A097779 -id:A097779 - cp's OEIS frontend

A090733 a(n) = 25*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 25.

2, 25, 623, 15550, 388127, 9687625, 241802498, 6035374825, 150642568127, 3760028828350, 93850078140623, 2342491924687225, 58468448039040002, 1459368709051312825, 36425749278243780623, 909184363247043202750

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18 2004

A Chebyshev T-sequence with Diophantine property.

a(n) gives the general (nonnegative integer) solution of the Pell equation a^2 - 69*(3*b)^2 =+4 together with the companion sequence b(n)=A097780(n-1), n>=0.

			(x,y) =(2,0), (25;1), (623;25), (15550;624), ... give the nonnegative integer solutions to x^2 - 69*(3*y)^2 =+4.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Indranil Ghosh, Table of n, a(n) for n = 0..714
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (25,-1)

Cf. A046069, A082974.
a(n)=sqrt(4 + 69*(3*A097780(n-1))^2), n>=1.
Cf. A077428, A078355 (Pell +4 equations).
Cf. A097779 for 2*T(n, 23/2).

a[0] = 2; a[1] = 25; a[n_] := 25a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)

[lucas_number2(n,25,1) for n in range(0,20)] # Zerinvary Lajos, Jun 26 2008

a(n) = S(n, 25) - S(n-2, 25) = 2*T(n, 25/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 25)=A097780(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120.
a(n) = ap^n + am^n, with ap := (25+3*sqrt(69))/2 and am := (25-3*sqrt(69))/2.
G.f.: (2-25*x)/(1-25*x+x^2).

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

A097098 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n and having a total of k level steps (1,0) to the left of the first (1,1) step and to the right of the last (1,-1) step (i.e., total length of the two tails is k; can be easily expressed in terms of RNA secondary structure terminology).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 0, 0, 1, 2, 2, 3, 0, 0, 1, 5, 4, 3, 4, 0, 0, 1, 11, 10, 6, 4, 5, 0, 0, 1, 25, 22, 15, 8, 5, 6, 0, 0, 1, 58, 50, 33, 20, 10, 6, 7, 0, 0, 1, 135, 116, 75, 44, 25, 12, 7, 8, 0, 0, 1, 317, 270, 174, 100, 55, 30, 14, 8, 9, 0, 0, 1, 750, 634, 405, 232, 125, 66, 35, 16, 9, 10, 0, 0, 1

Offset: 0

Emeric Deutsch, Sep 15 2004

Row sums yield the RNA secondary structure numbers (A004148). Column 0 is A097779.

			Triangle starts:
  1;
  0, 1;
  0, 0, 1;
  1, 0, 0, 1;
  1, 2, 0, 0, 1;
  2, 2, 3, 0, 0, 1;
  5, 4, 3, 4, 0, 0, 1;
Row n has n+1 terms.
T(6,3)=4 because we have (HHH)UHD, (HH)UHD(H), (H)UHD(HH) and UHD(HHH), where U=(1,1), H=(1,0) and D=(1,-1); the 3 required level steps 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, Sem. Loth. Comb. B08l (1984) 79-86. [Formerly: Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, p. 79-86.]

Cf. A004148, A097779.

a:=proc(n) if n=0 then 1 else sum(binomial(j,n-j)*binomial(j,n-j+1)/j,j=ceil((n+1)/2)..n) fi end: T:=proc(n,k) if k<0 then 0 elif kT(n-1,k-1): matrix(13,13,TT); # yields the sequence in matrix form:

(* c is A004148 *)
c[n_] := c[n] = If[n == 0, 1, c[n-1] + Sum[c[k]*c[n-2-k], {k, n-2}]];
T[n_, k_] := T[n, k] = Which[k < 0, 0, k < n-1, (k+1)*(c[n-k] - 2*c[n-k-1] + c[n-k-2]), k == n-1, 0, k == n, 1, True, 0];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 30 2024 *)

T(n,k) = (k+1)T(n-k, 0) for k < n; T(n,n) = 1. T(n,0) = a(n)-2a(n-1) + a(n-2) (n >= 2), where a(n) = Sum_{k=ceiling((n+1)/2)..n} binomial(k, n-k)*binomial(k, n-k+1)/k = A004148(n).

G.f.: 1/(1-tz) + (1-z)(g-1-zg)/(1-tz)^2, where g = (1 - z + z^2 - sqrt(1 - 2z - z^2 - 2z^3 + z^4))/(2z^2) is the g.f. of the RNA secondary structure numbers (A004148).

cp's OEIS Frontend

A090733 a(n) = 25*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 25.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions