A136211 -id:A136211 - cp's OEIS frontend

A041011 Denominators of continued fraction convergents to sqrt(8).

1, 1, 5, 6, 29, 35, 169, 204, 985, 1189, 5741, 6930, 33461, 40391, 195025, 235416, 1136689, 1372105, 6625109, 7997214, 38613965, 46611179, 225058681, 271669860, 1311738121, 1583407981, 7645370045, 9228778026, 44560482149

Offset: 0

N. J. A. Sloane

Sqrt(8) = 2 + continued fraction [0; 1, 4, 1, 4, 1, 4, ...] = 4/2 + 4/5 + 4/(5*29) + 4/(29*169) + 4/(169*985) + ... - Gary W. Adamson, Dec 21 2007
This is the sequence of Lehmer numbers U_n(sqrt(R),Q) with the parameters R = 4 and Q = -1. It is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all natural numbers n and m. - Peter Bala, May 12 2014
Apparently the same as A152118(n). - Georg Fischer, Jul 01 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..199 [shifted by _Georg Fischer_, Jul 01 2019]
Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
Hongshen Chua, A Study of Second-Order Linear Recurrence Sequences via Continuants, J. Int. Seq. (2023) Vol. 26, Art. 23.8.8.
J. L. Ramirez and F. Sirvent, A q-Analogue of the Bi-Periodic Fibonacci Sequence, J. Int. Seq. 19 (2016) # 16.4.6, t_n at a=1, b=4.
Eric Weisstein's World of Mathematics, Lehmer Number
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Cf. A000129, A010466, A041010. A089499, A136211, A152118.

I:=[1, 1, 5, 6]; [n le 4 select I[n] else 6*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 10 2013

with(combinat): a := n -> fibonacci(n + 1, 2)/2^(n mod 2):
seq(a(n), n = 0 .. 28); # Miles Wilson, Aug 04 2024

Denominator[NestList[(4/(4 + #))&, 0, 60]] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2010 *)
CoefficientList[Series[(x + x^2 - x^3)/(1 - 6 x^2 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 10 2013 *)
a0[n_] := ((3-2*Sqrt[2])^n*(2+Sqrt[2])-(-2+Sqrt[2])*(3+2*Sqrt[2])^n)/4 // Simplify
a1[n_] := (-(3-2*Sqrt[2])^n+(3+2*Sqrt[2])^n)/(4*Sqrt[2]) // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* Gerry Martens, Jul 11 2015 *)
LinearRecurrence[{0,6,0,-1},{1,1,5,6},40] (* Harvey P. Dale, Oct 21 2024 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,0,6,0]^n*[1;1;5;6])[1,1] \\ Charles R Greathouse IV, Nov 13 2015

my(x='x+O('x^99)); concat(0, Vec((1+x-x^2)/(1-6*x^2+x^4))) \\ Altug Alkan, Mar 27 2016

a(2n) = A000129(2n+1), a(2n+1) = A000129(2n+2)/2.
a(n) = 6*a(n-2) - a(n-4). Also:
a(2n) = a(2n-1)+a(2n-2), a(2n+1)=4*a(2n)+a(2n-1).
G.f.: (1+x-x^2)/(1-6*x^2+x^4).
From Peter Bala, May 12 2014: (Start)
For n even, a(n) = (alpha^n - beta^n)/(alpha - beta), and for n odd, a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2), where alpha = 1 + sqrt(2) and beta = 1 - sqrt(2).
a(n) = Product_{k = 1..floor((n-1)/2)} ( 4 + 4*cos^2(k*Pi/n) ) for n >= 1. (End)
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = ((3-2*sqrt(2))^n*(2+sqrt(2))-(-2+sqrt(2))*(3+2*sqrt(2))^n)/4.
a1(n) = (-(3-2*sqrt(2))^n+(3+2*sqrt(2))^n)/(4*sqrt(2)). (End)
a(n) = ((-(-1 - sqrt(2))^n - 3*(1-sqrt(2))^n + (-1+sqrt(2))^n + 3*(1+sqrt(2))^n))/(8*sqrt(2)). - Colin Barker, Mar 27 2016

Entry improved by Michael Somos

First term 0 in b-file, formulas and programs removed by Georg Fischer, Jul 01 2019

A176014 Decimal expansion of (3+sqrt(21))/6.

