A094954 -id:A094954 - cp's OEIS frontend

A049685 a(n) = L(4*n+2)/3, where L=A000032 (the Lucas sequence).

1, 6, 41, 281, 1926, 13201, 90481, 620166, 4250681, 29134601, 199691526, 1368706081, 9381251041, 64300051206, 440719107401, 3020733700601, 20704416796806, 141910183877041, 972666870342481, 6666757908520326, 45694638489299801, 313195711516578281

Offset: 0

Clark Kimberling

In general, Sum_{k=0..n} binomial(2*n-k,k)j^(n-k) = (-1)^n*U(2n, I*sqrt(j)/2), i=sqrt(-1). - Paul Barry, Mar 13 2005
a(n) = L(n,7), where L is defined as in A108299; see also A033890 for L(n,-7). - Reinhard Zumkeller, Jun 01 2005
Take 7 numbers consisting of 5 ones together with any two successive terms from this sequence. This set has the property that the sum of their squares is 7 times their product. (R. K. Guy, Oct 12 2005.) See also A111216.
Number of 01-avoiding words of length n on alphabet {0,1,2,3,4,5,6} which do not end in 0. - Tanya Khovanova, Jan 10 2007
For positive n, a(n) equals the permanent of the (2n) X (2n) tridiagonal matrix with sqrt(5)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
From Wolfdieter Lang, Feb 09 2021: (Start)
All positive solutions of the Diophantine equation x^2 + y^2 - 7*x*y = -5 are given by [x(n) = S(n, 7) - S(n-1, 7), y(n) = x(n-1)], for all integer numbers n, with the Chebyshev S-polynomials (A049310), with S(-1, 0) = 0, and S(-n, x) = -S(n-2, x), for n >= 2. x(n) = a(n), for n >= 0.
This indefinite binary quadratic form has discriminant D = +45. There is only this family representing -5 properly with x and y positive, and there are no improper solutions.
All proper and improper solutions of the generalized Pell equation X^2 - 45*Y^2 = +4 are given, up to a combined sign change in X and Y, in terms of x(n) = a(n) from the preceding comment, by X(n) = x(n) + x(n-1) = S(n-1, 7) - S(n-2, 7) and Y(n) = (x(n) - x(n-1))/3 = S(n-1, 7), for all integer numbers n. For positive integers X(n) = A056854(n) and Y(n) = A004187(n). X(-n) = X(n) and Y(-n) = - Y(n), for n >= 1.
The two conjugated proper family of solutions are given by [X(3*n+1), Y(3*n+1)] and [X(3*n+2), Y(3*n+2)], and the one improper family by [X(3*n), Y(3*n)], for all integer numbers n.
This comment is inspired by a paper by Robert K. Moniot (private communication). See his Oct 04 2020 comment in A027941 related to the case of x^2 + y^2 - 3*x*y = -1 (special Markov solutions). (End)

			a(3) = L(4*3 + 2)/3 = 843/3 = 281. - _Indranil Ghosh_, Feb 06 2017

Indranil Ghosh, Table of n, a(n) for n = 0..1193
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
Tanya Khovanova, Recursive Sequences
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory, Vol. 7, No. 5 (2011), pp. 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012), The John Selfridge Memorial Volume.
Index entries for linear recurrences with constant coefficients, signature (7,-1).

Row 7 of array A094954. First differences of A004187.
Cf. A002310, A002320, A049310, A049685, A056854.
Cf. similar sequences listed in A238379.

[Lucas(4*n+2)/3: n in [0..30]]; // G. C. Greubel, Dec 17 2017

Table[LucasL[4*n+2]/3, {n,0,50}] (* or *) LinearRecurrence[{7,-1}, {1,6}, 50] (* G. C. Greubel, Dec 17 2017 *)

a(n)=(fibonacci(4*n+1)+fibonacci(4*n+3))/3 \\ Charles R Greathouse IV, Jun 16 2014

[lucas_number1(n,7,1)-lucas_number1(n-1,7,1) for n in range(1, 20)] # Zerinvary Lajos, Nov 10 2009

Let q(n, x) = Sum_{i=0, n} x^(n-i)*binomial(2*n-i, i); then q(n, 5)=a(n); a(n) = 7a(n-1) - a(n-2). - Benoit Cloitre, Nov 10 2002
From Ralf Stephan, May 29 2004: (Start)
a(n+2) = 7a(n+1) - a(n).
G.f.: (1-x)/(1-7x+x^2).
a(n)*a(n+3) = 35 + a(n+1)*a(n+2). (End)
a(n) = Sum_{k=0..n} binomial(n+k, 2k)*5^k. - Paul Barry, Aug 30 2004
If another "1" is inserted at the beginning of the sequence, then A002310, A002320 and A049685 begin with 1, 2; 1, 3; and 1, 1; respectively and satisfy a(n+1) = (a(n)^2+5)/a(n-1). - Graeme McRae, Jan 30 2005
a(n) = (-1)^n*U(2n, i*sqrt(5)/2), U(n, x) Chebyshev polynomial of second kind, i=sqrt(-1). - Paul Barry, Mar 13 2005
[a(n), A004187(n+1)] = [1,5; 1,6]^(n+1) * [1,0]. - Gary W. Adamson, Mar 21 2008
a(n) = S(n, 7) - S(n-1, 7) with Chebyshev S polynomials S(n-1, 7) = A004187(n), for n >= 0. - Wolfdieter Lang, Feb 09 2021
E.g.f.: exp(7*x/2)*(3*cosh(3*sqrt(5)*x/2) + sqrt(5)*sinh(3*sqrt(5)*x/2))/3. - Stefano Spezia, Apr 14 2025
From Peter Bala, May 04 2025: (Start)
a(n) = sqrt(2/9) * sqrt(1 - T(2*n+1, -7/2)), where T(k, x) denotes the k-th Chebyshev polynomial of the first kind.
a(n) divides a(3*n+1); a(n) divides a(5*n+2); in general, for k >= 0, a(n) divides a((2*k+1)*n + k).
The aerated sequence [b(n)]n>=1 = [1, 0, 6, 0, 41, 0, 281, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -9, Q = 1 of the 3-parameter family of divisibility sequences found by Williams and Guy.
Sum_{n >= 1} 1/(a(n) - 1/a(n)) = 1/5 (telescoping series: for n >= 1, 1/(a(n) - 1/a(n)) = 1/A290903(n-1) - 1/A290903(n).) (End)

