A041059 -id:A041059 - cp's OEIS frontend

A077417 Chebyshev T-sequence with Diophantine property.

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

A192062 Square Array T(ij) read by antidiagonals (from NE to SW) with columns 2j being the denominators of continued fraction convergents to square root of (j^2 + 2j).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 3, 3, 1, 0, 1, 1, 4, 4, 5, 3, 0, 1, 1, 5, 5, 11, 8, 1, 0, 1, 1, 6, 6, 19, 15, 13, 4, 0, 1, 1, 7, 7, 29, 24, 41, 21, 1, 0, 1, 1, 8, 8, 41, 35, 91, 56, 34, 5, 0, 1, 1, 9, 9, 55, 48, 169, 115, 153, 55, 1, 0, 1, 1, 10, 10, 71, 63, 281, 204, 436, 209, 89, 6

Offset: 0

Kenneth J Ramsey, Jun 21 2011

Column j=1 is the Fibonacci sequence A000045.  Column 2 is A002530; column 4 is A041011; column 6 is A041023; column 8 is A041039, column 10 is A041059, column 12 is A041083, column 14 is A041111 corresponding the denominators of continued fraction convergents to square root of 3,8,15,24,35,48 and 63.
  T(2*i-1,j)*T(2*i,j)^2*T(2*i+1,j)*j/2 appears to be always a triangular number, T(j*T(2*i,j)^2).
  T(2*i,j)*T(2*i+1,j)^2*T(2*i+2)*j/2 appears to always equal a triangular number, T(j*T(2*i,j)*T(2*i+2,j)).
Conjecture re relation of A192062 to the sequence of primes: T(2*n,j) = A(n,j)*T(n,j) where A(n,j) is from the square array A191971. There, A(3*n,j) = A(n,j)*B(n,j) where B(n,j) are integers. It appears further that B(5*n,j)=B(n,j)*C(n,j); C(7*n,j)= C(n,j)*D(n,j); D(11*n,j) = D(n,j)*E(n,j); E(13*n,j) = E(n,j)*F(n,j) and F(17*n,j) = F(n,j)*G(n,j) where C(n,j), D(n,j) etc. are all integers. My conjecture is that this property continues indefinitely and follows the sequence of primes.

			Array as meant by the definition
First column has index j=0
0  0  0   0   0   0   0 ...
1  1  1   1   1   1   1 ...
1  1  1   1   1   1   1 ...
1  2  3   4   5   6   7 ...
2  3  4   5   6   7   8 ...
1  5 11  19  29  41  55 ...
3  8 15  24  35  48  63 ...
1 13 41  91 169 281 433 ...
4 21 56 115 204 329 496 ...
.
.
.

Kenneth J Ramsey,Triangular and Fibonacci Numbers

Cf. A191579, A191971.

Each column j is a recursive sequence defined by T(0,j)=0, T(1,j) = 1, T(2i,j)= T(2i-2,j)+T(2i-1,j) and T(2i+1,j) = T(2i-1,j)+j*T(2i,j). Also, T(n+2,j) = (j+2)*T(n,j)-T(n-2,j).
T(2n,j) = Sum(k=1 to n) C(k)*T(2*k,j-1) where the C(k) are the n-th row of the triangle A191579.
T(2*i,j) = T(i,j)*A(i,j) where A(i,j) is from the table A(i,j) of A191971.
T(4*i,j) = (T(2*i+1)^2 - T(2*i-1)^2)/j
T(4*i+2,j) = T(2*i+2,j)^2 - T(2*i,j)^2

Corrected and edited by Olivier Gérard, Jul 05 2011

A041058 Numerators of continued fraction convergents to sqrt(35).

Original entry on oeis.org

5, 6, 65, 71, 775, 846, 9235, 10081, 110045, 120126, 1311305, 1431431, 15625615, 17057046, 186196075, 203253121, 2218727285, 2421980406, 26438531345, 28860511751, 315043648855, 343904160606, 3754085254915

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,12,0,-1).

Cf. A010490, A041059.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[35], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011 *)
Numerator[Convergents[Sqrt[35], 30]] (* Vincenzo Librandi, Oct 23 2013 *)

a(n) = 12*a(n-2)-a(n-4). G.f.: (5+6*x+5*x^2-x^3)/(1-12*x^2+x^4). [Colin Barker, Apr 17 2012]

A177187 Union of A057080 and A001090.

