login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A007655 Standard deviation of A007654.
(Formerly M4948)
25
0, 1, 14, 195, 2716, 37829, 526890, 7338631, 102213944, 1423656585, 19828978246, 276182038859, 3846719565780, 53577891882061, 746243766783074, 10393834843080975, 144767444036350576, 2016350381665827089, 28084137899285228670, 391161580208327374291 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

a(n) corresponds also to one-sixth the area of Fleenor-Heronian triangle with middle side A003500(n). - Lekraj Beedassy, Jul 15 2002

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.

Number of units of a(n) belongs to a periodic sequence: 0, 1, 4, 5, 6, 9.We conclude that a(n) and a(n+6) have the same number of units. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 05 2009

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

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

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

REFERENCES

D. A. Benaron, personal communication.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=1..100

R. Flórez, R. A. Higuita, A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).

M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, 2014; http://matinf.pmfbl.org/wp-content/uploads/2015/01/za-arhiv-18.-1.pdf

Tanya Khovanova, Recursive Sequences

E. K. Lloyd, The standard deviation of 1, 2, .., n, Pell's equation and rational triangles, The Mathematical Gazette, Vol. 81, No. 491 (Jul., 1997), pp. 231-243.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (14,-1).

FORMULA

a(n) = 14*a(n-1) - a(n-2). G.f.: x^2/(1-14*x+x^2).

a(n+1) ~ 1/24*sqrt(3)*(2 + sqrt(3))^(2*n). - Joe Keane (jgk(AT)jgk.org), May 15 2002

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.

a(n+1) = ((7+4*sqrt(3))^n - (7-4*sqrt(3))^n)/(8*sqrt(3)).

a(n+1) = sqrt((A011943(n)^2 - 1)/48), n>=0.

Chebyshev's polynomials U(n-2, x) evaluated at x=7.

a(n) = A001353(2n)/4. - Lekraj Beedassy, Jul 15 2002

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

(4*a(n))^2 = A103974(n)^2 - A011922(n-1)^2. - Paul D. Hanna, Mar 06 2005

a(n) = 13*(a(n-1)+a(n-2))-a(n-3), a(n) = 15*(a(n-1)-a(n-2))+a(n-3). a(n)=14*a(n-1)-a(n-2). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 26 2007

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

a(n+2) = Sum_{k, 0<=k<=n} A101950(n,k)*13^k. - Philippe Deléham, Feb 10 2012

Product {n >= 1} (1 + 1/a(n)) = 1/3*(3 + 2*sqrt(3)). - Peter Bala, Dec 23 2012

Product {n >= 2} (1 - 1/a(n)) = 1/7*(3 + 2*sqrt(3)). - Peter Bala, Dec 23 2012

a(n) = (A028230(n) - A001570(n))/2. - Richard R. Forberg, Nov 14 2013

EXAMPLE

G.f. = x^2 + 14*x^3 + 195*x^4 + 2716*x^5 + 37829*x^6 + 526890*x^7 + ...

MAPLE

0, seq(orthopoly[U](n, 7), n=0..30); # Robert Israel, Feb 04 2016

MATHEMATICA

lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 7]], {n, 0, 8^2}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)

LinearRecurrence[{14, -1}, {0, 1}, 20] (* Vincenzo Librandi, Feb 02 2016 *)

PROG

(Sage) [lucas_number1(n, 14, 1) for n in xrange(0, 20)] # Zerinvary Lajos, Jun 25 2008

(MAGMA) [n le 2 select n-1 else 14*Self(n-1)-Self(n-2): n in [1..70]]; // Vincenzo Librandi, Feb 02 2016

(PARI) concat(0, Vec((x^2/(1-14*x+x^2) + O(x^30)))) \\ Michel Marcus, Feb 02 2016

CROSSREFS

Cf. A001353, A003500.

Cf. A011945, A067900.

Cf. A103974, A011922.

Sequence in context: A086946 A158530 A171319 * A208383 A208110 A208842

Adjacent sequences:  A007652 A007653 A007654 * A007656 A007657 A007658

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Chebyshev comments from Wolfdieter Lang, Nov 08 2002

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 25 20:30 EDT 2016. Contains 273298 sequences.