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Search: id:A083884
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| A083884 |
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a(n) = (3^(2n) + 1) / 2. |
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+0
10
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| 1, 5, 41, 365, 3281, 29525, 265721, 2391485, 21523361, 193710245, 1743392201, 15690529805, 141214768241, 1270932914165, 11438396227481, 102945566047325, 926510094425921, 8338590849833285, 75047317648499561, 675425858836496045, 6078832729528464401
(list; graph; listen)
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OFFSET
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0,2
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FORMULA
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a(0) = 1, a(n) = 9a(n-1) - 4.
a(n) = Sum_{ k, 0<=k<=n} binomial(2n, 2k)*4^k.
a(n) = A002438(n) / A000364(n); A000364(n) : Euler numbers.
G.f.: (1-5x)/((1-x)(1-9x)).
a(n)=(3^n+1^n+(-1)^n+(-3)^n)/4; e.g.f.: exp(3x)+exp(x)+exp(-x)+exp(-3x).
Each term expresses a Pythagorean relationship, along with (a(n)-1) and a power of 3, n>0, such that sqrt((a(n))^2 - (a(n)-1)^2) = 3^n. E.g. 365^2 - 364^2 - 3^3 = 27. (the Pythagorean triangle (365, 364, 27). - Gary W. Adamson (qntmpkt(AT)yahoogroups.com), Jun 25 2006
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CROSSREFS
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Cf. A000364, A002438, A083885, A086645.
Sequence in context: A067381 A155612 A145215 this_sequence A156153 A026000 A058475
Adjacent sequences: A083881 A083882 A083883 this_sequence A083885 A083886 A083887
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry (pbarry(AT)wit.ie), May 09 2003
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EXTENSIONS
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Additional comments from Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 10 2005
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