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%I #31 Sep 27 2022 07:12:18
%S 0,32,8160,2072640,526442432,133714305120,33962907058080,
%T 8626444678447232,2191082985418538880,556526451851630428320,
%U 141355527687328710254432,35903747506129640774197440,9119410511029241427935895360,2316294366053921193054943224032
%N a(n) are solutions of the 2 equations: 7*a(n) +1 = c(n)^2 and 9*a(n) +1 = b(n)^2.
%H Colin Barker, <a href="/A159679/b159679.txt">Table of n, a(n) for n = 1..400</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (255,-255,1).
%F G.f.: 32*x^2 / ((1-x)*(1-254*x+x^2)).
%F c(n) = A157456(n).
%F b(n) = A159678(n).
%F a(n+3) = 255*(a(n+2) -a(n+1)) + a(n).
%F a(n) = 2*A077412(n-2)*A077412(n-1). - _Johannes Boot_, Jan 17 2011
%F 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
%F a(n) = (8/63)*(-1 + ChebyshevU(n, 127) - 253*ChebyshevU(n-1, 127)). - _G. C. Greubel_, Sep 27 2022
%p for a from 1 by 2 to 100000 do b:=sqrt((9*a*a-2)/7): if (trunc(b)=b) then
%p n:=(a*a-1)/7: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: end if: end do:
%p # Second program
%p seq((8/63)*(simplify(ChebyshevU(n,127) -253*ChebyshevU(n-1,127)) -1), n=1..30); # _G. C. Greubel_, Sep 27 2022
%t 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 *)
%o (PARI) concat(0, Vec(32*x^2/(-x^3+255*x^2-255*x+1) + O(x^100))) \\ _Colin Barker_, Mar 18 2014
%o (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
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 50); [0] cat Coefficients(R!(32*x^2/((1-x)*(1-254*x+x^2)))); // _G. C. Greubel_, Jun 03 2018
%o (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
%Y Cf. A077412, A157456, A159678.
%K nonn,easy
%O 1,2
%A _Paul Weisenhorn_, Apr 19 2009
%E More terms from _Colin Barker_, Mar 18 2014
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