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A120694
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Sequence demonstrating the Pythagorean theorem.
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0
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1, 25, 1201, 58825, 2882401, 141237625, 6920643601, 339111536425, 16616465284801, 814206798955225, 39896133148806001, 1954910524291494025, 95790615690283207201, 4693740168823877152825, 229993268272369980488401, 11269670145346129043931625
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OFFSET
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0,2
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COMMENTS
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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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LINKS
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Table of n, a(n) for n=0..15.
Index entries for linear recurrences with constant coefficients, signature (50, -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, Aug 28 2008]
G.f.: (1-25*x)/(49*x^2-50*x+1) [From Harvey P. Dale, Dec 31 2011]
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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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MATHEMATICA
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LinearRecurrence[{50, -49}, {1, 25}, 20] (* Harvey P. Dale, Dec 31 2011 *)
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CROSSREFS
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Sequence in context: A012508 A112102 A012799 * A012809 A014769 A012851
Adjacent sequences: A120691 A120692 A120693 * A120695 A120696 A120697
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KEYWORD
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nonn
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AUTHOR
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Gary W. Adamson, Jun 28 2006
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EXTENSIONS
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More terms from Harvey P. Dale, Dec 31 2011
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STATUS
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approved
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