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Search: id:A072280
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| A072280 |
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Sequence arising from factorization of the Pell numbers and the Companion Pell numbers f(n)=A000129 and L(n)=A002203. f(n)=2f(n-1)+f(n-2) L(n)=f(n-1)+f(n+1). |
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+0
3
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| 2, 1, 7, 6, 41, 5, 239, 34, 199, 29, 8119, 33, 47321, 169, 961, 1154, 1607521, 197, 9369319, 1121, 32641, 5741, 318281039, 1153
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OFFSET
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1,1
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COMMENT
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For even n, f(n)=Product(d|n)a(d); for odd n, f(n)=Product( d divides n )a(2d); for odd prime p, a(p)=L(p)/2, a(2p)=f(p) a(1)=2, a(2)=1; a(2^(k+1))=L(2^k); for odd n, L(n)=Product( d divides n )a(d); for k>0 and odd n, L(n*2^k)=Product( d divides n )a(d*2^(k+1))
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FORMULA
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h=1+sqrt(2). h^2=1+2h. 1/h=h-2. K(n, x)=n-th cyclotomic polynomial, so that x^n-1=Product( d divides n )K(d, x). g(d) is the order of K(d, x). a(n)=(h-2)^g(n)*K(n, h^2)
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EXAMPLE
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f(12)=a(1)*a(2)*a(3)*a(4)*a(6)*a(12)=2*1*7*6*5*33=13860 for even n, f(9)=a(2)*a(6)*a(18)= 1*5*197=985 for odd n. L(12)=a(8)*a(24)=34*1153=39202 for even n, L(21)=a(1)*a(3)*a(7)*a(21)=2*7*239*32641=109216786.
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CROSSREFS
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Cf. A000129, A002203.
Adjacent sequences: A072277 A072278 A072279 this_sequence A072281 A072282 A072283
Sequence in context: A078301 A060583 A078104 this_sequence A086054 A011134 A157240
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KEYWORD
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nonn,uned
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AUTHOR
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M. Kristof (kristmikl(AT)freemail.hu), Jul 10 2002
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