A070997 a(n) = 8*a(n-1) - a(n-2), a(0)=1, a(-1)=1.

Original entry on oeis.org

1, 7, 55, 433, 3409, 26839, 211303, 1663585, 13097377, 103115431, 811826071, 6391493137, 50320119025, 396169459063, 3119035553479, 24556114968769, 193329884196673, 1522082958604615, 11983333784640247, 94344587318517361

Offset: 0

Joe Keane (jgk(AT)jgk.org), May 18 2002

A Pellian sequence.
In general, Sum_{k=0..n} binomial(2n-k,k)j^(n-k) = (-1)^n*U(2n,i*sqrt(j)/2), i=sqrt(-1). - Paul Barry, Mar 13 2005
a(n) = L(n,8), where L is defined as in A108299; see also A057080 for L(n,-8). - Reinhard Zumkeller, Jun 01 2005
Number of 01-avoiding words of length n on alphabet {0,1,2,3,4,5,6,7} which do not end in 0. - Tanya Khovanova, Jan 10 2007
Hankel transform of A158197. - Paul Barry, Mar 13 2009
For positive n, a(n) equals the permanent of the (2n) X (2n) tridiagonal matrix with sqrt(6)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
Values of x (or y) in the solutions to x^2 - 8xy + y^2 + 6 = 0. - Colin Barker, Feb 05 2014
From Klaus Purath, May 06 2025: (Start)
Nonnegative solutions to the Diophantine equation 3*b(n)^2 - 5*a(n)^2 = -2. The corresponding b(n) are A057080(n). Note that (b(n)*b(n+2) - b(n+1)^2)/2 = -5 and (a(n)*a(n+2) - a(n+1)^2)/2 = 3.
(a(n) + b(n))/2 = (b(n+1) - a(n+1))/2 = A001090(n+1) = Lucas U(8,1). Also b(n)*a(n+1) - b(n+1)*a(n) = -2.
a(n)=(t(i+2*n+1) + t(i))/(t(i+n+1) + t(i+n)) as long as t(i+n+1) + t(i+n) != 0 for any integer i and n >= 1 where (t) is a sequence satisfying t(i+3) = 7*t(i+2) - 7*t(i+1) + t(i) or t(i+2) = 8*t(i+1) - t(i) regardless of initial values and including this sequence itself. (End)

			1 + 7*x + 55*x^2 + 433*x^3 + 3409*x^4 + 26839*x^5 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, Contrib. Discr. Math. 3 (2) (2008), 76-114. See Section 13.
Tanya Khovanova, Recursive Sequences
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory, Vol. 7, No. 5 (2011), pp. 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012), The John Selfridge Memorial Volume.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (8,-1).

a(n) = sqrt((3*A057080(n)^2+2)/5) (cf. Richardson comment).
Cf. A057080, A001090, A001091.
Row 8 of array A094954.
Cf. A001090.
Cf. similar sequences listed in A238379.
Cf. A041023.

I:=[1, 7]; [n le 2 select I[n] else 8*Self(n-1) - Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jan 26 2013

CoefficientList[Series[(1 - x)/(1 - 8*x + x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 26 2013 *)
a[c_, n_] := Module[{},
   p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];
   d := Denominator[Convergents[Sqrt[c], n p]];
   t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];
   Return[t];
   ] (* Complement of A041023 *)
a[15, 20] (* Gerry Martens, Jun 07 2015 *)
LinearRecurrence[{8,-1},{1,7},20] (* Harvey P. Dale, Dec 04 2021 *)

{a(n) = subst( 9*poltchebi(n) - poltchebi(n-1), x, 4) / 5} /* Michael Somos, Jun 07 2005 */

{a(n) = if( n<0, n=-1-n); polcoeff( (1 - x) / (1 - 8*x + x^2) + x * O(x^n), n)} /* Michael Somos, Jun 07 2005 */

[lucas_number1(n,8,1)-lucas_number1(n-1,8,1) for n in range(1, 21)] # Zerinvary Lajos, Nov 10 2009

