A159669 Expansion of x*(1 + x)/(1 - 28*x + x^2).

1, 29, 811, 22679, 634201, 17734949, 495944371, 13868707439, 387827863921, 10845311482349, 303280893641851, 8481019710489479, 237165271000063561, 6632146568291290229, 185462938641156062851, 5186330135384078469599, 145031780852113041085921

Offset: 1

Author

Paul Weisenhorn, Apr 19 2009

Keywords

• nonn
• easy

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.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Index entries for linear recurrences with constant coefficients, signature (28,-1).

Crossrefs

Cf. A097311, A157456, A159668, A159673.

Programs

Magma

A097311:= func< n | Evaluate(ChebyshevSecond(n-1), 14) >;
[A097311(n+1) + A097311(n): n in [1..30]]; // G. C. Greubel, Sep 25 2022

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:
# Second program
seq(simplify(ChebyshevU(n, 14) +ChebyshevU(n-1, 14)), n=1..30); # G. C. Greubel, Sep 25 2022

Mathematica

CoefficientList[Series[(1+x)/(1-28x+x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 26 2014 *)
LinearRecurrence[{28,-1},{1,29},20] (* Harvey P. Dale, Jul 01 2019 *)

PARI

Vec(x*(x+1)/(x^2-28*x+1) + O(x^100)) \\ Colin Barker, Feb 24 2014

PARI

a(n) = round((14+sqrt(195))^(-n)*(-13-sqrt(195)+(-13+sqrt(195))*(14+sqrt(195))^(2*n))/26) \\ Colin Barker, Jul 25 2016

SageMath

def A159669(n): return chebyshev_U(n-1, 14) + chebyshev_U(n-2, 14)
[A159669(n) for n in range(1,30)] # G. C. Greubel, Sep 25 2022

Formula

The a(j) recurrence is a(1)=1, a(2)=27, a(t+2) = 28*a(t+1) - a(t) resulting in terms 1, 27, 755, 21113, ... (A159668).

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

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).

G.f.: x*(1+x)/(1-28*x+x^2). - Vincenzo Librandi, Feb 26 2014

a(n) = (14+sqrt(195))^(-n)*(-13-sqrt(195)+(-13+sqrt(195))*(14+sqrt(195))^(2*n))/26. - Colin Barker, Jul 25 2016

a(n) = chebyshev_U(n-1, 14) + chebyshev_U(n-2, 14) = A097311(n) + A097311(n-1). - G. C. Greubel, Sep 25 2022

Extensions

More terms and new name from Colin Barker, Feb 24 2014