A159679 a(n) are solutions of the 2 equations: 7*a(n) +1 = c(n)^2 and 9*a(n) +1 = b(n)^2.

0, 32, 8160, 2072640, 526442432, 133714305120, 33962907058080, 8626444678447232, 2191082985418538880, 556526451851630428320, 141355527687328710254432, 35903747506129640774197440, 9119410511029241427935895360, 2316294366053921193054943224032

Offset: 1

Author

Paul Weisenhorn, Apr 19 2009

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 1..400
• Index entries for linear recurrences with constant coefficients, signature (255,-255,1).

Crossrefs

Cf. A077412, A157456, A159678.

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); [0] cat Coefficients(R!(32*x^2/((1-x)*(1-254*x+x^2)))); // G. C. Greubel, Jun 03 2018

Maple

for a from 1 by 2 to 100000 do b:=sqrt((9*a*a-2)/7): if (trunc(b)=b) then
n:=(a*a-1)/7: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: end if: end do:
# Second program
seq((8/63)*(simplify(ChebyshevU(n,127) -253*ChebyshevU(n-1,127)) -1), n=1..30); # G. C. Greubel, Sep 27 2022

Mathematica

LinearRecurrence[{255,-255,1}, {0, 32, 8160}, 50] (* or *) CoefficientList[Series[32*x^2/((1-x)*(x^2-254*x+1)), {x,0,50}], x] (* G. C. Greubel, Jun 03 2018 *)

PARI

concat(0, Vec(32*x^2/(-x^3+255*x^2-255*x+1) + O(x^100))) \\ Colin Barker, Mar 18 2014

PARI

a(n) = round((-16+(8+3*sqrt(7))*(127+48*sqrt(7))^(-n)+(8-3*sqrt(7))*(127+48*sqrt(7))^n)/126) \\ Colin Barker, Jul 25 2016

SageMath

[(8/63)*(-1 + chebyshev_U(n, 127) - 253*chebyshev_U(n-1, 127)) for n in range(1,30)] # G. C. Greubel, Sep 27 2022

Formula

G.f.: 32*x^2 / ((1-x)*(1-254*x+x^2)).

c(n) = A157456(n).

b(n) = A159678(n).

a(n+3) = 255*(a(n+2) -a(n+1)) + a(n).

a(n) = 2*A077412(n-2)*A077412(n-1). - Johannes Boot, Jan 17 2011

a(n) = (-16+(8+3*sqrt(7))*(127+48*sqrt(7))^(-n)+(8-3*sqrt(7))*(127+48*sqrt(7))^n)/126. - Colin Barker, Jul 25 2016

a(n) = (8/63)*(-1 + ChebyshevU(n, 127) - 253*ChebyshevU(n-1, 127)). - G. C. Greubel, Sep 27 2022

Extensions

More terms from Colin Barker, Mar 18 2014