For all members x of the sequence, 15*x^2 - 6 is a square. Lim_{n->infinity} a(n)/a(n-1) = 4 + sqrt(15). - Gregory V. Richardson, Oct 12 2002
a(n) = (5+sqrt(15))/10 * (4+sqrt(15))^n + (5-sqrt(15))/10 * (4-sqrt(15))^n.
a(n) ~ 1/10*sqrt(10)*(1/2*(sqrt(10)+sqrt(6)))^(2*n+1)
a(n) = U(n, 4)-U(n-1, 4) = T(2*n+1, sqrt(5/2))/sqrt(5/2), with Chebyshev's U and T polynomials and U(-1, x) := 0. U(n, 4)=A001090(n+1), n>=-1.
Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n-i, i); then q(n, 6) = a(n) - Benoit Cloitre, Nov 10 2002
a(n)*a(n+3) = 48 + a(n+1)a(n+2). - Ralf Stephan, May 29 2004
a(n) = (-1)^n*U(2n, i*sqrt(6)/2), U(n, x) Chebyshev polynomial of second kind, i=sqrt(-1). - Paul Barry, Mar 13 2005
G.f.: (1-x)/(1-8*x+x^2).
a(n) = a(-1-n).
a(n) = Jacobi_P(n,-1/2,1/2,4)/Jacobi_P(n,-1/2,1/2,1). - Paul Barry, Feb 03 2006
[a(n), A001090(n+1)] = [1,6; 1,7]^(n+1) * [1,0]. - Gary W. Adamson, Mar 21 2008
For n>0, a(n) is the numerator of the continued fraction [2,3,2,3,...,2,3] with n repetitions of 2,3. For the denominators see A136325. - Greg Dresden, Sep 12 2019
From Peter Bala, Apr 30 2025: (Start)
a(n) = (1/sqrt(5)) * sqrt(1 - T(2*n+1, -4)), where T(k, x) denotes the k-th Chebyshev polynomial of the first kind.
a(n) divides a(3*n+1); a(n) divides a(5*n+2); in general, for k >= 0, a(n) divides a((2*k+1)*n + k).
The aerated sequence [b(n)]n>=1 = [1, 0, 7, 0, 55, 0, 433, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -10, Q = 1 of the 3-parameter family of divisibility sequences found by Williams and Guy.
Sum_{n >= 1} 1/(a(n) - 1/a(n)) = 1/6 (telescoping series: for n >= 1, 1/(a(n) - 1/a(n)) = 1/A291033(n-1) - 1/A291033(n).) (End)
In addition to the first formula above: In general, the following applies to all recurrences (a(n)) of the form (8,-1) with a(0) = 1 and arbitrary a(1): 15*a(n)^2 + y = b^2 where y = x^2 + 8*x + 1 and x = a(1) - 8. Also y = a(k+1)^2 - a(k)*a(k+1) for any k >=0. - Klaus Purath, May 06 2025

A079935 a(n) = 4*a(n-1) - a(n-2) with a(1) = 1, a(2) = 3.

Original entry on oeis.org

1, 3, 11, 41, 153, 571, 2131, 7953, 29681, 110771, 413403, 1542841, 5757961, 21489003, 80198051, 299303201, 1117014753, 4168755811, 15558008491, 58063278153, 216695104121, 808717138331, 3018173449203, 11263976658481, 42037733184721, 156886956080403

Offset: 1

Benoit Cloitre and Paul D. Hanna, Jan 20 2003

See A001835 for another version.
Greedy frac multiples of sqrt(3): a(1)=1, Sum_{n>0} frac(a(n)*x) < 1 at x=sqrt(3).
The n-th greedy frac multiple of x is the smallest integer that does not cause Sum_{k=1..n} frac(a(k)*x) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of x.
Binomial transform of A002605. - Paul Barry, Sep 17 2003
In general, Sum_{k=0..n} binomial(2n-k,k)*j^(n-k) = (-1)^n* U(2n, i*sqrt(j)/2), i=sqrt(-1). - Paul Barry, Mar 13 2005
The Hankel transform of this sequence is [1,2,0,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007
From Richard Choulet, May 09 2010: (Start)
This sequence is a particular case of the following situation:
a(0)=1, a(1)=a, a(2)=b with the recurrence relation a(n+3) = (a(n+2)*a(n+1)+q)/a(n)
where q is given in Z to have Q = (a*b^2 + q*b + a + q)/(a*b) itself in Z.
The g.f is f: f(z) = (1 + a*z + (b-Q)*z^2 + (a*b + q - a*Q)*z^3)/(1 - Q*z^2 + z^4);
so we have the linear recurrence: a(n+4) = Q*a(n+2) - a(n).
The general form of a(n) is given by:
a(2*m) = Sum_{p=0..floor(m/2)} (-1)^p*binomial(m-p,p)*Q^(m-2*p) + (b-Q)*Sum_{p=0..floor((m-1)/2)} (-1)^p*binomial(m-1-p,p)*Q^(m-1-2*p) and
a(2*m+1) = a*Sum_{p=0..floor(m/2)} (-1)^p*binomial(m-p,p)*Q^(m-2*p) + (a*b+q-a*Q)*Sum_{p=0..floor((m-1)/2)} (-1)^p*binomial(m-1-p,p)*Q^(m-1-2*p).
(End)
x-values in the solution to 3*x^2 - 2 = y^2. - Sture Sjöstedt, Nov 25 2011
From Wolfdieter Lang, Oct 12 2020: (Start)
[X(n) = S(n, 4) - S(n-1, 4), Y(n) = X(n-1)] gives all positive solutions of X^2 + Y^2 - 4*X*Y = -2, for n = -oo..+oo, where the Chebyshev S-polynomials are given in A049310, with S(-1, 0) = 0, and S(-|n|, x) = - S(|n|-2, x), for |n| >= 2.
This binary indefinite quadratic form has discriminant D = +12. There is only this family representing -2 properly with X and Y positive, and there are no improper solutions.
See also the preceding comment by Sture Sjöstedt.
See the formula for a(n) = X(n-1), for n >= 1, in terms of S-polynomials below.
This comment is inspired by a paper by Robert K. Moniot (private communication). See his Oct 04 2020 comment in A027941 related to the case of x^2 + y^2 - 3*x*y = -1 (special Markov solutions). (End)
a(n) is also the output of Tesler's formula for the number of perfect matchings of an m x n Mobius band where m and n are both even, specialized to m=2. (The twist is on the length-n side.) - Sarah-Marie Belcastro, Feb 15 2022

