A077413 -id:A077413 - cp's OEIS frontend

A077241 Combined Diophantine Chebyshev sequences A054488 and A077413.

1, 2, 8, 13, 47, 76, 274, 443, 1597, 2582, 9308, 15049, 54251, 87712, 316198, 511223, 1842937, 2979626, 10741424, 17366533, 62605607, 101219572, 364892218, 589950899, 2126747701, 3438485822, 12395593988, 20040964033, 72246816227, 116807298376

Offset: 0

Wolfdieter Lang, Nov 08 2002

-8*a(n)^2 + b(n)^2 = 17, with the companion sequence b(n)= A077242(n).
The number a > 0 belongs to the sequence A077241, if a^2 belongs to the sequence A034856. - Alzhekeyev Ascar M, Apr 27 2012
Numbers k such that k^2 + 2 is a triangular number (see A214838). - Alex Ratushnyak, Mar 07 2013

			8*a(2)^2 + 17 = 8*8^2+17 = 529 = 23^2 = A077242(2)^2.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

I:=[1,2,8,13]; [n le 4 select I[n] else 6*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 18 2014

LinearRecurrence[{0, 6, 0, -1}, {1, 2, 8, 13}, 30] (* Bruno Berselli, Mar 10 2013 *)
CoefficientList[Series[(1 + x) (1 + x + x^2)/(1 - 6 x^2 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 18 2014 *)

makelist(expand((-1)^n*((4-5*sqrt(2))*(1-(-1)^n*sqrt(2))^(2*floor((n+1)/2))+(4+5*sqrt(2))*(1+(-1)^n*sqrt(2))^(2*floor((n+1)/2)))/8), n, 0, 30); /* Bruno Berselli, Mar 10 2013 */

a(2k) = A054488(k) and a(2k+1)= A077413(k) for k>=0.
G.f.: (1+x)*(1+x+x^2)/(1-6*x^2+x^4).
a(n) = (-1)^n*((4-5*sqrt(2))*(1-(-1)^n*sqrt(2))^(2*floor((n+1)/2))+(4+5*sqrt(2))*(1+(-1)^n*sqrt(2))^(2*floor((n+1)/2)))/8. [Bruno Berselli, Mar 10 2013]

A156035 Decimal expansion of 3 + 2*sqrt(2).

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Feb 02 2009

Limit_{n -> oo} b(n+1)/b(n) = 3+2*sqrt(2) for b = A155464, A155465, A155466.
Limit_{n -> oo} b(n)/b(n-1) = 3+2*sqrt(2) for b = A001652, A001653, A002315, A156156, A156157, A156158. - Klaus Brockhaus, Sep 23 2009
From Richard R. Forberg, Aug 14 2013: (Start)
Ratios b(n+1)/b(n) for all sequences of the form b(n) = 6*b(n-1) - b(n-2), for any initial values of b(0) and b(1), converge to this ratio.
Ratios b(n+1)/b(n) for all sequences of the form b(n) = 5*b(n-1) + 5*b(n-2) + b(n-3), for all b(0), b(1) and b(2) also converge to 3 + 2*sqrt(2). For example see A084158 (Pell Triangles).
Ratios of alternating values, b(n+2)/b(n), for all sequences of the form b(n) = 2*b(n-1) + b(n-2), also converge to 3 + 2*sqrt(2). These include A000129 (Pell Numbers). Also see A014176. (End)
Let ABCD be a square inscribed in a circle. When P is the midpoint of the arc AB, then the ratio (PC*PD)/(PA*PB) is equal to 3+2*sqrt(2). See the Mathematical Reflections link. - Michel Marcus, Jan 10 2017
Limit of ratios of successive terms of A001652 when n-> infinity. - Harvey P. Dale, Jun 16 2017; improved by Bernard Schott, Feb 28 2022
A quadratic integer with minimal polynomial x^2 - 6x + 1. - Charles R Greathouse IV, Jul 11 2020
Ratio between radii of the large circumscribed circle R and the small internal circle r drawn on the Sangaku tablet at Isaniwa Jinjya shrine in Ehime Prefecture (pictures in links). - Bernard Schott, Feb 25 2022

			3 + 2*sqrt(2) = 5.828427124746190097603377448...

