A089927 -id:A089927 - cp's OEIS frontend

A099025 Expansion of 1 / ((1+x) * (1-5*x+x^2)).

1, 4, 20, 95, 456, 2184, 10465, 50140, 240236, 1151039, 5514960, 26423760, 126603841, 606595444, 2906373380, 13925271455, 66719983896, 319674648024, 1531653256225, 7338591633100, 35161304909276, 168467932913279, 807178359657120, 3867423865372320

Offset: 0

Ralf Stephan, Sep 26 2004

			1 + 4*x + 20*x^2 + 95*x^3 + 456*x^4 + 2184*x^5 + 10465*x^6 + ...

Colin Barker, Table of n, a(n) for n = 0..1000
R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., Vol. 58, No. 2 (2020), 140-142.
Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
Index entries for linear recurrences with constant coefficients, signature (4,4,-1).

First differences of A089927. First differences are in A003769 and A005386. Pairwise sums are in A004254.

I:=[1, 4, 20]; [n le 3 select I[n] else 4*Self(n-1) + 4*Self(n-2) - Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 31 2017

CoefficientList[Series[1/((1+x)*(1-5*x+x^2)), {x,0,50}], x] (* or *) LinearRecurrence[{4,4,-1}, {1,4,20}, 30] (* G. C. Greubel, Dec 31 2017 *)

Vec(1/(1+x)/(1-5*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

{a(n) = (3 * (-1)^n + 38 * subst( poltchebi(n), x, 5/2) - 8 * subst( poltchebi(n-1), x, 5/2)) / 21} /* Michael Somos, Jan 25 2013 */

a(n) = (1/7)*[A030221(n+2) - A003501(n+2) + (-1)^n].
a(n) = 5*a(n-1) -a(n-2) +(-1)^n, a(0)=1, a(1)=4. - Vincenzo Librandi, Mar 22 2011
G.f.: 1 / ((1 + x) * (1 - 5*x + x^2)).
a(-3-n) = -a(n). - Michael Somos, Jan 25 2013
a(n) = (2^(-n)*(3*(-2)^n+(9-2*sqrt(21))*(5-sqrt(21))^n+(5+sqrt(21))^n*(9+2*sqrt(21))))/21. - Colin Barker, Nov 02 2016

A160695 Integers m such that 3m+1 and 7m+1 are both perfect squares.

Original entry on oeis.org

0, 5, 120, 2760, 63365, 1454640, 33393360, 766592645, 17598237480, 403992869400, 9274237758725, 212903475581280, 4887505700610720, 112199727638465285, 2575706229984090840, 59129043561995624040, 1357392295695915262085, 31160893757444055403920

Offset: 1

Paul Weisenhorn, May 24 2009

The ansatz 3*a(n)+1=A^2, 7*a(n)+1=B^2 is equivalent to the Pell equation x^2-21*y^2=1 (see A077232 for d=21), with x=(21*a(n)+5)/2 and y=A*B/2.
The associated A are in A004253, the B in A030221.
Bisection of A089927. - R. J. Mathar, Jul 10 2009

Michael De Vlieger, Table of n, a(n) for n = 1..736
Francesca Arici and Jens Kaad, Gysin sequences and SU(2)-symmetries of C*-algebras, arXiv:2012.11186 [math.OA], 2020.
Index entries for linear recurrences with constant coefficients, signature (24,-24,1).

Cf. A004253, A004254, A030221, A089927, A160682, A245031, A334673.

j:=0: for n from 0 to 1000000 do a:=sqrt(3*n+1): b:=sqrt(7*n+1):
if (trunc(a)=a) and (trunc(b)=b) then j:=j+1: print(j,n,a,b): end if:
end do:

LinearRecurrence[{24,-24,1},{0,5,120},30] (* Harvey P. Dale, Dec 17 2013 *)

a(n) = 24*a(n-1) - 24*a(n-2) + a(n-3).
a(n) = (A004253(n)^2 - 1)/3 = (A030221(n)^2 - 1)/7.
a(n) = ((5+w)/2*((23+5*w)/2)^(n-1) + (5-w)/2*((23-5*w)/2)^(n-1) - 5)/21; where w=sqrt(21). [Corrected by Kevin Ryde, Sep 11 2020]
G.f.: 5*x^2/((1-x)*(x^2-23*x+1)). - R. J. Mathar, Jul 10 2009
From Francesca Arici, Sep 12 2020: (Start)
a(n) = 23*a(n-1) - a(n-2) + 5.
a(n) = A004254(n)* A004254(n+1). (End)
a(n) = 5*A334673(n-1). - Hugo Pfoertner, Apr 07 2021

Edited and extended by R. J. Mathar, Jul 10 2009

Name edited by Michel Marcus, Sep 12 2020

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A099025 Expansion of 1 / ((1+x) * (1-5*x+x^2)).

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A160695 Integers m such that 3m+1 and 7m+1 are both perfect squares.

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cp's OEIS Frontend

A099025 Expansion of 1 / ((1+x) * (1-5*x+x^2)).

Views

Author

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Examples

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Magma

Mathematica

PARI

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A160695 Integers m such that 3*m+1 and 7*m+1 are both perfect squares.

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Maple

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A160695 Integers m such that 3m+1 and 7m+1 are both perfect squares.