			a(4) = 41 since frac(1*x) + frac(3*x) + frac(11*x) + frac(41*x) < 1, while frac(1*x) + frac(3*x) + frac(11*x) + frac(k*x) > 1 for all k > 11 and k < 41.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Dalen Dockery, Marie Jameson, and Samuel Wilson, d-Fold Partition Diamonds, arXiv:2307.02579 [math.NT], 2023.
Tanya Khovanova, Recursive Sequences
Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9:3 (2005), pp. 609-640.
G. Tesler, Matchings in graphs on non-orientable surfaces, Journal of Combinatorial Theory B, 78(2000), 198-231.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory, Vol. 7, No. 5 (2011), pp. 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012), The John Selfridge Memorial Volume.
Index entries for linear recurrences with constant coefficients, signature (4,-1).

Cf. A002530 (denominators of convergents to sqrt(3)), A079934, A079936, A001353.
Cf. A001835 (same except for the first term).
Row 4 of array A094954.
Cf. similar sequences listed in A238379.

a079935 n = a079935_list !! (n-1)
a079935_list =
   1 : 3 : zipWith (-) (map (4 *) $ tail a079935_list) a079935_list
-- Reinhard Zumkeller, Aug 14 2011

I:=[1,3]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jun 06 2015

f:= gfun:-rectoproc({a(n) = 4*a(n-1) - a(n-2),a(1)=1,a(2)=3}, a(n), remember):
seq(f(n),n=1..30); # Robert Israel, Jun 05 2015

a[n_] := (MatrixPower[{{1, 2}, {1, 3}}, n].{{1}, {1}})[[1, 1]]; Table[ a[n], {n, 0, 23}] (* Robert G. Wilson v, Jan 13 2005 *)
LinearRecurrence[{4,-1},{1,3},30] (* or *) CoefficientList[Series[ (1-x)/(1-4x+x^2),{x,0,30}],x]  (* Harvey P. Dale, Apr 26 2011 *)
a[n_] := Sqrt[2/3] Cosh[(-1 - 2 n) ArcCsch[Sqrt[2]]];
Table[Simplify[a[n-1]], {n, 1, 12}] (* Peter Luschny, Oct 13 2020 *)

a(n)=([0,1; -1,4]^(n-1)*[1;3])[1,1] \\ Charles R Greathouse IV, Mar 18 2017

my(x='x+O('x^30)); Vec((1-x)/(1-4*x+x^2)) \\ G. C. Greubel, Feb 25 2019

[lucas_number1(n,4,1)-lucas_number1(n-1,4,1) for n in range(1, 25)] # Zerinvary Lajos, Apr 29 2009

For n > 0, a(n) = ceiling( (2+sqrt(3))^n/(3+sqrt(3)) ).
From Paul Barry, Sep 17 2003: (Start)
G.f.: (1-x)/(1-4*x+x^2).
E.g.f.: exp(2*x)*(sinh(sqrt(3)*x)/sqrt(3) + cosh(sqrt(3)*x)).
a(n) = ( (3+sqrt(3))*(2+sqrt(3))^n + (3-sqrt(3))*(2-sqrt(3))^n )/6 (offset 0). (End)
a(n) = Sum_{k=0..n} binomial(2*n-k, k)*2^(n-k). - Paul Barry, Jan 22 2005 [offset 0]
a(n) = (-1)^n*U(2*n, i*sqrt(2)/2), U(n, x) Chebyshev polynomial of second kind, i=sqrt(-1). - Paul Barry, Mar 13 2005 [offset 0]
a(n) = Jacobi_P(n,-1/2,1/2,2)/Jacobi_P(n,-1/2,1/2,1). - Paul Barry, Feb 03 2006 [offset 0]
a(n) = sqrt(2+(2-sqrt(3))^(2*n-1) + (2+sqrt(3))^(2*n-1))/sqrt(6). - Gerry Martens, Jun 05 2015
a(n) = (1/2 + sqrt(3)/6)*(2-sqrt(3))^n + (1/2 - sqrt(3)/6)*(2+sqrt(3))^n. - Robert Israel, Jun 05 2015
a(n) = S(n-1,4) - S(n-2,4) = (-1)^(n-1)*S(2*(n-1), i*sqrt(2)), with Chebyshev S-polynomials (A049310), the imaginary unit i, S(-1, x) = 0, for n >= 1. See also the formula above by Paul Barry (with offset 0). - Wolfdieter Lang, Oct 12 2020
a(n) = sqrt(2/3)*cosh((-1 - 2*n) arccsch(sqrt(2))), where arccsch is the inverse hyperbolic cosecant function (with offset 0). - Peter Luschny, Oct 13 2020
From Peter Bala, May 04 2025: (Start)
a(n) = (1/sqrt(3)) * sqrt(1 - T(2*n-1, -2)), where T(k, x) denotes the k-th Chebyshev polynomial of the first kind.
a(n) divides a(3*n-1); a(n) divides a(5*n-2); in general, for k >= 0, a(n) divides a((2*k+1)*n - k).
The aerated sequence [b(n)]n>=1 = [1, 0, 3, 0, 11, 0, 41, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -6, Q = 1 of the 3-parameter family of divisibility sequences found by Williams and Guy.
Sum_{n >= 2} 1/(a(n) - 1/a(n)) = 1/2 (telescoping series: for n >= 1, 1/(a(n) - 1/a(n)) = 1/A052530(n-1) - 1/A052530(n).) (End)

A077417 Chebyshev T-sequence with Diophantine property.