Original entry on oeis.org

1, 2, 6, 3, 7, 6, 2, 6, 1, 5, 8, 2, 5, 9, 7, 3, 3, 3, 4, 4, 3, 1, 3, 4, 1, 1, 9, 8, 9, 5, 4, 6, 6, 8, 0, 8, 1, 4, 9, 7, 4, 0, 9, 4, 2, 9, 4, 6, 1, 3, 2, 8, 6, 5, 0, 4, 3, 4, 5, 4, 0, 3, 5, 3, 9, 8, 4, 4, 7, 8, 0, 7, 0, 9, 2, 4, 6, 2, 8, 4, 8, 1, 1, 0, 0, 7, 2, 6, 9, 2, 6, 5, 8, 2, 2, 4, 0, 8, 3, 8, 7, 7, 9, 6, 0

Offset: 1

Klaus Brockhaus, Apr 06 2010

Continued fraction expansion of (3+sqrt(21))/6 is A010684.

Also greatest eigenvalue of the 6 X 6 matrix [[3 0 0 3 0 0][0 0 0 0 1 0][0 3 0 0 3 0][0 0 0 0 1 0][0 0 3 0 0 3][0 0 0 0 1 0]]/3. It is conjectured that this is lim_{k->infinity} A005186(k+1)/A005186(k), i.e., the asymptotic growth rate of the number of numbers with the same total stopping time in the Collatz iteration. - Hugo Pfoertner, Sep 28 2020

			(3+sqrt(21))/6 = 1.26376261582597333443...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Hugo Pfoertner, Ratio of successive terms in A005186, illustration of deviation from (3+sqrt(21))/6.
Index entries for algebraic numbers, degree 2.

Cf. A010477 (decimal expansion of sqrt(21)).
Cf. A010684 (repeat 1, 3), A136210, A136211.
Cf. A005186, A006577, A127824, A176866.

RealDigits[(3+Sqrt[21])/6,10,120][[1]] (* Harvey P. Dale, Jul 21 2023 *)

vecmax(mateigen([1,0,0,1,0,0; 0,0,0,0,1/3,0; 0,1,0,0,1,0; 0,0,0,0,1/3,0; 0,0,1,0,0,1; 0,0,0,0,1/3,0],1)[1]) \\ Hugo Pfoertner, Sep 28 2020

A136210 Numerators in continued fraction [0; 1, 3, 1, 3, 1, 3, ...].

Original entry on oeis.org

1, 3, 4, 15, 19, 72, 91, 345, 436, 1653, 2089, 7920, 10009, 37947, 47956, 181815, 229771, 871128, 1100899, 4173825, 5274724, 19997997, 25272721, 95816160, 121088881, 459082803, 580171684, 2199597855, 2779769539, 10538906472

Offset: 1

Gary W. Adamson, Dec 21 2007

A136210(n)/A136211(n) tends to 0.7912878474... = (sqrt(21) - 3)/2 = continued fraction [0; 1, 3, 1, 3, 1, 3, ...] = the inradius of a right triangle with hypotenuse 5, legs 2 and sqrt(21).

This is a strong divisibility sequence, that is, GCD(a(n),a(m)) = a(GCD(n,m)) for all natural numbers n and m. - Peter Bala, May 14 2014

			a(4) = 15 = 3*a(3) + a(2) = 3*4 + 3.
a(5) = 19 = a(4) + a(3) = 15 + 4.
T^3 = [19, 72; 24, 91], where [19, 72] = [a(5), a(6)]. [24, 91] = [A136211(5), A136211(6)].
G.f. = x + 3*x^2 + 4*x^3 + 15*x^4 + 19*x^5 + 72*x^6 + 91*x^7 + 345*x^8 + ...

J. L. Ramirez, F. Sirvent, A q-Analogue of the Bi-Periodic Fibonacci Sequence, J. Int. Seq. 19 (2016) # 16.4.6, t_n at a=3, b=1.
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-1).

