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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A023039 a(n) = 18*a(n-1) - a(n-2).

Original entry on oeis.org

1, 9, 161, 2889, 51841, 930249, 16692641, 299537289, 5374978561, 96450076809, 1730726404001, 31056625195209, 557288527109761, 10000136862780489, 179445175002939041, 3220013013190122249, 57780789062419261441
Offset: 0

Views

Author

David W. Wilson

Keywords

Comments

The primitive Heronian triangle 3*a(n) +- 2, 4*a(n) has the latter side cut into 2*a(n) +- 3 by the corresponding altitude and has area 10*a(n)*A060645(n). - Lekraj Beedassy, Jun 25 2002
Chebyshev polynomials T(n,x) evaluated at x=9.
{a(n)} gives all (unsigned, integer) solutions of Pell equation a(n)^2 - 80*b(n)^2 = +1 with b(n) = A049660(n), n >= 0.
{a(n)} gives all possible solutions for x in Pell equation x^2 - D*y^2 = 1 for D=5, D=20 and D=80. The corresponding values for y are A060645 (D=5), A207832 (D=20) and A049660 (D=80). - Herbert Kociemba, Jun 05 2022
Also gives solutions to the equation x^2 - 1 = floor(x*r*floor(x/r)) where r=sqrt(5). - Benoit Cloitre, Feb 14 2004
Appears to give all solutions > 1 to the equation: x^2 = ceiling(x*r*floor(x/r)) where r=sqrt(5). - Benoit Cloitre, Feb 24 2004
For all terms x of the sequence, 5*x^2 - 5 is a square, A004292(n)^2.
The a(n) are the x-values in the nonnegative integer solutions of x^2 - 5y^2 = 1, see A060645(n) for the corresponding y-values. - Sture Sjöstedt, Nov 29 2011
Rightmost digits alternate repeatedly: 1 and 9 in fact, a(2) = 18*9 - 1 == 1 (mod 10); a(3) = 18*1 - 9 == 9 (mod 10) therefore a(2n) == 1 (mod 10), a(2n+1) == 9 (mod 10). - Carmine Suriano, Oct 03 2013

Examples

			G.f. = 1 + 9*x + 161*x^2 + 2889*x^3 + 51841*x4 + 930249*x^5 + 16692641*x^6 + ...

Links

Crossrefs

Cf. A001077, A115032.
Row 2 of array A188645.
Row 4 of A322790.

Programs

  • Magma
    I:=[1, 9]; [n le 2 select I[n] else 18*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 13 2012
  • Maple
    a := n -> hypergeom([n, -n], [1/2], -4):
seq(simplify(a(n)), n=0..16); # Peter Luschny, Jul 26 2020
  • Mathematica
    LinearRecurrence[{18, -1}, {1, 9}, 50] (* Sture Sjöstedt, Nov 29 2011 *)
CoefficientList[Series[(1-9*x)/(1-18*x+x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 19 2017 *)
  • PARI
    {a(n) = fibonacci(6*n) / 2 + fibonacci(6*n - 1)}; /* Michael Somos, Aug 11 2009 */
  • PARI
    x='x+O('x^30); Vec((1-9*x)/(1-18*x+x^2)) \\ G. C. Greubel, Dec 19 2017

