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A057093
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Scaled Chebyshev U-polynomials evaluated at i*sqrt(10)/2. Generalized Fibonacci sequence.
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7
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1, 10, 110, 1200, 13100, 143000, 1561000, 17040000, 186010000, 2030500000, 22165100000, 241956000000, 2641211000000, 28831670000000, 314728810000000, 3435604800000000, 37503336100000000, 409389409000000000, 4468927451000000000, 48783168600000000000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| This is the m=10 member of the m-family of sequences a(m,n)= S(n,i*sqrt(m))*(-i*sqrt(m))^n, with S(n,x) given in Formula and g.f.: 1/(1-m*x-m*x^2). The instances m=1..9 are A000045 (Fibonacci), A002605, A030195, A057087-92.
With the roots rp(m) := (m+sqrt(m*(m+4)))/2 and rm(m) := (m-sqrt(m*(m+4)))/2 the Binet form of these m-sequences is a(n,m)= (rp(m)^(n+1)-rm(m)^(n+1))/(rp(m)-rm(m)).
a(n) gives the length of the word obtained after n steps with the substitution rule 0->1^10, 1->(1^10)0, starting from 0. The number of 1's and 0's of this word is 10*a(n-1) and 10*a(n-2), resp.
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REFERENCES
| 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.
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs.(39) and (45),rhs, m=10.
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| a(n) = 10*(a(n-1)+a(n-2)), a(-1)=0, a(0)=1.
a(n)= S(n, i*sqrt(10))*(-i*sqrt(10))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
G.f.: 1/(1-10*x-10*x^2).
a(n)=Sum_{k, 0<=k<=n}9^k*A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 03 2006
a(n)=-(1/70)*[5-sqrt(35)]^(n+1)*sqrt(35)+(1/70)*sqrt(35)*[5+sqrt(35)]^(n+1), with n>=0 [From Paolo P. Lava (paoloplava(AT)gmail.com), Nov 20 2008]
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MATHEMATICA
| Join[{a=0, b=1}, Table[c=10*b+10*a; a=b; b=c, {n, 100}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 17 2011*)
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PROG
| (Other) sage: [lucas_number1(n, 10, -10) for n in xrange(1, 19)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2009]
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CROSSREFS
| Sequence in context: A102092 A189788 A105279 * A055276 A143749 A049398
Adjacent sequences: A057090 A057091 A057092 * A057094 A057095 A057096
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 11 2000
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EXTENSIONS
| Extended by T. D. Noe (noe(AT)sspectra.com), May 23 2011
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