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%I
%S 0,1,7,48,329,2255,15456,105937,726103,4976784,34111385,233802911,
%T 1602508992,10983760033,75283811239,516002918640,3536736619241,
%U 24241153416047,166151337293088,1138818207635569,7805576116155895,53500214605455696,366695926122033977
%N a(n) = 7*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.
%C 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).
%C A004187 == One third of Fibonacci numbers (Integers Only) [From _Vladimir Joseph Stephan Orlovsky_, Oct 25 2009]
%C a(n) = A167816(4*n). [From _Reinhard Zumkeller_, Nov 13 2009]
%C This is a divisibility sequence.
%C 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]
%D 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;.
%D 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.
%D 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.
%H Vincenzo Librandi, <a href="/A004187/b004187.txt">Table of n, a(n) for n = 0..1000</a>
%H Zvonko Cerin, <a href="http://dx.doi.org/10.2478/BF02475651">Some alternating sums of Lucas numbers</a>, Centr. Eur. J. Math. vol 3 no 1 (2005) 1-13.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Rea#recLCC">Index entries for sequences related to linear recurrences with constant coefficients</a>, signature (7,-1).
%F G.f.: x/(1-7*x+x^2).
%F a(n) = F(4*n)/3, where F=A000045 (the Fibonacci sequence).
%F 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.
%F 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
%F [A049685(n-1), a(n)] = [1,5; 1,6]^n * [1,0]. - _Gary W. Adamson_, Mar 21 2008
%F a(n) = (((7+sqrt(45))/2)^n-((7-sqrt(45))/2)^n)/sqrt(45). - Noureddine Chair, Aug 31 2011
%F a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k)*6^k. - Philippe Deléham, Feb 10 2012
%F a(n) = (A081072(n)/3)-1. - _Martin Ettl_, Nov 11 2012
%F Product {n >= 1} (1 + 1/a(n)) = 1/5*(5 + 3*sqrt(5)). - _Peter Bala_, Dec 23 2012
%F Product {n >= 2} (1 - 1/a(n)) = 1/14*(5 + 3*sqrt(5)). - _Peter Bala_, Dec 23 2012
%t LinearRecurrence[{7,-1},{0,1},30] (* From Harvey P. Dale, Jul 13 2011 *)
%t CoefficientList[Series[x/(1 - 7*x + x^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Dec 23 2012 *)
%o (Mupad) numlib::fibonacci(4*n)/3 $ n = 0..25; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2008
%o (Sage) [lucas_number1(n,7,1) for n in range(27)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
%o (Sage) [fibonacci(4*n)/3 for n in xrange(0, 21)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009]
%o (MAGMA) [Fibonacci(4*n)/3 : n in [0..30]]; // Vincenzo Librandi, Jun 07 2011
%o (PARI) a(n)=fibonacci(4*n)/3 \\ _Charles R Greathouse IV_, Mar 09, 2012
%o (Maxima)
%o 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 */
%o (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
%Y Cf. A000027, A001906, A001353, A004254, A001109, A049685, A033888. a(n)=sqrt((A056854(n)^2 - 4)/45).
%Y Second column of array A028412.
%K nonn,easy,mult,changed
%O 0,3
%A _N. J. A. Sloane_, _R. K. Guy_
%E Entry improved by comments from Michael Somos and _Wolfdieter Lang_, Aug 02 2000
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