Original entry on oeis.org

1, 11, 131, 1561, 18601, 221651, 2641211, 31472881, 375033361, 4468927451, 53252096051, 634556225161, 7561422605881, 90102515045411, 1073668757939051, 12793922580223201, 152453402204739361, 1816646903876649131, 21647309444315050211

Offset: 0

Wolfdieter Lang, Nov 29 2002

7*a(n)^2 - 5*b(n)^2 = 2 with companion sequence b(n) = A077416(n), n>=0.
a(n) = L(n,12), where L is defined as in A108299; see also A077416 for L(n,-12). - Reinhard Zumkeller, Jun 01 2005
[a(n), A004191(n)] = the 2 X 2 matrix [1,10; 1,11]^(n+1) * [1,0]. - Gary W. Adamson, Mar 19 2008
Hankel transform of A174227. - Paul Barry, Mar 12 2010
Alternate denominators of the continued fraction convergents to sqrt(35), see A041059. - James R. Buddenhagen, May 20 2010
For positive n, a(n) equals the permanent of the (2n)X(2n) tridiagonal matrix with sqrt(10)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
Positive values of x (or y) satisfying x^2 - 12xy + y^2 + 10 = 0. - Colin Barker, Feb 09 2014
a(n) = a(-1-n) for all n in Z. - Michael Somos, Jun 29 2019

			G.f. = 1 + 11*x + 131*x^2 + 1561*x^3 + 18601*x^4 221651*x^5 + 2641211*x^6 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
Tanya Khovanova, Recursive Sequences
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (12,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A072256(n) with companion A054320(n-1), n>=1.
Row 12 of array A094954.
Cf. A004191.
Cf. A041059. [James R. Buddenhagen, May 20 2010]
Cf. similar sequences listed in A238379.

I:=[1,11]; [n le 2 select I[n] else 12*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 10 2014

CoefficientList[Series[(1 - x)/(1 - 12 x + x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 10 2014 *)
LinearRecurrence[{12,-1},{1,11},30] (* Harvey P. Dale, Apr 09 2015 *)
a[ n_] := With[{x = Sqrt[7/2]}, ChebyshevT[2 n + 1, x]/x] // Expand; (* Michael Somos, Jun 29 2019 *)

my(x='x+O('x^30)); Vec((1-x)/(1-12*x+x^2)) \\ G. C. Greubel, Jan 18 2018

{a(n) = my(x = quadgen(56)/2); simplify(polchebyshev(2*n + 1, 1, x)/x)}; /* Michael Somos, Jun 29 2019 */

a(n) = 12*a(n-1) - a(n-2), a(-1)=1, a(0)=1.
a(n) = S(n, 12) - S(n-1, 12) = T(2*n+1, sqrt(14)/2)/(sqrt(14)/2) with S(n, x) := U(n, x/2), resp. T(n, x), Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120. S(-1, x)=0, S(n, 12)=A004191(n).
G.f.: (1-x)/(1-12*x+x^2).
a(n) = (ap^(2*n+1) + am^(2*n+1))/sqrt(14) with ap := (sqrt(7)+sqrt(5))/sqrt(2) and am := (sqrt(7)-sqrt(5))/sqrt(2).
a(n) = sqrt((5*A077416(n)^2 + 2)/7).
a(n)*a(n+3) = 120 + a(n+1)*a(n+2). - Ralf Stephan, May 29 2004
E.g.f.: exp(6*x)*(7*cosh(sqrt(35)*x) + sqrt(35)*sinh(sqrt(35)*x))/7. - Stefano Spezia, Aug 29 2025

More terms from Vincenzo Librandi, Feb 10 2014

A070998 a(n) = 9*a(n-1) - a(n-2) for n > 0, a(0)=1, a(-1)=1.

Original entry on oeis.org

1, 8, 71, 631, 5608, 49841, 442961, 3936808, 34988311, 310957991, 2763633608, 24561744481, 218292066721, 1940066856008, 17242309637351, 153240719880151, 1361924169284008, 12104076803675921, 107574767063799281, 956068826770517608

Offset: 0

Joe Keane (jgk(AT)jgk.org), May 18 2002

A Pellian sequence.
In general, Sum_{k=0..n} binomial(2n-k,k)j^(n-k) = (-1)^n*U(2n, i*sqrt(j)/2), i=sqrt(-1). - Paul Barry, Mar 13 2005
a(n) = L(n,9), where L is defined as in A108299; see also A057081 for L(n,-9). - Reinhard Zumkeller, Jun 01 2005
Number of 01-avoiding words of length n on alphabet {0,1,2,3,4,5,6,7,8} which do not end in 0. - Tanya Khovanova, Jan 10 2007
For positive n, a(n) equals the permanent of the (2n) X (2n) tridiagonal matrix with sqrt(7)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
Positive values of x (or y) satisfying x^2 - 9xy + y^2 + 7 = 0. - Colin Barker, Feb 09 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
Tanya Khovanova, Recursive Sequences
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (9,-1).

Cf. A057081, A056918.
Row 9 of array A094954.
Cf. similar sequences listed in A238379.

