

A057093


Scaled Chebyshev Upolynomials evaluated at i*sqrt(10)/2. Generalized Fibonacci sequence.


10



1, 10, 110, 1200, 13100, 143000, 1561000, 17040000, 186010000, 2030500000, 22165100000, 241956000000, 2641211000000, 28831670000000, 314728810000000, 3435604800000000, 37503336100000000, 409389409000000000, 4468927451000000000, 48783168600000000000
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OFFSET

0,2


COMMENTS

This is the m=10 member of the mfamily 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/(1m*xm*x^2). The instances m=1..9 are A000045 (Fibonacci), A002605, A030195, A057087, A057088, A057089, A057090, A057091, A057092.
With the roots rp(m) := (m+sqrt(m*(m+4)))/2 and rm(m) := (msqrt(m*(m+4)))/2 the Binet form of these msequences 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(n1) and 10*a(n2), resp.


LINKS

Colin Barker, Table of n, a(n) for n = 0..963
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the nanacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424434. Case n>n+1, a=0,b=1; p=10, q=10.
Tanya Khovanova, Recursive Sequences
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408419. Eqs.(39) and (45),rhs, m=10.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (10,10).


FORMULA

a(n) = 10*(a(n1) + a(n2)), 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..n} 9^k*A063967(n,k).  Philippe Deléham, Nov 03 2006
a(n) = (1/70)*(5sqrt(35))^(n+1)*sqrt(35) + (1/70)*sqrt(35)*(5+sqrt(35))^(n+1), with n>=0.  Paolo P. Lava, Nov 20 2008


MATHEMATICA

Join[{a=0, b=1}, Table[c=10*b+10*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)


PROG

(Sage) [lucas_number1(n, 10, 10) for n in range(1, 19)] # Zerinvary Lajos, Apr 26 2009
(PARI) Vec(1/(110*x10*x^2) + O(x^30)) \\ Colin Barker, Jun 14 2015


CROSSREFS

Sequence in context: A289933 A289967 A343331 * A055276 A264915 A289414
Adjacent sequences: A057090 A057091 A057092 * A057094 A057095 A057096


KEYWORD

nonn,easy


AUTHOR

Wolfdieter Lang, Aug 11 2000


EXTENSIONS

Extended by T. D. Noe, May 23 2011


STATUS

approved



