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A004187
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a(n) = 7*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.
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41
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0, 1, 7, 48, 329, 2255, 15456, 105937, 726103, 4976784, 34111385, 233802911, 1602508992, 10983760033, 75283811239, 516002918640, 3536736619241, 24241153416047, 166151337293088, 1138818207635569, 7805576116155895, 53500214605455696, 366695926122033977
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OFFSET
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0,3
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COMMENTS
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Define the sequence T(a_0,a_1) by a_{n+2} is the greatest integer such that a_{n+2}/a_{n+1}<a_{n+1}/a_n for n >= 0 . A004178 (with initial 0 omitted) is T(1,7).
A004187 == One third of Fibonacci numbers (Integers Only) [From Vladimir Joseph Stephan Orlovsky, Oct 25 2009]
a(n) = A167816(4*n). [From Reinhard Zumkeller, Nov 13 2009]
This is a divisibility sequence.
For n>=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 7's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). [From John M. Campbell, Jul 08 2011]
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REFERENCES
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D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993;.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=7, q=-1.
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq.(44), lhs, m=9.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Zvonko Cerin, Some alternating sums of Lucas numbers, Centr. Eur. J. Math. vol 3 no 1 (2005) 1-13.
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index to divisibility sequences
Index entries for sequences related to linear recurrences with constant coefficients, signature (7,-1).
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FORMULA
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G.f.: x/(1-7*x+x^2).
a(n) = F(4*n)/3, where F=A000045 (the Fibonacci sequence).
a(n) = S(2*n-1, sqrt(9))/sqrt(9) = S(n-1, 7); S(n, x) := U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310.
a(n)=sum(i=0..n-1, C(2*n-1-i, i)*5^(n-i-1) ). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
[A049685(n-1), a(n)] = [1,5; 1,6]^n * [1,0]. - Gary W. Adamson, Mar 21 2008
a(n) = (((7+sqrt(45))/2)^n-((7-sqrt(45))/2)^n)/sqrt(45). - Noureddine Chair, Aug 31 2011
a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k)*6^k. - Philippe Deléham, Feb 10 2012
a(n) = (A081072(n)/3)-1. - Martin Ettl, Nov 11 2012
Product {n >= 1} (1 + 1/a(n)) = 1/5*(5 + 3*sqrt(5)). - Peter Bala, Dec 23 2012
Product {n >= 2} (1 - 1/a(n)) = 1/14*(5 + 3*sqrt(5)). - Peter Bala, Dec 23 2012
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MATHEMATICA
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LinearRecurrence[{7, -1}, {0, 1}, 30] (* From Harvey P. Dale, Jul 13 2011 *)
CoefficientList[Series[x/(1 - 7*x + x^2), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 23 2012 *)
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PROG
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(Mupad) numlib::fibonacci(4*n)/3 $ n = 0..25; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2008
(Sage) [lucas_number1(n, 7, 1) for n in range(27)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
(Sage) [fibonacci(4*n)/3 for n in xrange(0, 21)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009]
(MAGMA) [Fibonacci(4*n)/3 : n in [0..30]]; // Vincenzo Librandi, Jun 07 2011
(PARI) a(n)=fibonacci(4*n)/3 \\ Charles R Greathouse IV, Mar 09, 2012
(Maxima)
a[0]:0$ a[1]:1$ a[n]:=7*a[n-1] - a[n-2]$ A004187(n):=a[n]$ makelist(A004187(n), n, 0, 30); /* Martin Ettl, Nov 11 2012 */
(MAGMA) /* By definition: */ [n le 2 select n-1 else 7*Self(n-1)-Self(n-2): n in [1..23]]; // Bruno Berselli, Dec 24 2012
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CROSSREFS
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Cf. A000027, A001906, A001353, A004254, A001109, A049685, A033888. a(n)=sqrt((A056854(n)^2 - 4)/45).
Second column of array A028412.
Sequence in context: A081106 A036829 A164591 * A180167 A186653 A024092
Adjacent sequences: A004184 A004185 A004186 * A004188 A004189 A004190
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KEYWORD
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nonn,easy,mult
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AUTHOR
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N. J. A. Sloane, R. K. Guy
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EXTENSIONS
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Entry improved by comments from Michael Somos and Wolfdieter Lang, Aug 02 2000
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STATUS
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approved
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