I:=[1,8]; [n le 2 select I[n] else 9*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 10 2014

CoefficientList[Series[(1 - x)/(1 - 9 x + x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 10 2014 *)
LinearRecurrence[{9,-1},{1,8},30] (* Harvey P. Dale, Sep 24 2015 *)

[lucas_number1(n, 9, 1) - lucas_number1(n-1, 9, 1) for n in range(1, 19)]  # Zerinvary Lajos, Nov 10 2009

a(n) ~ (1/11)*sqrt(11)*((1/2)*(sqrt(11) + sqrt(7)))^(2*n+1).
Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n-i, i); then q(n, 7) = a(n). - Benoit Cloitre, Nov 10 2002
a(n)*a(n+3) = 63 + a(n+1)*a(n+2). - Ralf Stephan, May 29 2004
a(n) = (-1)^n*U(2n, i*sqrt(7)/2), U(n, x) Chebyshev polynomial of second kind, i=sqrt(-1). - Paul Barry, Mar 13 2005
G.f.: (1-x)/(1-9*x+x^2). - Philippe Deléham, Nov 03 2008
a(n) = A018913(n+1) - A018913(n). - R. J. Mathar, Jun 07 2013

More terms from Vincenzo Librandi, Feb 10 2014

A077420 Bisection of Chebyshev sequence T(n,3) (odd part) with Diophantine property.

Original entry on oeis.org

1, 33, 1121, 38081, 1293633, 43945441, 1492851361, 50713000833, 1722749176961, 58522759015841, 1988051057361633, 67535213191279681, 2294209197446147521, 77935577499977736033, 2647515425801796877601

Offset: 0

Wolfdieter Lang, Nov 29 2002

(3*a(n))^2 - 2*(2*b(n))^2 = 1 with companion sequence b(n)= A046176(n+1), n>=0 (special solutions of Pell equation).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Z. Cerin and G. M. Gianella, On sums of squares of Pell-Lucas Numbers, INTEGERS 6 (2006) #A15
Tanya Khovanova, Recursive Sequences
S. Vidhyalakshmi, V. Krithika, and K. Agalya, On The Negative Pell Equation y^2 = 72x^2 - 8, International Journal of Emerging Technologies in Engineering Research (IJETER) Volume 4, Issue 2, February (2016).
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (34,-1).

Cf. A056771 (even part).
Row 34 of array A094954.
Row 3 of array A188646.
Cf. similar sequences listed in A238379.
Similar sequences of the type cosh((2*n+1)*arccosh(k))/k are listed in A302329. This is the case k=3.

I:=[1,33]; [n le 2 select I[n] else 34*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 22 2011

LinearRecurrence[{34,-1},{1,33},20] (* Vincenzo Librandi, Nov 22 2011 *)
a[c_, n_] := Module[{},
   p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];
   d := Denominator[Convergents[Sqrt[c], n p]];
   t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];
   Return[t];
] (* Complement of A041027 *)
a[18, 20] (* Gerry Martens, Jun 07 2015 *)

makelist(expand(((1+sqrt(2))^(4*n+2)+(1-sqrt(2))^(4*n+2))/6),n,0,14);  /* _Bruno Berselli, Nov 22 2011 */

Vec((1-x)/(1-34*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Nov 22 2011

a(n) = 34*a(n-1) - a(n-2), a(-1)=1, a(0)=1.
a(n) = T(2*n+1, 3)/3 = S(n, 34) - S(n-1, 34), with S(n, x) := U(n, x/2), resp. T(n, x), Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120. S(-1, x)=0, S(n, 34)= A029547(n), T(n, 3)=A001541(n).
G.f.: (1-x)/(1-34*x+x^2).
a(n) = sqrt(8*A046176(n+1)^2 + 1)/3.
a(n) = (k^n)+(k^(-n))-a(n-1) = A003499(2*n)-a(n-1), where k = (sqrt(2)+1)^4 = 17+12*sqrt(2) and a(0)=1. - Charles L. Hohn, Apr 05 2011
a(n) = a(-n-1) = A029547(n)-A029547(n-1) = ((1+sqrt(2))^(4n+2)+(1-sqrt(2))^(4n+2))/6. - Bruno Berselli, Nov 22 2011

A299045 Rectangular array: A(n,k) = Sum_{j=0..k} (-1)^floor(j/2)binomial(k-floor((j+1)/2), floor(j/2))(-n)^(k-j), n >= 1, k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, -1, -1, 1, -2, 1, 1, 1, -3, 5, -1, 0, 1, -4, 11, -13, 1, -1, 1, -5, 19, -41, 34, -1, 1, 1, -6, 29, -91, 153, -89, 1, 0, 1, -7, 41, -169, 436, -571, 233, -1, -1, 1, -8, 55, -281, 985, -2089, 2131, -610, 1, 1, 1, -9, 71, -433, 1926, -5741, 10009, -7953, 1597, -1, 0

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Feb 01 2018

This array is used to compute A269252: A269252(n) = least k such that |A(n,k)| is a prime, or -1 if no such k exists.
For detailed theory, see [Hone].
The array can be extended to k<0 with A(n, k) = -A(n, -k-1) for all k in Z. - Michael Somos, Jun 19 2023

			Array begins:
1   0  -1     1     0      -1       1         0        -1           1
1  -1   1    -1     1      -1       1        -1         1          -1
1  -2   5   -13    34     -89     233      -610      1597       -4181
1  -3  11   -41   153    -571    2131     -7953     29681     -110771
1  -4  19   -91   436   -2089   10009    -47956    229771    -1100899
1  -5  29  -169   985   -5741   33461   -195025   1136689    -6625109
1  -6  41  -281  1926  -13201   90481   -620166   4250681   -29134601
1  -7  55  -433  3409  -26839  211303  -1663585  13097377  -103115431
1  -8  71  -631  5608  -49841  442961  -3936808  34988311  -310957991
1  -9  89  -881  8721  -86329  854569  -8459361  83739041  -828931049

Robert Price, Table of n, a(n) for n = 1..5050
Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269251, A269252, A269253, A269254, A294099, A298675, A298677, A298878, A299045, A299071.
Cf. A094954 (unsigned version of this array, but missing the first row).
Cf. Rows: A057078, A033999, A099496, A079935 (or A001835), A004253, A001653, A049685, A070997, A070998, A138288 (or A072256), ...
Cf. Columns: A000012, A001477 (A000027), A110331 (A165900), A123972, A192398, ...