Cf. A004253, A004254, A136211.

a = {1, 3}; Do[If[EvenQ[n], AppendTo[a, 3*a[[ -1]] + a[[ -2]]], AppendTo[a, a[[ -1]] + a[[ -2]]]], {n, 3, 30}]; a (* Stefan Steinerberger, Dec 31 2007 *)
a[n_] := FromContinuedFraction[ Join[{0}, 3 - 2*Array[Mod[#, 2]&, n]]] // Numerator; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, May 15 2014 *)

{a(n) = (-1)^((n+1) * (n<0)) * polcoeff( x * (1 + 3*x - x^2) / (1 - 5*x^2 + x^4) + x * O(x^abs(n)), abs(n))}; /* Michael Somos, May 15 2014 */

a(0) = 0, a(1) = 1, a(2n) = 3*a(2n-1) + a(2n-2); a(2n-1) = a(2n-2) + a(2n-3). Given the 2 X 2 matrix [1, 3; 1, 4] = T, [a(2n-1), a(2n)] = top row of T^n.
g.f.: x*(1+3*x-x^2)/(1-5*x^2+x^4). - Colin Barker, Jan 04 2012
a(-n) = -(-1)^n * a(n). a(2*n - 1) = A004253(n). a(2*n) = 3 * A004254(n). - Michael Somos, May 15 2014
a(n+1) - a(n-1) = a(n) * (2 - (-1)^n) for all n in Z. - Michael Somos, May 15 2014

More terms from Stefan Steinerberger, Dec 31 2007

A152118 a(n) = Product_{k=1..floor((n-1)/2)} (4 + 4cos(kPi/n)^2).

Original entry on oeis.org

1, 1, 1, 5, 6, 29, 35, 169, 204, 985, 1189, 5741, 6930, 33461, 40391, 195025, 235416, 1136689, 1372105, 6625109, 7997214, 38613965, 46611179, 225058681, 271669860, 1311738121, 1583407981, 7645370045, 9228778026, 44560482149, 53789260175, 259717522849

Offset: 0

Roger L. Bagula and Gary W. Adamson, Nov 24 2008

Product_{k=1..floor((n-1)/2)} (m + 4*cos(k*Pi/n)^2) for m=1,2,3,4 respectively give A000045, A002530, A136211 and this sequence.

Apparently the same as A041011 after the initial term. - R. J. Mathar, Nov 27 2008

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Cf. A000045, A002530, A136211.

Cf. A041011 (essentially the same).

with(combinat); a := n -> `if`(n = 0, 1, fibonacci(n, 2)/2^((n + 1) mod 2)); seq(a(n), n = 0 .. 31); # Miles Wilson, Aug 04 2024

a = Table[Product[4 + 4*Cos[k*Pi/n]^2, {k, 1, (n - 1)/2}], {n, 0, 30}]; FullSimplify[ExpandAll[a]] Round[%]
Join[{1}, LinearRecurrence[{0, 6, 0, -1}, {1, 1, 5, 6}, 20]] (* G. C. Greubel, Mar 28 2016 *)

a(n) = round(prod(k=1, (n-1)/2, 4 + 4*cos(k*Pi/n)^2)) \\ Colin Barker, Oct 23 2013

Vec((x^4-x^3-5*x^2+x+1)/((x^2-2*x-1)*(x^2+2*x-1)) + O(x^50)) \\ Colin Barker, Mar 28 2016

From Colin Barker, Oct 23 2013: (Start)
a(n) = 6*a(n-2)-a(n-4) for n>4.
G.f.: (x^4-x^3-5*x^2+x+1) / ((x^2-2*x-1)*(x^2+2*x-1)). (End)
a(n) = ((-(-1 - sqrt(2))^n - 3*(1-sqrt(2))^n + (-1+sqrt(2))^n + 3*(1+sqrt(2))^n)) / (8*sqrt(2)) for n>0. - Colin Barker, Mar 28 2016
E.g.f.: (1/(2*sqrt(2)))*(2*sqrt(2) + (2*cosh(x) + sinh(x))*sinh(sqrt(2)*x)). - G. C. Greubel, Mar 28 2016

A180062 Irregular triangle by rows derived from variants of Cartan matrices: 1's in the super and subdiagonals and 3,4,4,4,... in the main diagonal alternating with 4,4,4,...

