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%I #12 Jan 16 2024 16:54:53
%S 1,3,1,1,5,5,1,7,11,1,9,19,1,11,29,1,13,41,1,15,55,1,17,71,1,19,89,1,
%T 21,109,1,23,131,1,25,155,1,27,181,1,29,209,1,31,239,1,33,271,1,35,
%U 305,1,37,341,1,39,379,1,41,419,1,43,461,1,45,505,1,47,551,1,49,599,1,51,649,1,53
%N Interlaced triples: a(3n+1)=1, a(3n+2) = 2n+3, a(3n+3)= A028387(n).
%C Three consecutive terms of the sequence define a polynomial a(3n+1)*x^2 - a(3n+2)*x +a(3n+3) which has a root x = 1 + n + golden ratio.
%H G. C. Greubel, <a href="/A124925/b124925.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,3,0,0,-3,0,0,1).
%F a(n) = 3*a(n-3) -3*a(n-6) +a(n-9).
%F G.f.: x*(-1-3*x-x^2+2*x^3+4*x^4-2*x^5-x^6-x^7+x^8)/((x-1)^3*(1+x+x^2)^3).
%p seq(coeff(series(x*(-1-3*x-x^2+2*x^3+4*x^4-2*x^5-x^6-x^7+x^8)/((x-1)^3*( 1+x+x^2)^3), x, n+1), x, n), n = 1 ..80); # _G. C. Greubel_, Nov 19 2019
%t Rest@CoefficientList[Series[x*(-1-3*x-x^2+2*x^3+4*x^4-2*x^5-x^6-x^7+x^8 )/((x-1)^3*(1+x+x^2)^3), {x, 0, 80}], x] (* _G. C. Greubel_, Nov 19 2019 *)
%t LinearRecurrence[{0,0,3,0,0,-3,0,0,1},{1,3,1,1,5,5,1,7,11},80] (* _Harvey P. Dale_, Jan 16 2024 *)
%o (PARI) my(x='x+O('x^80)); Vec(x*(-1-3*x-x^2+2*x^3+4*x^4-2*x^5-x^6-x^7+ x^8)/((x-1)^3*(1+x+x^2)^3)) \\ _G. C. Greubel_, Nov 19 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 80); Coefficients(R!( x*(-1-3*x-x^2+2*x^3+4*x^4-2*x^5-x^6-x^7+x^8)/((x-1)^3*(1+x+x^2)^3) )); // _G. C. Greubel_, Nov 19 2019
%o (Sage)
%o def A124925_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P(x*(-1-3*x-x^2+2*x^3+4*x^4-2*x^5-x^6-x^7+x^8)/((x-1)^3*(1+x+x^2)^3)).list()
%o a=A124925_list(80); a[1:] # _G. C. Greubel_, Nov 19 2019
%o (GAP) a:=[1,3,1,1,5,5,1,7,11];; for n in [10..80] do a[n]:=3*a[n-3]-3*a[n-6]+a[n-9]; od; a; # _G. C. Greubel_, Nov 19 2019
%Y Cf. A001622.
%K nonn,easy,less
%O 1,2
%A _Gary W. Adamson_, Nov 12 2006
%E Edited and extended by _R. J. Mathar_, Mar 28 2010
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