Search: id:A057086
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%I A057086
%S A057086 1,10,90,800,7100,63000,559000,4960000,44010000,390500000,3464900000,
%T A057086 30744000000,272791000000,2420470000000,21476790000000,190563200000000,
%U A057086 1690864100000000,15003009000000000,133121449000000000
%N A057086 Scaled Chebyshev U-polynomials evaluated at sqrt(10)/2.
%C A057086 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 A057086 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.
%D A057086 A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. 
               Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=10, q=-10.
%D A057086 W. Lang, On polynomials related to powers of the generating function 
               of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs.(38) and 
               (45),lhs, m=10.
%H A057086 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A057086 a(n) = 10*(a(n-1)-a(n-2)), a(-1)=0, a(0)=1.
%F A057086 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 A057086 a(2*k)= A057080(k)*10^k, a(2*k+1)=A001090(k)*10^(k+1).
%F A057086 G.f.: 1/(1-10*x+10*x^2).
%F A057086 a(n)=Sum_[k, 0<=k<=n} A109466(n,k)*10^k. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 28 2008]
%F A057086 a(n)=-(1/30)*[5-sqrt(15)]^(n+1)*sqrt(15)+(1/30)*sqrt(15)*[5+sqrt(15)]^(n+1), 
               with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Nov 20 2008]
%o A057086 (Other) sage: [lucas_number1(n,10,10) for n in xrange(1, 20)]# [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2009]
%Y A057086 Sequence in context: A009454 A004985 A164552 this_sequence A092420 A010579 
               A010576
%Y A057086 Adjacent sequences: A057083 A057084 A057085 this_sequence A057087 A057088 
               A057089
%K A057086 nonn,easy
%O A057086 0,2
%A A057086 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 11 2000

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