Original entry on oeis.org

1, 1, 9, 8, 71, 63, 559, 496, 4401, 3905, 34649, 30744, 272791, 242047, 2147679, 1905632, 16908641, 15003009, 133121449, 118118440, 1048062951, 929944511, 8251382159, 7321437648, 64962994321, 57641556673, 511452572409, 453811015736, 4026657584951, 3572846569215

Offset: 1

Mark Dols, May 04 2010

Column sums of shifted Pascal-like array:
  1..1..10..10..100..100.1000.1000
  ......-1..-2..-30..-40.-500.-600
  ................1....3...60..100
  .........................-1...-4
  -------------------------------- +
  1..1...9...8...71...63..559..496
Decimal expansion of ratio n/(n+1) is accumulation of Catalan numbers; (5 +/- sqrt(15)).

Index entries for linear recurrences with constant coefficients, signature (0,8,0,-1).

Cf. A057080, A001090, A041059, A004189, A000108.

Cf. A010472.

For n odd a(n) = 10*a(n-1) - a(n-2), for n even a(n) = a(n-1) - a(n-2); with a(0) = 0, a(1) = 1.

G.f.: x*(1+x+x^2) / ( 1-8*x^2+x^4 ). - R. J. Mathar, Nov 11 2011

A162671 For n even a(n) = a(n-1) + a(n-2), for n odd a(n) = 100*a(n-1) + a(n-2), with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 1, 101, 102, 10301, 10403, 1050601, 1061004, 107151001, 108212005, 10928351501, 11036563506, 1114584702101, 1125621265607, 113676711262801, 114802332528408, 11593909964103601, 11708712296632009, 1182465139627304501, 1194173851923936510, 120599850332020955501

Offset: 0

Mark Dols, Jul 10 2009

Index entries for linear recurrences with constant coefficients, signature (0,102,0,-1).

Partly same as A041059 (and its palindromic partner-sequence A015446). A007318.

a:= proc(n) a(n):= `if`(n<2, n, a(n-1)*(1+99*(n mod 2))+a(n-2)) end:
seq(a(n), n=0..22);  # Alois P. Heinz, Jan 20 2025

LinearRecurrence[{0,102,0,-1},{1,1,101,102},20] (* Harvey P. Dale, May 08 2020 *)

a(n) = 102*a(n-2)-a(n-4). G.f.: x*(1+x-x^2)/((x^2+10*x-1)*(x^2-10*x-1)). - R. J. Mathar, Jul 14 2009

More terms from R. J. Mathar, Jul 14 2009

A164828 Generalized Fibonacci numbers.

Original entry on oeis.org

1, 1, 11, 120, 122, 1331, 1462, 15960, 17413, 190081, 207503, 2265120, 2472614, 26991251, 29463874, 321630000, 351093865, 3832568641, 4183662515, 45669193800, 49852856306, 544197756851, 594050613166, 6484703888520, 7078754501677, 77272248905281

Offset: 1

Mark Dols, Aug 27 2009

Index entries for linear recurrences with constant coefficients, signature (0,11,0,11,0,-1).

Cf. A000045, A041059.

LinearRecurrence[{0,11,0,11,0,-1},{1,1,11,120,122,1331},30] (* Harvey P. Dale, Jun 14 2014 *)

For n odd a(n) = 11a(n-2) + a(n-3), for n even a(n) = 11a(n-1)- a(n-3); with a(1,2) = 1 (For Fibonacci sequence: factor 2 instead of 11).

G.f.: x*(1+x+109*x^3-10*x^4)/((1+x^2)*(1-12*x^2+x^4)). [Colin Barker, Jul 27 2012]

cp's OEIS Frontend

A077417 Chebyshev T-sequence with Diophantine property.

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A192062 Square Array T(ij) read by antidiagonals (from NE to SW) with columns 2j being the denominators of continued fraction convergents to square root of (j^2 + 2j).

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A041058 Numerators of continued fraction convergents to sqrt(35).

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A177187 Union of A057080 and A001090.

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A162671 For n even a(n) = a(n-1) + a(n-2), for n odd a(n) = 100*a(n-1) + a(n-2), with a(0) = 0, a(1) = 1.

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A164828 Generalized Fibonacci numbers.

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