(* Array: *)
Grid[Table[LinearRecurrence[{-n, -1}, {1, 1 - n}, 10], {n, 10}]]
(*Array antidiagonals flattened (gives this sequence):*)
A299045[n_, k_] := Sum[(-1)^(Floor[j/2]) Binomial[k - Floor[(j + 1)/2], Floor[j/2]] (-n)^(k - j), {j, 0, k}]; Flatten[Table[A299045[n - k, k], {n, 11}, {k, 0, n - 1}]]

{A(n, k) = sum(j=0, k, (-1)^(j\2)*binomial(k-(j+1)\2, j\2)*(-n)^(k-j))}; /* Michael Somos, Jun 19 2023 */

G.f. for row n: (1 + x)/(1 + n*x + x^2), n >= 1.

A(n, k) = B(-n, k) where B = A294099. - Michael Somos, Jun 19 2023

A078922 a(n) = 11*a(n-1) - a(n-2) with a(1)=1, a(2) = 10.

Original entry on oeis.org

1, 10, 109, 1189, 12970, 141481, 1543321, 16835050, 183642229, 2003229469, 21851881930, 238367471761, 2600190307441, 28363725910090, 309400794703549, 3375045015828949, 36816094379414890, 401601993157734841

Offset: 1

Nick Renton (ner(AT)nickrenton.com), Jan 11 2003

All positive integer solutions of Pell equation (3*b(n))^2 - 13*a(n)^2 = -4 together with b(n)=A097783(n-1), n >= 1.
a(n) = L(n-1,11), where L is defined as in A108299; see also A097783 for L(n,-11). - Reinhard Zumkeller, Jun 01 2005
Number of 01-avoiding words of length n on alphabet {0,1,2,3,4,5,6,7,8,9, A} which do not end in 0. - Tanya Khovanova, Jan 10 2007

			All positive solutions of the Pell equation x^2 - 13*y^2 = -4 are
(x,y)= (3=3*1,1), (36=3*12,10), (393=3*131,109), (4287=3*1429,1189 ), ...

G. C. Greubel, Table of n, a(n) for n = 1..960
R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.
Sergio Falcon, Relationships between Some k-Fibonacci Sequences, Applied Mathematics, 2014, 5, 2226-2234.
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
Tanya Khovanova, Recursive Sequences
Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
Index entries for sequences related to Chebyshev polynomials..
Index entries for linear recurrences with constant coefficients, signature (11,-1).

Row 11 of array A094954.

Cf. similar sequences listed in A238379.

a:=[1,10];; for n in [3..30] do a[n]:=11*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 12 2019

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( x*(1-x)/(1-11*x+x^2) )); // G. C. Greubel, Jan 12 2019

LinearRecurrence[{11,-1},{1,10},20] (* Harvey P. Dale, Jan 26 2014 *)
Table[Fibonacci[2n-1, 3], {n, 1, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)

a(n)=([0,1;-1,11]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Jun 11 2015

my(x='x+O('x^30)); Vec(x*(1-x)/(1-11*x+x^2)) \\ G. C. Greubel, Jan 12 2019

(x*(1-x)/(1-11*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 12 2019

a(1)=1, a(2)=10 and for n > 2, a(n) = ceiling(g*f^n) where f = (11+sqrt(117))/2 and g = (1-3/sqrt(13))/2. - Benoit Cloitre, Jan 12 2003
a(n)*a(n+3) = 99 + a(n+1)*a(n+2). - Ralf Stephan, May 29 2004
a(n) = S(n-1, 11) - S(n-2, 11) = T(2*n-1, sqrt(13)/2)/(sqrt(13)/2).
a(n+1) = ((-1)^n)*S(2*n, i*3), n >= 0, with the imaginary unit i and S(n, x) = U(n, x/2) Chebyshev's polynomials of the second kind, A049310.
G.f.: x*(1-x)/(1-11*x+x^2).
a(n) = A006190(2*n-1). - Vladimir Reshetnikov, Sep 16 2016

More terms from Benoit Cloitre, Jan 12 2003

Definition adapted to offset by Georg Fischer, Jun 18 2021

A075839 Numbers k such that 11*k^2 - 2 is a square.

Original entry on oeis.org

1, 19, 379, 7561, 150841, 3009259, 60034339, 1197677521, 23893516081, 476672644099, 9509559365899, 189714514673881, 3784780734111721, 75505900167560539, 1506333222617099059, 30051158552174420641, 599516837820871313761, 11960285597865251854579

Offset: 1

Gregory V. Richardson, Oct 14 2002

Lim_{n -> infinity} a(n)/a(n-1) = 10 + 3*sqrt(11).

Positive values of x (or y) satisfying x^2 - 20xy + y^2 + 18 = 0. - Colin Barker, Feb 18 2014

A. H. Beiler, "The Pellian", ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.
L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.
Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147.

G. C. Greubel, Table of n, a(n) for n = 1..750 (terms 1..200 from Vincenzo Librandi)
Tanya Khovanova, Recursive Sequences
J. J. O'Connor and E. F. Robertson, Pell's Equation
Eric Weisstein's World of Mathematics, Pell Equation.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (20,-1).