Formula

a(n) ~ (1/2)*(sqrt(5) + 2)^(2*n). - Joe Keane (jgk(AT)jgk.org), May 15 2002
Limit_{n->infinity} a(n)/a(n-1) = phi^6 = 9 + 4*sqrt(5). - Gregory V. Richardson, Oct 13 2002
a(n) = T(n, 9) = (S(n, 18) - S(n-2, 18))/2, with S(n, x) := U(n, x/2) and T(n, x), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(-2, x) := -1, S(-1, x) := 0, S(n, 18)=A049660(n+1).
a(n) = sqrt(80*A049660(n)^2 + 1) (cf. Richardson comment).
a(n) = ((9 + 4*sqrt(5))^n + (9 - 4*sqrt(5))^n)/2.
G.f.: (1 - 9*x)/(1 - 18*x + x^2).
a(n) = cosh(2*n*arcsinh(2)). - Herbert Kociemba, Apr 24 2008
a(n) = A001077(2*n). - Michael Somos, Aug 11 2009
From Johannes W. Meijer, Jul 01 2010: (Start)
a(n) = 2*A167808(6*n+1) - A167808(6*n+3).
Limit_{k->infinity} a(n+k)/a(k) = a(n) + A060645(n)*sqrt(5).
Limit_{n->infinity} a(n)/A060645(n) = sqrt(5).
(End)
a(n) = (1/2)*A087215(n) = (1/2)*(sqrt(5) + 2)^(2*n) + (1/2)*(sqrt(5) - 2)^(2*n).
Sum_{n >= 1} 1/( a(n) - 5/a(n) ) = 1/8. Compare with A005248, A002878 and A075796. - Peter Bala, Nov 29 2013
a(n) = 2*A115032(n-1) - 1 = S(n, 18) - 9*S(n-1, 18), with A115032(-1) = 1, and see the above formula with S(n, 18) using its recurrence. - Wolfdieter Lang, Aug 22 2014
a(n) = A128052(3n). - A.H.M. Smeets, Oct 02 2017
a(n) = A049660(n+1) - 9*A049660(n). - R. J. Mathar, May 24 2018
a(n) = hypergeom([n, -n], [1/2], -4). - Peter Luschny, Jul 26 2020
a(n) = L(6*n)/2 for L(n) the Lucas sequence A000032(n). - Greg Dresden, Dec 07 2021
a(n) = cosh(6*n*arccsch(2)). - Peter Luschny, May 25 2022

Extensions

Chebyshev and Pell comments from Wolfdieter Lang, Nov 08 2002
Sture Sjöstedt's comment corrected and reformulated by Wolfdieter Lang, Aug 24 2014

A041041 Denominators of continued fraction convergents to sqrt(26).

Original entry on oeis.org

1, 10, 101, 1020, 10301, 104030, 1050601, 10610040, 107151001, 1082120050, 10928351501, 110365635060, 1114584702101, 11256212656070, 113676711262801, 1148023325284080, 11593909964103601, 117087122966320090, 1182465139627304501, 11941738519239365100
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Generalized Fibonacci sequence.
Sqrt(26) = 10/2 + 10/101 + 10/(101*10301) + 10/(10301*1050601) + ... - Gary W. Adamson, Jun 13 2008
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 10's along the main diagonal and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
a(n) equals the number of words of length n on alphabet {0, 1, ..., 10} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
From Bruno Berselli, May 03 2018: (Start)
Numbers k for which m*k^2 + (-1)^k is a perfect square:
m = 2: 0, 1, 2, 5, 12, 29, 70, 169, ... (A000129);
m = 3: 0, 4, 56, 780, 10864, 151316, ... (4*A007655);
m = 5: 0, 1, 4, 17, 72, 305, 1292, ... (A001076);
m = 6: 0, 2, 20, 198, 1960, 19402, ... (A001078);
m = 7: 0, 48, 12192, 3096720, ... (2*A175672);
m = 8: 0, 6, 204, 6930, 235416, ... (A082405);
m = 10: 0, 1, 6, 37, 228, 1405, 8658, ... (A005668);
m = 11: 0, 60, 23880, 9504180, ... [°];
m = 12: 0, 2, 28, 390, 5432, 75658, ... (A011944);
m = 13: 0, 5, 180, 6485, 233640, ... (5*A041613);
m = 14: 0, 4, 120, 3596, 107760, ... (A068204);
m = 15: 0, 8, 496, 30744, 1905632, ... [°];
m = 17: 0, 1, 8, 65, 528, 4289, 34840, ... (A041025);
m = 18: 0, 4, 136, 4620, 156944, ... (A202299);
m = 19: 0, 13260, 1532829480, ... [°];
m = 20: 0, 2, 36, 646, 11592, 208010, ... (A207832);
m = 21: 0, 12, 1320, 145188, ... (A174745);
m = 22: 0, 42, 16548, 6519870, ... (A174766);
m = 23: 0, 240, 552480, 1271808720, ... [°];
m = 24: 0, 10, 980, 96030, 9409960, ... (A168520);
m = 26: 0, 1, 10, 101, 1020, 10301, ... (this sequence);
m = 27: 0, 260, 702520, 1898208780, ... [°];
m = 28: 0, 24, 6096, 1548360, ... (A175672);
m = 29: 0, 13, 1820, 254813, 35675640, ... [°];
m = 30: 0, 2, 44, 966, 21208, 465610, ... (2*A077421), etc.
[°] apparently without related sequences in the OEIS.
(End)
From Michael A. Allen, Mar 12 2023: (Start)
Also called the 10-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 10 kinds of squares available. (End)

Links

Crossrefs

Cf. A099374 (squares).
Cf. A041040.
Cf. A000045, A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371, A243399.
Row n=10 of A073133, A172236 and A352361.