Diogo Queiros-Condé and Michel Feidt, Fractal and Trans-scale Nature of Entropy, Iste Press and Elsevier, 2018, page 45.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Mathematical Reflections, Solution to Problem J286, Issue 1, 2014, p. 5.
Bernard Schott, Sangaku at Isaniwa Jinya, The six circles.
Terakoya Suzu, Sangaku (mathematics tablet) II, Sangaku at Isaniwa Jinya shrine.
Wikipedia, Sangaku.
Bernard Ycart, Les Sangakus, Sangaku du Temple Isaniwa Jinya (in French).
Index entries for algebraic numbers, degree

Cf. A002193 (sqrt(2)), A090488, A010466, A014176.
Cf. A155464, A155465, A155466.
Cf. A104178 (decimal expansion of log_10(3+2*sqrt(2))).
Cf. A000129, A001109, A001541, A001542, A001652, A001653, A002315, A005319, A075870, A038723, A038725, A038761, A054488, A054489, A075848, A077413, A084158, A106328, A156156, A156157, A156158.
Cf. A242412 (sangaku).

SetDefaultRealField(RealField(100));  3 + 2*Sqrt(2); // G. C. Greubel, Aug 21 2018

RealDigits[N[3+2*Sqrt[2],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)

3+sqrt(8) \\ Charles R Greathouse IV, Jun 10 2011

Equals 1 + A090488 = 3 + A010466. - R. J. Mathar, Feb 19 2009
Equals exp(arccosh(3)), since arccosh(x) = log(x+sqrt(x^2-1)). - Stanislav Sykora, Nov 01 2013
Equals (1+sqrt(2))^2, that is, A014176^2. - Michel Marcus, May 08 2016
The periodic continued fraction is [5; [1, 4]]. - Stefano Spezia, Mar 17 2024

A054488 Expansion of (1+2x)/(1-6x+x^2).

Original entry on oeis.org

1, 8, 47, 274, 1597, 9308, 54251, 316198, 1842937, 10741424, 62605607, 364892218, 2126747701, 12395593988, 72246816227, 421085303374, 2454265004017, 14304504720728, 83372763320351, 485932075201378, 2832219687887917

Offset: 0

Barry E. Williams, May 04 2000

Bisection (even part) of Chebyshev sequence with Diophantine property.
b(n)^2 - 8*a(n)^2 = 17, with the companion sequence b(n)= A077240(n).
The odd part is A077413(n) with Diophantine companion A077239(n).

			8 = a(1) = sqrt((A077240(1)^2 - 17)/8) = sqrt((23^2 - 17)/8)= sqrt(64) = 8.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N. Y., 1964, pp. 122-125, 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,-1).

Cf. A000129, A002203, A002315, A038761, A077239, A077240, A077413.

Cf. A077241 (even and odd parts).

a:=[1,8];; for n in [3..30] do a[n]:=6*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 19 2020

I:=[1,8]; [n le 2 select I[n] else 6*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 19 2020

R:=PowerSeriesRing(Integers(), 21); Coefficients(R!( (1+2*x)/(1-6*x+x^2))); // Marius A. Burtea, Jan 20 2020

a[0]:=1: a[1]:=8: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], n=0..20); # Zerinvary Lajos, Jul 26 2006

LinearRecurrence[{6,-1},{1,8},30] (* Harvey P. Dale, Oct 09 2017 *)
Table[(LucasL[2*n+1, 2] + Fibonacci[2*n, 2])/2, {n,0,30}] (* G. C. Greubel, Jan 19 2020 *)

my(x='x+O('x^30)); Vec((1+2*x)/(1-6*x+x^2)) \\ G. C. Greubel, Jan 19 2020

apply( {A054488(n)=[1,8]*([0,-1;1,6]^n)[,1]}, [0..30]) \\ M. F. Hasler, Feb 27 2020

[(lucas_number2(2*n+1,2,-1) + lucas_number1(2*n,2,-1))/2 for n in (0..30)] # G. C. Greubel, Jan 19 2020

a(n) = 6*a(n-1) - a(n-2), a(0)=1, a(1)=8.
a(n) = ((3 + 2*sqrt(2))^(n+1) - (3 - 2*sqrt(2))^(n+1) + 2*((3 + 2*sqrt(2))^n - (3 - 2*sqrt(2))^n))/(4*sqrt(2)).
a(n) = S(n, 6) + 2*S(n-1, 6), with S(n, x) Chebyshev's polynomials of the second kind, A049310. S(n, 6) = A001109(n+1).
a(n) = (-1)^n*Sum_{k = 0..n} A238731(n,k)*(-9)^k. - Philippe Deléham, Mar 05 2014
a(n) = (Pell(2*n+2) + 2*Pell(2*n))/2 = (Pell-Lucas(2*n+1) + Pell(2*n))/2. - G. C. Greubel, Jan 19 2020
E.g.f.: (1/4)*exp(3*x)*(4*cosh(2*sqrt(2)*x) + 5*sqrt(2)*sinh(2*sqrt(2)*x)). - Stefano Spezia, Jan 27 2020

