2007
%C a(n+1) is the Hankel transform of the aerated large Schroeder numbers. a(n) and a(n+1) both satisfy the trivial Somos-4 recurrence u(n)=4*u(n-2)^2/u(n-4). Associated with the elliptic curve y^2=1-6x^2+x^4 via Schroeder numbers. - _Paul Barry_, Dec 08 2009
%C Hankel transform of A089324. - _Paul Barry_, Mar 01 2010
%C a(n+1) is the number of n X n binary matrices that are symmetric about both diagonals (bisymmetric). For the derivation of this result, see the link below. - _Dennis P. Walsh_, Apr 03 2014
%C 1 followed by {a(n-1)}_(n>=1) is the Somos-3 sequence: b(0)=b(1)=b(2)=1;for n>=3, b(n)=2*b(n-1)*b(n-2)/b(n-3) (cf. comment in A078495). - _Vladimir Shevelev_, Apr 20 2016
%C If the Hankel transform is defined as in the link 'Sequence transformations' then a(n) is the Hankel transform of A151374. - _Peter Luschny_, Nov 30 2016
%H Harry J. Smith, <a href="/A060656/b060656.txt">Table of n, a(n) for n = 0..100</a>
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SequenceTransformations#Hankel">Sequence transformations</a>
%H Dennis P. Walsh, <a href="http://capone.mtsu.edu/dwalsh/BINRMAT.pdf">Notes on binary bisymmetric matrices</a>
%F a(n) = 2^floor( n^2/4 ) = a(n - 1) * 2^floor( n/2 ) = a(n - 2) * 2^(n - 1) = a(n - 1) * A016116(n) = 2^A002620(n).
%F 0 = a(n) * a(n+3) + a(n+1) * ( -2*a(n+2) ) for all n in Z. - _Michael Somos_, Jan 24 2014
%F 0 = a(n) * a(n+4) + a(n+2) * ( -4*a(n+2) ) for all n in Z. - _Michael Somos_, Jan 24 2014
%e a(6) = 2*64*16/4 = 512.
%e G.f. = 1 + x + 2*x^2 + 4*x^3 + 16*x^4 + 64*x^5 + 512*x^6 + 4096*x^7 + ...
%p A060656:=n->2^floor(n^2/4); seq(A060656(n), n=0..20); # _Wesley Ivan Hurt_, Apr 30 2014
%t a[ n_] := 2^Quotient[n^2, 4]; (* _Michael Somos_, Jan 24 2014 *)
%t nxt[{a_,b_,c_}]:={b,c,(2c*b)/a}; NestList[nxt,{1,1,2},20][[All,1]] (* _Harvey P. Dale_, Nov 26 2017 *)
%o (PARI) { for (n=0, 100, write("b060656.txt", n, " ", 2^(n^2\4)); ) } \\ _Harry J. Smith_, Jul 09 2009
%o (PARI) {a(n) = 2^(n^2\4)}; /* _Michael Somos_, Jan 24 2014 */
%Y Cf. A002416, A002620, A016116, A038754, A089324, A135052, A262666, A078495, A151374.
%K nonn,easy
%O 0,3
%A _Henry Bottomley_, Apr 18 2001
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