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%I M4948 #143 Feb 28 2024 06:33:03
%S 0,1,14,195,2716,37829,526890,7338631,102213944,1423656585,
%T 19828978246,276182038859,3846719565780,53577891882061,
%U 746243766783074,10393834843080975,144767444036350576,2016350381665827089,28084137899285228670,391161580208327374291,5448177985017298011404
%N Standard deviation of A007654.
%C a(n) corresponds also to one-sixth the area of Fleenor-Heronian triangle with middle side A003500(n). - _Lekraj Beedassy_, Jul 15 2002
%C a(n) give all (nontrivial, integer) solutions of Pell equation b(n+1)^2 - 48*a(n+1)^2 = +1 with b(n+1)=A011943(n), n>=0.
%C For n>=3, a(n) equals the permanent of the (n-2) X (n-2) tridiagonal matrix with 14's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - _John M. Campbell_, Jul 08 2011
%C For n>1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,13}. - _Milan Janjic_, Jan 25 2015
%C 6*a(n)^2 = 6*S(n-1, 14)^2 is the triangular number Tri((T(n, 7) - 1)/2) with Tri = A000217 and T = A053120. This is instance k = 3 of the general k-identity given in a comment to A001109. - _Wolfdieter Lang_, Feb 01 2016
%D D. A. Benaron, personal communication.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Indranil Ghosh, <a href="/A007655/b007655.txt">Table of n, a(n) for n = 1..874</a> (terms 1..100 from T. D. Noe)
%H R. Flórez, R. A. Higuita and A. Mukherjee, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mukherjee/mukh2.html">Alternating Sums in the Hosoya Polynomial Triangle</a>, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
%H D. S. Hale, <a href="https://www.jstor.org/stable/3614690">3165. Perfect Squares of the Form 48n^2+1</a>, Math. Gaz., Oct. 1966, page 307.
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Murray S. Klamkin, <a href="https://www.jstor.org/stable/2689119">Perfect Squares of the Form (m^2 - 1)a_n^2 + t</a>, Math. Mag., 1969, page 111.
%H E. K. Lloyd, <a href="http://www.jstor.org/stable/3619201">The standard deviation of 1, 2, ..., n, Pell's equation and rational triangles</a>, The Mathematical Gazette, Vol. 81, No. 491 (Jul., 1997), pp. 231-243.
%H Dino Lorenzini and Z. Xiang, <a href="http://alpha.math.uga.edu/~lorenz/IntegralPoints.pdf">Integral points on variable separated curves</a>, Preprint 2016.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (14,-1).
%F a(n) = 14*a(n-1) - a(n-2).
%F G.f.: x^2/(1-14*x+x^2).
%F a(n+1) ~ 1/24*sqrt(3)*(2 + sqrt(3))^(2*n). - Joe Keane (jgk(AT)jgk.org), May 15 2002
%F a(n+1) = S(n-1, 14), n>=0, with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. S(-1, x) := 0. See A049310.
%F a(n+1) = ( (7+4*sqrt(3))^n - (7-4*sqrt(3))^n )/(8*sqrt(3)).
%F a(n+1) = sqrt((A011943(n)^2 - 1)/48), n>=0.
%F Chebyshev's polynomials U(n-2, x) evaluated at x=7.
%F a(n) = A001353(2n)/4. - _Lekraj Beedassy_, Jul 15 2002
%F 4*a(n+1) + A046184(n) = A055793(n+2) + A098301(n+1) 4*a(n+1) + A098301(n+1) + A055793(n+2) = A046184(n+1) (4*a(n+1))^2 = A098301(2n+1) (conjectures). - _Creighton Dement_, Nov 02 2004
%F (4*a(n))^2 = A103974(n)^2 - A011922(n-1)^2. - _Paul D. Hanna_, Mar 06 2005
%F From _Mohamed Bouhamida_, May 26 2007: (Start)
%F a(n) = 13*( a(n-1) + a(n-2) ) - a(n-3).
%F a(n) = 15*( a(n-1) - a(n-2) ) + a(n-3). (End)
%F a(n) = b such that (-1)^n/4*Integral_{x=-Pi/2..Pi/2} (sin((2*n-2)*x))/(2-sin(x)) dx = c+b*log(3). - _Francesco Daddi_, Aug 02 2011
%F a(n+2) = Sum_{k=0..n} A101950(n,k)*13^k. - _Philippe Deléham_, Feb 10 2012
%F Product {n >= 1} (1 + 1/a(n)) = 1/3*(3 + 2*sqrt(3)). - _Peter Bala_, Dec 23 2012
%F Product {n >= 2} (1 - 1/a(n)) = 1/7*(3 + 2*sqrt(3)). - _Peter Bala_, Dec 23 2012
%F a(n) = (A028230(n) - A001570(n))/2. - _Richard R. Forberg_, Nov 14 2013
%F E.g.f.: 1 - exp(7*x)*(12*cosh(4*sqrt(3)*x) - 7*sqrt(3)*sinh(4*sqrt(3)*x))/12. - _Stefano Spezia_, Dec 11 2022
%e G.f. = x^2 + 14*x^3 + 195*x^4 + 2716*x^5 + 37829*x^6 + 526890*x^7 + ...
%p 0,seq(orthopoly[U](n,7),n=0..30); # _Robert Israel_, Feb 04 2016
%t Table[GegenbauerC[n, 1, 7], {n,0,20}] (* _Vladimir Joseph Stephan Orlovsky_, Sep 11 2008 *)
%t LinearRecurrence[{14,-1}, {0,1}, 20] (* _Vincenzo Librandi_, Feb 02 2016 *)
%t ChebyshevU[Range[21] -2, 7] (* _G. C. Greubel_, Dec 23 2019 *)
%t Table[Sum[Binomial[n, 2 k - 1]*7^(n - 2 k + 1)*48^(k - 1), {k, 1, n}], {n, 0, 15}] (* _Horst H. Manninger_, Jan 16 2022 *)
%o (Sage) [lucas_number1(n,14,1) for n in range(0,20)] # _Zerinvary Lajos_, Jun 25 2008
%o (Sage) [chebyshev_U(n,7) for n in (-1..20)] # _G. C. Greubel_, Dec 23 2019
%o (Magma) [n le 2 select n-1 else 14*Self(n-1)-Self(n-2): n in [1..70]]; // _Vincenzo Librandi_, Feb 02 2016
%o (PARI) concat(0, Vec((x^2/(1-14*x+x^2) + O(x^30)))) \\ _Michel Marcus_, Feb 02 2016
%o (PARI) vector(21, n, polchebyshev(n-2, 2, 7) ) \\ _G. C. Greubel_, Dec 23 2019
%o (GAP) m:=7;; a:=[0,1];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Dec 23 2019
%Y Cf. A000217, A001570, A003500, A011922, A011943, A011945, A028230, A046184, A049310, A053120, A055793, A067900, A098301, A101950, A103974.
%Y Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), A004191 (m=6), this sequence (m=7), A077412 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33).
%Y Cf. A323182.
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_
%E Chebyshev comments from _Wolfdieter Lang_, Nov 08 2002
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