More terms from James Sellers, May 05 2000

Chebyshev comments from Wolfdieter Lang, Nov 08 2002

A077239 Bisection (odd part) of Chebyshev sequence with Diophantine property.

Original entry on oeis.org

7, 37, 215, 1253, 7303, 42565, 248087, 1445957, 8427655, 49119973, 286292183, 1668633125, 9725506567, 56684406277, 330380931095, 1925601180293, 11223226150663, 65413755723685, 381259308191447, 2222142093424997, 12951593252358535, 75487417420726213

Offset: 0

Wolfdieter Lang, Nov 08 2002

a(n)^2 - 8*b(n)^2 = 17, with the companion sequence b(n)= A077413(n).

The even part is A077240(n) with Diophantine companion A054488(n).

			37 = a(1) = sqrt(8*A077413(1)^2 +17) = sqrt(8*13^2 + 17)= sqrt(1369) = 37.

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,-1).

Cf. A077242 (even and odd parts).

Table[2*ChebyshevT[n+1, 3] + ChebyshevT[n, 3], {n, 0, 19}]  (* Jean-François Alcover, Dec 19 2013 *)

Vec((7-5*x)/(1-6*x+x^2) + O(x^40)) \\ Colin Barker, Oct 12 2015

a(n) = 6*a(n-1) - a(n-2), a(-1) := 5, a(0)=7.
a(n) = 2*T(n+1, 3)+T(n, 3), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 3)= A001541(n).
G.f.: (7-5*x)/(1-6*x+x^2).
a(n) = (((3-2*sqrt(2))^n*(-8+7*sqrt(2))+(3+2*sqrt(2))^n*(8+7*sqrt(2))))/(2*sqrt(2)). - Colin Barker, Oct 12 2015

A077240 Bisection (even part) of Chebyshev sequence with Diophantine property.

Original entry on oeis.org

5, 23, 133, 775, 4517, 26327, 153445, 894343, 5212613, 30381335, 177075397, 1032071047, 6015350885, 35060034263, 204344854693, 1191009093895, 6941709708677, 40459249158167, 235813785240325, 1374423462283783, 8010726988462373, 46689938468490455

Offset: 0

Wolfdieter Lang, Nov 08 2002

a(n)^2 - 8*b(n)^2 = 17, with the companion sequence b(n)= A054488(n).

The odd part is A077239(n) with Diophantine companion A077413(n).

			23 = a(1) = sqrt(8*A054488(1)^2 + 17) = sqrt(8*8^2 + 17)= sqrt(529) = 23.

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,-1).

Cf. A077242 (even and odd parts).

Table[ChebyshevT[n+1, 3] + 2*ChebyshevT[n, 3], {n, 0, 19}]  (* Jean-François Alcover, Dec 19 2013 *)
LinearRecurrence[{6,-1},{5,23},30] (* Harvey P. Dale, Mar 29 2017 *)

Vec((5-7*x)/(1-6*x+x^2) + O(x^40)) \\ Colin Barker, Oct 12 2015

a(n) = 6*a(n-1) - a(n-2), a(-1) = 7, a(0) = 5.
a(n) = T(n+1, 3)+2*T(n, 3), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 3)= A001541(n).
G.f.: (5-7*x)/(1-6*x+x^2).
a(n) = (((3-2*sqrt(2))^n*(-4+5*sqrt(2))+(3+2*sqrt(2))^n*(4+5*sqrt(2))))/(2*sqrt(2)). - Colin Barker, Oct 12 2015

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A077241 Combined Diophantine Chebyshev sequences A054488 and A077413.

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A156035 Decimal expansion of 3 + 2*sqrt(2).

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A054488 Expansion of (1+2*x)/(1-6*x+x^2).

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A077239 Bisection (odd part) of Chebyshev sequence with Diophantine property.

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A077240 Bisection (even part) of Chebyshev sequence with Diophantine property.

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A054488 Expansion of (1+2x)/(1-6x+x^2).