Programs

  • Magma
    I:=[1,10]; [n le 2 select I[n] else 10*Self(n-1) + Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018
  • Maple
    seq(combinat:-fibonacci(n+1, 10), n=0..19); # Peter Luschny, May 04 2018
  • Mathematica
    Denominator[Convergents[Sqrt[26], 30]] (* Vincenzo Librandi, Dec 10 2013 *)
LinearRecurrence[{10,1}, {1,10}, 30] (* G. C. Greubel, Jan 24 2018 *)
  • PARI
    x='x+O('x^30); Vec(1/(1-10*x-x^2)) \\ G. C. Greubel, Jan 24 2018
  • Sage
    [lucas_number1(n,10,-1) for n in range(1, 19)] # Zerinvary Lajos, Apr 26 2009

Formula

G.f.: 1/(1 - 10*x - x^2).
a(n) = 10*a(n-1) + a(n-2), n>=1; a(-1):=0, a(0)=1.
a(n) = S(n, 10*i)*(-i)^n where i^2:=-1 and S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind. See A049310.
a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = 5+sqrt(26), am = -1/ap = 5-sqrt(26).
a(n) = F(n+1, 10), the (n+1)-th Fibonacci polynomial evaluated at x=10. - T. D. Noe, Jan 19 2006
a(n) = Sum_{i=0..floor(n/2)} binomial(n-i,i)*10^(n-2*i). - Sergio Falcon, Sep 24 2007

Extensions

Extended by T. D. Noe, May 23 2011

A132584 a(0)=0, a(1)=4; for n > 1, a(n) = 18*a(n-1) - a(n-2) + 8.

Original entry on oeis.org

0, 4, 80, 1444, 25920, 465124, 8346320, 149768644, 2687489280, 48225038404, 865363202000, 15528312597604, 278644263554880, 5000068431390244, 89722587501469520, 1610006506595061124, 28890394531209630720, 518417095055178291844, 9302617316461999622480
Offset: 0

Views

Author

Mohamed Bouhamida, Nov 14 2007

Keywords

Comments

The old definition given for this sequence was "Sequence allows us to find X values of the equation: X(X + 1) - 5*Y^2 = 0".
With this old definition, if X = a(n), then Y = A207832(n). Now, with u = 2X+1, this Diophantine equation becomes the Pell-Fermat equation u^2 - 20*Y^2 = 1, and then, u = A023039(n) and Y = A207832(n). - Bernard Schott, Jan 25 2023

Links

Crossrefs

Cf. A000032, A000045, A000217, A007654, A023039, A049683, A292443, A207832.

Programs

  • Magma
    I:=[0,4,80]; [n le 3 select I[n] else 18*Self(n-1)-Self(n-2)+8: n in [1..30]]; // Vincenzo Librandi, Dec 24 2018
  • Mathematica
    LinearRecurrence[{19, -19, 1}, {0, 4, 80}, 40] (* Vincenzo Librandi, Dec 24 2018 *)
nxt[{a_,b_}]:={b,18b-a+8}; NestList[nxt,{0,4},20][[;;,1]] (* Harvey P. Dale, Aug 25 2024 *)

Formula

a(n) = (A023039(n) - 1)/2. - Max Alekseyev, Nov 13 2009
G.f.: -4*x*(x+1)/((x-1)*(x^2-18*x+1)). - Colin Barker, Oct 24 2012
From Amiram Eldar, Jan 11 2022: (Start)
a(n) = 5*Fibonacci(3*n)^2/4 - 1 if n is odd and 5*Fibonacci(3*n)^2/4 if n is even.
A000217(a(n)) = A292443(n). (End)
a(n) = (Lucas(6*n)-2)/4. - Jeffrey Shallit, Jan 20 2023
a(n) = 4 * A049683(n). - Alois P. Heinz, Jan 20 2023

Extensions

More terms from Max Alekseyev, Nov 13 2009
New definition by Antti Karttunen, Oct 24 2012
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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