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A057087
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Scaled Chebyshev U-polynomials evaluated at i. Generalized Fibonacci sequence.
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26
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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; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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), respectively.
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, May 16 2003
For positive n, a(n) equals the permanent of the nXn tridiagonal matrix with 4's along the main diagonal, and 2's along the superdiagonal and the subdiagonal. [From John M. Campbell, Jul 19 2011]
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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=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
| 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) = 4*(a(n-1)+a(n-2)), a(-1)=0, a(0)=1.
G.f.: 1/(1-4*x-4*x^2).
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.
a(n)=Sum_{k, 0<=k<=n}3^k*A063967(n,k) . - Philippe DELEHAM, Nov 03 2006
a(n)=-(1/8)*sqrt(2)*[2-2*sqrt(2)]^(n+1)+(1/8)*[2+2*sqrt(2)]^(n+1)*sqrt(2), [From Paolo P. Lava, Nov 20 2008]
((2+sqrt(8))^(n+1)-(2-sqrt(8))^(n+1))/sqrt(32). [From Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009]
a(n)=A000129(n+1)*A000079(n). [From R. J. Mathar, Jul 08 2009]
Contribution from Johannes W. Meijer, Aug 01 2010: (Start)
Limit(a(n+k)/a(k), k=infinity) = A084128(n) + 2*A057087(n-1)*sqrt(2)
Limit(A084128(n)/A057087(n-1), n=infinity) = sqrt(2)
(End)
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MATHEMATICA
| Join[{a=1, b=4}, Table[c=4*b+4*a; a=b; b=c, {n, 100}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 16 2011*)
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PROG
| (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 */
(Sage) [lucas_number1(n, 4, -4) for n in xrange(1, 23)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
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CROSSREFS
| Pairwise sums are in A086347.
Appears in A086346, A086347 and A086348. [Johannes W. Meijer, Aug 01 2010]
Sequence in context: A094971 A099025 A008353 * A151254 A098225 A073532
Adjacent sequences: A057084 A057085 A057086 * A057088 A057089 A057090
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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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