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A248217 a(n) = 8^n - 2^n. 2
0, 6, 60, 504, 4080, 32736, 262080, 2097024, 16776960, 134217216, 1073740800, 8589932544, 68719472640, 549755805696, 4398046494720, 35184372056064, 281474976645120, 2251799813554176, 18014398509219840, 144115188075331584, 1152921504605798400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

If 2^(n+1) is the length of the even leg of a primitive Pythagorean triangle (PPT) then it constrains the odd leg to have a length of 4^n-1 and the hypotenuse to have a length of 4^n+1. The resulting triangle has a semiperimeter of 4^n+2^n, an area of 8^n-2^n and an inradius of 2^n-1. For n > 0, a(n) is the area of such triangles. - Frank M Jackson, Sep 07 2018

LINKS

Table of n, a(n) for n=0..20.

Index entries for linear recurrences with constant coefficients, signature (10,-16).

FORMULA

G.f.: 6*x/((1-2*x)*(1-8*x)).

a(n) = 10*a(n-1) - 16*a(n-2).

a(n) = 2^n*(4^n-1) = A000079(n) * A024036(n) = A001018(n) - A000079(n).

E.g.f.: exp(2*x)*(-1 + exp(6*x)). - Stefano Spezia, Sep 07 2018

MATHEMATICA

Table[8^n - 2^n, {n, 0, 25}] (* or *) CoefficientList[Series[6 x /((1 - 2 x) (1 - 8 x)), {x, 0, 30}], x]

LinearRecurrence[{10, -16}, {0, 6}, 30] (* Harvey P. Dale, Mar 29 2015 *)

CoefficientList[Series[E^(2 x) (-1 + E^(6 x)), {x, 0, 50}], x]*Table[n!, {n, 0, 50}] (* Stefano Spezia, Sep 07 2018 *)

PROG

(MAGMA) [8^n-2^n: n in [0..25]];

(PARI) a(n) = 8^n-2^n; \\ Altug Alkan, Sep 07 2018

CROSSREFS

Cf. similar sequences listed in A248216.

Cf. A000079, A024036.

Sequence in context: A179711 A061495 A220411 * A102232 A121113 A213269

Adjacent sequences:  A248214 A248215 A248216 * A248218 A248219 A248220

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, Oct 04 2014

STATUS

approved

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Last modified July 20 05:43 EDT 2019. Contains 325168 sequences. (Running on oeis4.)