A123004 - OEIS

%I #26 Sep 16 2024 06:37:38

%S 0,1,2,29,108,941,4582,32689,179928,1177081,6852362,43131749,

%T 257572548,1593438821,9626191342,59088353209,358831489968,

%U 2194871810161,13360530869522,81592856993069,497198985724188,3034219396275101

%N Expansion of g.f. x^2/(1 - 2*x - 25*x^2).

%D Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.

%H G. C. Greubel, <a href="/A123004/b123004.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2, 25).

%F a(n) = 2*a(n-1) + 25*a(n-2).

%F a(n+1) = ((1+sqrt(26))^n - (1-sqrt(26))^n)/(2*sqrt(26)). - _Rolf Pleisch_, Jul 06 2009

%F a(n) = (5*i)^(n-2)*ChebyshevU(n-2, -i/5). - _G. C. Greubel_, Jul 12 2021

%t Rest@CoefficientList[Series[x^2/(1 -2*x -25*x^2), {x,0,40}], x]

%t Join[{a=0,b=1},Table[c=2*b+25*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 01 2011 *)

%o (Magma) [n le 2 select n-1 else 2*Self(n-1) +25*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jul 12 2021

%o (Sage) [(5*i)^(n-2)*chebyshev_U(n-2, -i/5) for n in [1..30]] # _G. C. Greubel_, Jul 12 2021

%Y Sequences of the form (m*i)^(n-2)*ChebyshevU(n-2, -i/m): A131577 (m=0), A000129 (m=1), A085449 (m=2), A002534 (m=3), A161007 (m=4), this sequence (m=5), A123005 (m=7), A123006 (m=11).

%K nonn,easy

%O 1,3

%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 23 2006

%E Definition replaced by generating function - the Assoc. Eds. of the OEIS, Mar 27 2010