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A004189
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a(n) = 10*a(n-1)-a(n-2); a(0) = 0, a(1) = 1.
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27
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0, 1, 10, 99, 980, 9701, 96030, 950599, 9409960, 93149001, 922080050, 9127651499, 90354434940, 894416697901, 8853812544070, 87643708742799, 867583274883920, 8588189040096401, 85014307126080090, 841554882220704499
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Indices of square numbers which are also generalized pentagonal numbers.
If t(n) denotes the n-th triangular number, t(A105038(n))=a(n)*a(n+1). - Robert Phillips (bobanne(AT)bellsouth.net), May 25 2008
The n:th term is a(n)=((5+Sqrt(24))^n-(5-Sqrt(24))^n)/(2*Sqrt(24)) [From Sture Sjoestedt (sture.sjostedt(AT)spray.se), May 31 2009]
Number of units of a(n) belongs to a periodic sequence: 0, 1, 0, 9.We conclude that a(n) and a(n+4) have the same number of units. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 05 2009]
For n>=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 10'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
| 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=10, q=-1.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pps. 231-242.
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=12.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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) = S(2*n-1, sqrt(12))/sqrt(12) = S(n-1, 10); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(-1, x) := 0.
a(n)={[(5+2*sqrt(6))^n - (5-2*sqrt(6))^n]}/4*sqrt(6). G.f.(x)=x/(1-10*x+x^2). - Barry E. Williams, Aug 18 2000
G.f.: x/(1-10*x+x^2). a(-n)=-a(n). - Michael Somos Sep 05 2006
a(n) = 9*(a(n-1)+a(n-2))-a(n-3), a(n) = 11*(a(n-1)-a(n-2))+a(n-3). a(n)=10*a(n-1)-a(n-2). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 26 2007
a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k)*9^k. - DELEHAM Philippe, Feb 10 2012
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EXAMPLE
| a(2)=10 and (3(-8)^2-(-8))/2=10^2, a(3)=99 and (3(81)^2-(81))/2=99^2. - Michael Somos Sep 05 2006
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MATHEMATICA
| lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 5]], {n, 0, 8^2}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 11 2008]
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PROG
| (PARI) {a(n)=subst(poltchebi(n+1)-5*poltchebi(n), 'x, 5)/24} /* Michael Somos Sep 05 2006 */
sage: [lucas_number1(n, 10, 1) for n in range(22)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
(MAGMA) [ n eq 1 select 0 else n eq 2 select 1 else 10*Self(n-1)-Self(n-2): n in [1..20] ]; // Vincenzo Librandi, Aug 19 2011
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CROSSREFS
| Cf. A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913.
A001079(n) = sqrt{24*[a(n)^2]+1}, that is a(n) = sqrt((A001079(n)^2-1)/24).
A046173(n)=a(2n-1).
Sequence in context: A129542 A171315 A081109 * A179558 A179556 A179477
Adjacent sequences: A004186 A004187 A004188 * A004190 A004191 A004192
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KEYWORD
| easy,nonn,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 07 2000
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