Row 20 of array A094954.
Cf. A075844, A221762, A041015.
Cf. similar sequences listed in A238379.

a:=[1,19];; for n in [3..20] do a[n]:=20*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 06 2019

I:=[1,19]; [n le 2 select I[n] else 20*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 20 2014

seq(coeff(series( x*(1-x)/(1-20*x+x^2), x, n+1), x, n), n = 1..20); # G. C. Greubel, Dec 06 2019

LinearRecurrence[{20,-1},{1,19},20] (* Harvey P. Dale, Apr 13 2012 *)
Rest@CoefficientList[Series[x*(1-x)/(1-20x+x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Feb 20 2014 *)
a[c_, n_] := Module[{},
   p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];
   d := Denominator[Convergents[Sqrt[c], n p]];
   t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];
   Return[t];
] (* Complement of A041015 *)
a[11, 20] (* Gerry Martens, Jun 07 2015 *)

a(n)=subst(poltchebi(n+1)+poltchebi(n),x,10)/11

def A075839_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1-x)/(1-20*x+x^2) ).list()
a=A075839_list(20); a[1:] # G. C. Greubel, Dec 06 2019

11*a(n)^2 - 9*A083043(n)^2 = 2.
a(n) = ((3+sqrt(11))*(10+3*sqrt(11))^(n-1) - (3-sqrt(11))*(10-3*sqrt(11))^(n-1) )/(2*sqrt(11)). - Dean Hickerson, Dec 09 2002
From Michael Somos, Oct 29 2002: (Start)
G.f.: x*(1-x)/(1-20*x+x^2).
a(n) = 20*a(n-1) - a(n-2), n>1. (End)
Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n-i, i) then a(n) = q(n, 18). - Benoit Cloitre, Dec 06 2002
a(-n+1) = a(n). - Michael Somos, Apr 18 2003
E.g.f.: (1/11)*exp(10*x)*(11*cosh(3*sqrt(11)*x) - 3*sqrt(11)*sinh(3*sqrt(11)*x)) - 1. - Stefano Spezia, Dec 06 2019

More terms from Colin Barker, Feb 18 2014

Offset changed to 1 by G. C. Greubel, Dec 06 2019

A085260 Ratio-determined insertion sequence I(0.0833344) (see the link below).

Original entry on oeis.org

1, 12, 155, 2003, 25884, 334489, 4322473, 55857660, 721827107, 9327894731, 120540804396, 1557702562417, 20129592507025, 260127000028908, 3361521407868779, 43439651302265219, 561353945521579068

Offset: 1

John W. Layman, Jun 23 2003

nonn

This sequence is the ratio-determined insertion sequence (RDIS) "twin" to A078362 (see the link for an explanation of "twin"). See A082630 or A082981 for recent examples of RDIS sequences.
a(n) = L(n,13), where L is defined as in A108299. - Reinhard Zumkeller, Jun 01 2005
For n >= 2, a(n) equals the permanent of the (2n-2) X (2n-2) tridiagonal matrix with sqrt(11)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
Seems to be positive values of x (or y) satisfying x^2 - 13xy + y^2 + 11 = 0. - Colin Barker, Feb 10 2014
It appears that the b-file, formulas and programs are based on the conjectured, so far apparently unproved recurrence relation. - M. F. Hasler, Nov 05 2018
Nonnegative y values in solutions to the Diophantine equation 11*x^2 - 15*y^2 = -4. The corresponding x values are in A126866. Note that a(n+1)^2 - a(n)*a(n+2) = -11. - Klaus Purath, Mar 21 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..900
A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, Contrib. Discr. Math. 3 (2) (2008), pp. 76-114. See Section 13.
Tanya Khovanova, Recursive Sequences
John W. Layman, Ratio-Determined Insertion Sequences and the Tree of their Recurrence Types
John W. Layman, Ratio-Determined Insertion Sequences and the Tree of their Recurrence Types [local copy, corrected]
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (13,-1).

Cf. A078362, A082630, A082981.
Row 13 of array A094954.
Cf. similar sequences listed in A238379.

I:=[1,12]; [n le 2 select I[n] else 13*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 18 2018

CoefficientList[Series[(1 - x)/(1 - 13 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
LinearRecurrence[{13,-1}, {1,12}, 30] (* G. C. Greubel, Jan 18 2018 *)

my(x='x+O('x^30)); Vec(x*(1-x)/(1-13*x+x^2)) \\ G. C. Greubel, Jan 18 2018

It appears that the sequence satisfies a(n+1) = 13*a(n) - a(n-1). [Corrected by M. F. Hasler, Nov 05 2018]
If the recurrence a(n+2) = 13*a(n+1) - a(n) holds then for n > 0, a(n)*a(n+3) = 143 + a(n+1)*a(n+2). - Ralf Stephan, May 29 2004
G.f.: x*(1-x)/(1 - 13*x + x^2). - Philippe Deléham, Nov 17 2008
For n>1, a(n) is the numerator of the continued fraction [1,11,1,11,...,1,11] with (n-1) repetitions of 1,11. - Greg Dresden, Sep 10 2019

cp's OEIS Frontend

A049685 a(n) = L(4*n+2)/3, where L=A000032 (the Lucas sequence).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A070997 a(n) = 8*a(n-1) - a(n-2), a(0)=1, a(-1)=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

A079935 a(n) = 4*a(n-1) - a(n-2) with a(1) = 1, a(2) = 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A077417 Chebyshev T-sequence with Diophantine property.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

Extensions

A070998 a(n) = 9*a(n-1) - a(n-2) for n > 0, a(0)=1, a(-1)=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A077420 Bisection of Chebyshev sequence T(n,3) (odd part) with Diophantine property.

Views

Author

Keywords

Comments

A299045 Rectangular array: A(n,k) = Sum_{j=0..k} (-1)^floor(j/2)binomial(k-floor((j+1)/2), floor(j/2))(-n)^(k-j), n >= 1, k >= 0, read by antidiagonals.