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A120694 Sequence demonstrating the Pythagorean theorem. 0
1, 25, 1201, 58825, 2882401, 141237625, 6920643601, 339111536425, 16616465284801, 814206798955225, 39896133148806001, 1954910524291494025, 95790615690283207201, 4693740168823877152825, 229993268272369980488401, 11269670145346129043931625 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..15.

Index entries for linear recurrences with constant coefficients, signature (50, -49).

FORMULA

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]

EXAMPLE

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].

MATHEMATICA

LinearRecurrence[{50, -49}, {1, 25}, 20] (* Harvey P. Dale, Dec 31 2011 *)

CROSSREFS

Sequence in context: A012508 A112102 A012799 * A012809 A014769 A012851

Adjacent sequences:  A120691 A120692 A120693 * A120695 A120696 A120697

KEYWORD

nonn

AUTHOR

Gary W. Adamson, Jun 28 2006

EXTENSIONS

More terms from Harvey P. Dale, Dec 31 2011

STATUS

approved

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Last modified February 25 22:37 EST 2021. Contains 341618 sequences. (Running on oeis4.)