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Search: id:A057087
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| A057087 |
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Scaled Chebyshev U-polynomials evaluated at i. Generalized Fibonacci sequence. |
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+0
12
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| 1, 4, 20, 96, 464, 2240, 10816, 52224, 252160, 1217536, 5878784, 28385280, 137056256, 661766144, 3195289600, 15428222976, 74494050304, 359689093120, 1736732573696, 8385686667264, 40489676963840, 195501454524416
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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a(n) gives the length of the word obtained after n steps with the substitution rule 0->1111, 1->11110, starting from 0. The number of 1's and 0's of this word is 4*a(n-1) and 4*a(n-2), resp.
Inverse binomial transform of odd Pell bisection A001653. With a leading zero, inverse binomial transform of even Pell bisection A001542, divided by 2. - Paul Barry (pbarry(AT)wit.ie), May 16 2003
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REFERENCES
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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=4, q=4.
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=4.
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LINKS
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Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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a(n) = 4*(a(n-1)+a(n-2)), a(-1)=0, a(0)=1.
a(n)= S(n, 2*i)*(-2*i)^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
G.f.: 1/(1-4*x-4*x^2).
a(n)=Sum_{k, 0<=k<=n}3^k*A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 03 2006
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PROGRAM
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(PARI) a(n)=if(n<0, 0, (2*I)^n*subst(I*poltchebi(n+1)+poltchebi(n), 'x, -I)/2) /* Michael Somos Sep 16 2005 */
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CROSSREFS
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Pairwise sums are in A086347.
Adjacent sequences: A057084 A057085 A057086 this_sequence A057088 A057089 A057090
Sequence in context: A094971 A099025 A008353 this_sequence A098225 A073532 A103771
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 11 2000
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