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%I #35 Dec 30 2023 23:50:49
%S 1,10,90,800,7100,63000,559000,4960000,44010000,390500000,3464900000,
%T 30744000000,272791000000,2420470000000,21476790000000,
%U 190563200000000,1690864100000000,15003009000000000,133121449000000000,1181184400000000000,10480629510000000000
%N Scaled Chebyshev U-polynomials evaluated at sqrt(10)/2.
%C This is the m=10 member of the m-family of sequences S(n,sqrt(m))*(sqrt(m))^n; for S(n,x) see Formula. The m=4..9 instances are A001787, A030191, A030192, A030240, A057084-5 and the m=1..3 signed sequences are A010892, A009545, A057083.
%C The characteristic roots are rp(m) := (m + sqrt(m*(m-4)))/2 and rm(m) := (m-sqrt(m*(m-4)))/2 and a(n,m)= (rp(m)^(n+1) - rm(m)^(n+1))/(rp(m) - rm(m)) is the Binet form of these m-sequences.
%H Colin Barker, <a href="/A057086/b057086.txt">Table of n, a(n) for n = 0..1000</a>
%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/5-5/horadam.pdf">Special properties of the sequence W_n(a,b; p,q)</a>, Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=10, q=-10.
%H Wolfdieter Lang, <a href="http://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan's numbers</a>, Fib. Quart. 38 (2000) 408-419. Eqs.(38) and (45),lhs, m=10.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-10).
%F a(n) = 10*(a(n-1) - a(n-2)), a(-1)=0, a(0)=1.
%F a(n) = S(n, sqrt(10))*(sqrt(10))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
%F a(2*k) = A057080(k)*10^k, a(2*k+1) = A001090(k)*10^(k+1).
%F G.f.: 1/(1-10*x+10*x^2).
%F a(n) = Sum_{k=0..n} A109466(n,k)*10^k. - _Philippe Deléham_, Oct 28 2008
%t Join[{a=1,b=10},Table[c=10*b-10*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 20 2011 *)
%o (Sage) [lucas_number1(n,10,10) for n in range(1, 20)] # _Zerinvary Lajos_, Apr 26 2009
%o (PARI) Vec(1/(1-10*x+10*x^2) + O(x^30)) \\ _Colin Barker_, Jun 14 2015
%o (Magma) [(10)^n*Evaluate(DicksonSecond(n, 1/10), 1): n in [0..30]]; # _G. C. Greubel_, May 02 2022
%Y Cf. A001090, A001787, A030191, A030192, A030240, A049310.
%Y Cf. A009545, A010892, A057080, A057083, A057084, A057085.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 11 2000