A001081 -id:A001081 - cp's OEIS frontend

A099272 Unsigned member r=-14 of the family of Chebyshev sequences S_r(n) defined in A092184.

0, 1, 14, 225, 3584, 57121, 910350, 14508481, 231225344, 3685097025, 58730327054, 936000135841, 14917271846400, 237740349406561, 3788928318658574, 60385112749130625, 962372875667431424, 15337580897929772161, 244438921491208923150, 3895685162961412998241

Offset: 0

Wolfdieter Lang, Oct 18 2004

((-1)^(n+1))*a(n) = S_{-14}(n), n>=0, defined in A092184.

Robert Israel, Table of n, a(n) for n = 0..831
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (15, 15, -1).

Cf. A001081.

f:= n -> (orthopoly[T](n,8)-(-1)^n)/9:
map(f, [$0..20]); # Robert Israel, Jun 04 2018

CoefficientList[Series[x (1-x)/(1-15 x-15 x^2+x^3),{x,0,33}],x] (* Vincenzo Librandi, Jun 05 2018 *)

a(n) = (T(n, 8)-(-1)^n)/9, with Chebyshev's polynomials of the first kind evaluated at x=8: T(n, 8)=A001081(n)=((8+3*sqrt(7))^n + (8-3*sqrt(7))^n)/2.
a(n) = 16*a(n-1)-a(n-2)+2*(-1)^(n+1), n>=2, a(0)=0, a(1)=1.
a(n) = 15*a(n-1) + 15*a(n-2) - a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=14.
G.f.: x*(1-x)/((1+x)*(1-16*x+x^2)) = x*(1-x)/(1-15*x-15*x^2+x^3) (from the Stephan link, see A092184).

A118894 Numbers m such that the Pell equation x^2-m*y^2=1 has fundamental solution with x even.

Original entry on oeis.org

3, 7, 11, 15, 19, 23, 27, 31, 35, 43, 47, 51, 59, 63, 67, 71, 75, 79, 83, 87, 91, 99, 103, 107, 115, 119, 123, 127, 131, 135, 139, 143, 151, 159, 163, 167, 171, 175, 179, 187, 191, 195, 199, 211, 215, 219, 223, 227, 231, 235, 239, 243, 247, 251, 255, 263, 267

Offset: 1

T. D. Noe, May 04 2006

nonn

Numbers m such that A002350(m) is even. These m can be used to generate consecutive odd powerful numbers, as in A076445. As shown by Lang, the solution of Pell's equation is greatly simplified by Chebyshev polynomials of the first kind T(n,x), which is illustrated in A001075 for the case m=3. In that case, the solutions are x=T(n,2), for integer n>0. For any m in this sequence, let E(k)=T(m+2mk,A002350(m)). Then E(k)-1 and E(k)+1 are consecutive odd powerful numbers for k=0,1,2,...

Wolfdieter Lang, Chebyshev Polynomials and Certain Quadratic Diophantine Equations
H. W. Lenstra Jr., Solving the Pell equation, Notices AMS, 49 (2002), 182-192.

Cf. A001075, A001091, A023038, A001081, A001085, A077424, A097310 (x solutions for m=3, 15, 35, 63, 99, 143, 195).

A140746 Numbers n such that n^2 + 3 is powerful, (i.e., is of the form a^2*b^3, with a>=1, b>=1).

Original entry on oeis.org

1, 37, 79196, 177833, 5290738, 9667939010

Offset: 1

Lekraj Beedassy, Jul 12 2008

Florian Luca proved that this sequence is infinite, by showing that 37*x(7*k) + 98*y(7*k) is in the sequence, where x(k) = A001081(k) and y(k) = A001080(k) are solutions of the Pell equation x^2 - 7*y^2 = 1. The sequence of these numbers is 37, 9667939010, 2524807950507510523, 659360302164952911361460078, ... - Amiram Eldar, Aug 22 2018

a(7) <= 457189690981. - Giovanni Resta, Aug 23 2018

			37 is the sequence since 37^2 + 3 = 1372 = 2^2 * 7^3 is powerful.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 37, pp 14, Ellipses, Paris 2008.

Cf. A001694 (powerful), A001080, A001081, A117950 (n^2+3).

powerfulQ[n_] := Min@FactorInteger[n][[All, 2]] > 1; Select[Range[100000], powerfulQ[#^2 + 3] &] (* Amiram Eldar, Aug 22 2018 *)

isok(n) = vecmin(factor(n^2+3)[,2]) > 1; \\ Michel Marcus, Aug 24 2018

a(5) corrected and a(6) removed by Amiram Eldar, Aug 22 2018

a(6) from Giovanni Resta, Aug 23 2018

A132594 Values X satisfying the equation: X(X + 1) - 7*Y^2 = 0.

Original entry on oeis.org

0, 63, 16128, 4096575, 1040514048, 264286471743, 67127723308800, 17050177433963583, 4330677940503441408, 1099975146710440154175, 279389356586511295719168, 70963796597827158672514623

Offset: 0

Mohamed Bouhamida, Nov 14 2007

nonn

The full set of integer solutions to this equation consists of the pairs [X(i),Y(i)] = [1+-A001081(i), Y(i)=A001080(i)]. The present generates every second one of them: a(n) = [A001081(2n)-1]/2. - R. J. Mathar, Nov 20 2007

Index entries for linear recurrences with constant coefficients, signature (255, -255, 1).

Cf. A007654.

Cf. A001080, A001081.

LinearRecurrence[{255,-255,1},{0,63,16128},20] (* Harvey P. Dale, Dec 15 2012 *)

a(0)=0, a(1)=63 and a(n)=254*a(n-1) - a(n-2) + 126.
G.f.: -63*x*(1+x)/(-1+x)/(1-254*x+x^2). a(n) = [A001081(2n)-1]/2. - R. J. Mathar, Nov 20 2007
a(0)=0, a(1)=63, a(2)=16128, a(n)=255*a(n-1)-255*a(n-2)+a(n-3). - Harvey P. Dale, Dec 15 2012

More terms from Max Alekseyev, Nov 13 2009

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A099272 Unsigned member r=-14 of the family of Chebyshev sequences S_r(n) defined in A092184.

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A118894 Numbers m such that the Pell equation x^2-m*y^2=1 has fundamental solution with x even.

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A140746 Numbers n such that n^2 + 3 is powerful, (i.e., is of the form a^2*b^3, with a>=1, b>=1).

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A132594 Values X satisfying the equation: X(X + 1) - 7*Y^2 = 0.

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