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 1, 7, 11, 1, 8, 15, 1, 11, 38, 41, 1, 12, 46, 56, 1, 15, 81, 186, 153, 1, 16, 93, 232, 209, 1, 19, 140, 49, 859, 571, 1, 20, 156, 592, 1091, 780, 1, 23, 215, 1044, 2774, 3821, 2131, 1, 24, 235, 1200, 3366, 4912, 2911, 1, 27, 306, 1885, 6810, 14418

Offset: 1

Gary W. Adamson, Aug 08 2010

Row sums starting with row 2 = A136211: (1, 4, 5, 19, 24, ...) = denominators in convergents to [1, 3, 1, 3, 1, 3, ...].
Rightmost terms in each row = A002530, denominators in convergents to [1, 2, 1, 2, 1, 2, ...], prefaced with a 1 for row 1. Odd-indexed row rightmost terms = Product_{k=1..(n-1)/2} (2 + 4*cos^2(k*2*Pi/n))
Example: x^3 - 11x^2 + 38x + 41 = row 7 relating to the heptagon, with roots = 5.246979..., 3.554958..., and 2.19806226, product = 41 (same result as using the product formula).
Even-indexed rows related to even-sided regular polygons; but use the product formula: rightmost terms in even rows >2 = Product_{k=1..(n-2)/2} (2 + 4*cos^2(k*Pi/n)).
Using the product formula or root products with row 8 relating to the octagon, we obtain 5.414..., * 4 * 2.585... = 56, rightmost term of row 8.
Shifted columns of A180062 = triangle A180063.

			First few rows of the triangle:
  1;
  1;
  1,  3;
  1,  4;
  1,  7,  11;
  1,  8,  15;
  1, 11,  38,   41;
  1, 12,  46,   56;
  1, 15,  81,  186,   153;
  1, 16,  93,  232,   209;
  1, 19, 140,  499,   859,    571;
  1, 20, 156,  592,  1091,    780;
  1, 23, 215, 1044,  2774,   3821,   2131;
  1, 24, 235, 1200,  3366,   4912,   2911;
  1, 27, 306, 1885,  6810,  14418,  26556,   7953;
  1, 28, 330, 2120,  8010,  17784,  21468,  10864;
  1, 31, 413, 3086, 14135,  40614,  71454,  70356,  29681;
  1, 32, 441, 3416, 16255,  48624,  89238,  91824,  40545;
  1, 35, 536, 4711, 26173,  95269, 227100, 341754, 294549, 110771;
  1, 36, 568, 5152, 29589, 111524, 275724, 430992, 386373, 151316;
  ...
Examples:
Row 7 = x^3 - 11 x^2 + 38x + 41, charpoly of the 3 X 3 matrix [3,1,0; 1,4,1; 0,1,4], then changing (-) signs to (+).
Row 8 = x^3 - 12x^2 + 46x - 56, = charpoly of [4,1,0; 1,4,1; 0,1,4].

Cf. A002530, A136211, A180063, A180063.

Triangle read by rows generated from Cartan-like matrices, 1's in the super and subdiagonals, with alternates of (3,4,4,4,...) for odd-indexed rows and (4,4,4,...) for even-indexed rows. The first nontrivial matrix = [3,1; 1,4] with charpoly x^2 - 7x + 11, becoming row 5: (1, 7, 11); generating row 3: (x^2 - 7x + 11). Rows begin 1; 1; 1,3; 1,4;...
The first few rows can be constructed using the following set of rules:
Rightmost terms in each row = A002530, denominators in continued fraction [1, 2, 1, 2, 1, 2,...] = (1, 3, 4, 11, 15,...), while row sums = A136211, denominators in [1, 3, 1, 3, 1, 3,...] = (1, 4, 5, 19, 24,...) given row 1 = 1.
Negative signs in the charpolys are changed to + in the triangle.

cp's OEIS Frontend

A041011 Denominators of continued fraction convergents to sqrt(8).

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A176014 Decimal expansion of (3+sqrt(21))/6.

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A136210 Numerators in continued fraction [0; 1, 3, 1, 3, 1, 3, ...].

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A152118 a(n) = Product_{k=1..floor((n-1)/2)} (4 + 4*cos(k*Pi/n)^2).

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A180062 Irregular triangle by rows derived from variants of Cartan matrices: 1's in the super and subdiagonals and 3,4,4,4,... in the main diagonal alternating with 4,4,4,...

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A152118 a(n) = Product_{k=1..floor((n-1)/2)} (4 + 4cos(kPi/n)^2).