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A060656
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a(n) = 2*a(n-1)*a(n-2)/a(n-3), with a(0)=a(1)=1.
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10
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1, 1, 2, 4, 16, 64, 512, 4096, 65536, 1048576, 33554432, 1073741824, 68719476736, 4398046511104, 562949953421312, 72057594037927936, 18446744073709551616, 4722366482869645213696, 2417851639229258349412352
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OFFSET
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0,3
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COMMENTS
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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
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
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
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
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LINKS
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FORMULA
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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).
0 = a(n) * a(n+3) + a(n+1) * ( -2*a(n+2) ) for all n in Z. - Michael Somos, Jan 24 2014
0 = a(n) * a(n+4) + a(n+2) * ( -4*a(n+2) ) for all n in Z. - Michael Somos, Jan 24 2014
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EXAMPLE
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a(6) = 2*64*16/4 = 512.
G.f. = 1 + x + 2*x^2 + 4*x^3 + 16*x^4 + 64*x^5 + 512*x^6 + 4096*x^7 + ...
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MAPLE
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MATHEMATICA
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nxt[{a_, b_, c_}]:={b, c, (2c*b)/a}; NestList[nxt, {1, 1, 2}, 20][[All, 1]] (* Harvey P. Dale, Nov 26 2017 *)
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PROG
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(PARI) { for (n=0, 100, write("b060656.txt", n, " ", 2^(n^2\4)); ) } \\ Harry J. Smith, Jul 09 2009
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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