A097777 -id:A097777 - cp's OEIS frontend

A092499 Chebyshev polynomials S(n-1,21) with Diophantine property.

0, 1, 21, 440, 9219, 193159, 4047120, 84796361, 1776676461, 37225409320, 779956919259, 16341869895119, 342399310878240, 7174043658547921, 150312517518628101, 3149388824232642200, 65986852791366858099

Offset: 0

Rainer Rosenthal, Apr 05 2004

Sequence R_21: Starts with 0,1,21 and satisfies A*C=B^2-1 for successive A,B,C.
The natural numbers a(n)=n satisfy the recurrence a(n-1)*a(n+1)=a(n)^2-1. Let R_r denote the sequence starting with 0,1,r and with this recurrence. We see that R_2 = "the natural numbers" and we find R_3 = A001906. These R_r form a "family" of sequences, which coincides with the m-family (r=m-2, n -> n+1) provided by Wolfdieter Lang (see A078368). This sequence R_21 is strongly related to A041833, which gives the denominators in the continued fraction of sqrt(437).
All positive integer solutions of Pell equation b(n)^2 - 437*a(n)^2 = +4 together with b(n)=A097777(n), n>=0.
For n>=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 21'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,...,20}. - Milan Janjic, Jan 25 2015

			a(3)=440 because a(1)*440 = a(2)^2-1.

Indranil Ghosh, Table of n, a(n) for n = 0..756
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (21,-1).

Cf. R_3=A001906, R_4=A001353, R_5=A004254, R_6=A001109, R_7=A004187, R_8=A001090, R_9=A018913, R_10=A004189, R_11=A004190, R_12=A004191, R_13=A078362, R_14=A007655, R_15=A078364, R_16=A077412, R_17=A078366, R_18=A049660, R_19=A078368, R_20=A075843, R_21=this, sequence, R_22=A077421. See also A041219 and A041917.

LinearRecurrence[{21,-1},{0,1},30] (* Harvey P. Dale, Apr 23 2015 *)

[lucas_number1(n,21,1) for n in range(0,20)] # Zerinvary Lajos, Jun 25 2008

a(0)=0, a(1)=1, a(2)=21 and a(n-1)*a(n+1) = a(n)^2-1
a(n) = S(n-1, 21)=U(n-1, 21/2) with S(n, x)=U(n, x/2) Chebyshev's polynomials of the 2nd kind, A049310. S(-1, x)= 0 = U(-1, x).
a(n) = S(2*n-1, sqrt(23))/sqrt(23), n>=1.
a(n) = 21*a(n-1)-a(n-2), n >= 1; a(0)=0, a(1)=1.
a(n) = (ap^n-am^n)/(ap-am) with ap := (21+sqrt(437))/2 and am := (21-sqrt(437))/2.
G.f.: x/(1-21*x+x^2).
a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k)*20^k. - Philippe Deléham, Feb 10 2012
Product {n >= 1} (1 + 1/a(n)) = 1/19*(19 + sqrt(437)). - Peter Bala, Dec 23 2012
Product {n >= 2} (1 - 1/a(n)) = 1/42*(19 + sqrt(437)). - Peter Bala, Dec 23 2012

Extension, Chebyshev and Pell comments from Wolfdieter Lang, Aug 31 2004

Corrected by T. D. Noe, Nov 07 2006

A098056 Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n containing k subwords of the type uh^ju, dH^jd, or dh^ju for some j>0, where u=(1,1), d=(1,-1) and h=(1,0) (can be easily expressed using RNA secondary structure terminology).

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 15, 2, 27, 9, 1, 48, 29, 5, 84, 80, 21, 147, 198, 74, 4, 257, 463, 230, 27, 1, 451, 1033, 667, 125, 7, 796, 2235, 1811, 488, 43, 1413, 4727, 4694, 1676, 219, 6, 2526, 9828, 11700, 5317, 946, 54, 1, 4544, 20192, 28252, 15813, 3696, 326, 9, 8226, 41100

Offset: 0

Emeric Deutsch, Sep 11 2004

Row sums are the RNA secondary structure numbers (A004148).
T(n,0) = A098057(n).
Sum(k*T(n,k),k>=0) = A187259(n).

			Triangle starts:
  1;
  1;
  1;
  2;
  4;
  8;
  15,2;
  27,9,1;
  48,29,5;
  84.80,21;
  147,198,74,7;
  ...
It seems that the number r(n) of terms in row n>=3 is given by r(n)=n/2-1 if n=2 (mod 4) and r(n)=2*round(n/4)-1 otherwise (here round(m) is the nearest integer to m).
T(7,1)=9 because we have h(uhu)hdd, (uhhu)hdd, (uhu)hhdd, (uhu)hddh, uh(dhu)hd and the reflections of the first four paths in a vertical axis; here u=(1,1), h=(1,0), d=(1,-1) and the pertinent subwords are shown between parentheses.

I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.
M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08; Sem. Loth. Comb. B08l (1984) 79-86.

Cf. A041048, A098057, A187259.

G.f.=G=G(t, z) satisfies G = 1 + zG + z^2*[H + 2tzH/(1-z)+t^2*z^2*H/(1-z)^2+ z/(1-z)][G-(1-t)zH/(1-z)^2], where H=(1-z)^2*G-1+z.
The 4-variate g.f. G(t,s,v,z) of peakless Motzkin paths, where t, s, v mark subwords of the types uH^ju, dH^jd, dH^ju, respectively, and z marks length, satisfies the equation
G = 1+zG+z^2*[H + (t+s)zH/(1-z)+tsz^2*H/(1-z)^2+z/(1-z)][G-(1-v)zH/(1-z)^2],
where H = (1-z)[(1-z)G-1]. As special cases we get the current sequence A098056 and the sequences A097777 and A098083.

A187257 Number of UH^jU's for some j>0, where U=(1,1) and H=(1,1), in all peakless Motzkin paths of length n (can be easily expressed using RNA secondary structure terminology).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 5, 17, 52, 150, 417, 1135, 3047, 8103, 21409, 56303, 147569, 385808, 1006775, 2623477, 6828941, 17761182, 46165507, 119937807, 311485907, 808731993, 2099358057, 5448906369, 14141429587, 36699034884, 95237147804, 247149109444, 641388458961

Offset: 0

Emeric Deutsch, May 05 2011

nonn

a(n)=Sum(k*A097777(n,k), k>=0).

			a(6)=1 because among the 17 (=A004148(6)) peakless Motzkin paths of length 6 only UHUHDD contains subwords of type UH^jU for some j>0 (here D=(1,-1)).

Cf. A097777, A004148

G.f.=z^5*g^3*(g-1)/[(1-z)(1-z^2*g^2)], where g=1+zg+z^2*g(g-1)=A004148.

cp's OEIS Frontend

A092499 Chebyshev polynomials S(n-1,21) with Diophantine property.

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A098056 Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n containing k subwords of the type uh^ju, dH^jd, or dh^ju for some j>0, where u=(1,1), d=(1,-1) and h=(1,0) (can be easily expressed using RNA secondary structure terminology).

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A187257 Number of UH^jU's for some j>0, where U=(1,1) and H=(1,1), in all peakless Motzkin paths of length n (can be easily expressed using RNA secondary structure terminology).

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