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Search: id:A120694
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| A120694 |
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Sequence demonstrating the Pythagorean theorem. |
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+0
1
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| 1, 25, 1201, 58825, 2882401, 141237625, 6920643601, 339111536425
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Each of the two terms in M^n * [1,0] are the hypotenuse (left term); and longest leg in a Pythagorean triple: e.g. M^3 * [1,0] = [58825, 58824]; such that 58825^2 - 58824^2 = 7^3; generally: sqrt:((a(n)^2 - (a(n)-1)^2)) = 7^n. Characteristic polynomial of M = x^2 - 50x + 49.
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FORMULA
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a(n) = 50*a(n-1) - 49*a(n-2), n>1. Let M = the 2 X 2 matrix [25, 24; 24, 25]. Then, a(n) = left term in M^n * [1,0].
a(n)=(1/2)*(1+*49^n), with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Aug 28 2008]
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EXAMPLE
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58825 = a(3) = 50*a(2) - 49*a(1) = 50*1201 - 49*25.
58825 = a(3) = left term in M^3 * [1,0] = [58825, 58824].
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CROSSREFS
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Adjacent sequences: A120691 A120692 A120693 this_sequence A120695 A120696 A120697
Sequence in context: A012508 A112102 A012799 this_sequence A012809 A014769 A012851
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KEYWORD
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nonn
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AUTHOR
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Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 28 2006
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