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A159668 Expansion of (1 - x)/(1 - 28*x + x^2). 5
1, 27, 755, 21113, 590409, 16510339, 461699083, 12911063985, 361048092497, 10096435525931, 282339146633571, 7895399670214057, 220788851619360025, 6174192445671866643, 172656599627192905979, 4828210597115729500769, 135017240119613233115553 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Previous name was: The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 13*n(j)+1=a(j)*a(j) and 15*n(j)+1=b(j)*b(j) with positive integer numbers.

Positive values of x (or y) satisfying x^2 - 28xy + y^2 + 26 = 0. - Colin Barker, Feb 23 2014

REFERENCES

Mohammad K. Azarian, Diophantine Pair, Problem B-881, Fibonacci Quarterly, Vol. 37, No. 3, August 1999, pp. 277-278.  Solution published in Vol. 38, No. 2, May 2000, pp. 183-184.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..199

Index entries for linear recurrences with constant coefficients, signature (28,-1).

FORMULA

G.f.: (1 - x)/(1 - 28*x + x^2).

The a(j) recurrence is a(0)=1; a(1)=27; a(t+2)=28*a(t+1)-a(t) resulting in terms 1, 27, 755, 21113.. (this sequence)

The b(j) recurrence is b(0)=1; b(1)=29; b(t+2)=28*b(t+1)-b(t) resulting in terms 1, 29, 811, 22679... (A159669)

The n(j) recurrence is n(0)=n(1)=0; n(2)=56; n(t+3)=783*(n(t+2)-n(t+1))+n(t) resulting in terms 0, 0, 56, 43848, 34289136... (A159673)

a(n) = (1/30)*(15-sqrt(195))*(1+(14+sqrt(195))^(2*n+1))/(14+sqrt(195))^n. [Bruno Berselli, Feb 25 2014]

a(0)=1, a(1)=27, a(n)=28*a(n-1)-a(n-2). - Harvey P. Dale, Apr 09 2014

MAPLE

for a from 1 by 2 to 100000 do b:=sqrt((15*a*a-2)/13): if (trunc(b)=b) then

n:=(a*a-1)/13: La:=[op(La), a]:Lb:=[op(Lb), b]:Ln:=[op(Ln), n]: endif: enddo:

MATHEMATICA

CoefficientList[Series[(-x + 1)/(x^2 - 28 x + 1), {x, 0, 20}], x] (* Vincenzo Librandi, Feb 25 2014 *)

LinearRecurrence[{28, -1}, {1, 27}, 40] (* Harvey P. Dale, Apr 09 2014 *)

PROG

(PARI) Vec((-x+1)/(x^2-28*x+1) + O(x^100)) \\ Colin Barker, Feb 23 2014

(MAGMA) I:=[1, 27]; [n le 2 select I[n] else 28*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 25 2014

CROSSREFS

Cf. A157456, A159669, A159673.

Cf. similar sequences listed in A238379.

Sequence in context: A042406 A279652 A284039 * A158645 A232951 A138979

Adjacent sequences:  A159665 A159666 A159667 * A159669 A159670 A159671

KEYWORD

nonn,easy

AUTHOR

Paul Weisenhorn, Apr 19 2009

EXTENSIONS

More terms from Colin Barker, Feb 23 2014

New name and offset changed to 0 from Joerg Arndt, Feb 23 2014

STATUS

approved

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Last modified February 25 11:05 EST 2018. Contains 299653 sequences. (Running on oeis4.)