A266698 -id:A266698 - cp's OEIS frontend

A041008 Numerators of continued fraction convergents to sqrt(7).

2, 3, 5, 8, 37, 45, 82, 127, 590, 717, 1307, 2024, 9403, 11427, 20830, 32257, 149858, 182115, 331973, 514088, 2388325, 2902413, 5290738, 8193151, 38063342, 46256493, 84319835, 130576328, 606625147, 737201475, 1343826622, 2081028097, 9667939010, 11748967107, 21416906117

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
C. Elsner, Series of Error Terms for Rational Approximations of Irrational Numbers, J. Int. Seq. 14 (2011) # 11.1.4.
C. Elsner, M. Stein, On Error Sum Functions Formed by Convergents of Real Numbers, J. Int. Seq. 14 (2011) # 11.8.6.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,16,0,0,0,-1).

Cf. A010465, A041009 (denominators), A266698 (quadrisection), A001081 (quadrisection).

Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041010 (m=8), A005667 (m=10), A041014 (m=11), A041016 (m=12), ..., A042934 (m=999), A042936 (m=1000).

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[7],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)
Numerator[Convergents[Sqrt[7], 30]] (* Vincenzo Librandi, Oct 28 2013 *)
LinearRecurrence[{0,0,0,16,0,0,0,-1},{2,3,5,8,37,45,82,127},40] (* Harvey P. Dale, Jul 23 2021 *)

A041008=contfracpnqn(c=contfrac(sqrt(7)),#c)[1,][^-1] \\ Discard possibly incorrect last element. NB: a(n)=A041008[n+1]! For more terms use:
A041008(n)={n<#A041008|| A041008=extend(A041008, [4, 16; 8, -1], n\.8); A041008[n+1]}
extend(A,c,N)={for(n=#A+1, #A=Vec(A, N), A[n]=[A[n-i]|i<-c[,1]]*c[,2]); A} \\ (End)

G.f.: (2 + 3*x + 5*x^2 + 8*x^3 + 5*x^4 - 3*x^5 + 2*x^6 - x^7)/(1 - 16*x^4 + x^8).

A159678 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2-equation problem 7n(j) + 1 = a(j)a(j) and 9n(j) + 1 = b(j)b(j) with positive integer numbers.

Original entry on oeis.org

1, 17, 271, 4319, 68833, 1097009, 17483311, 278635967, 4440692161, 70772438609, 1127918325583, 17975920770719, 286486814005921, 4565813103324017, 72766522839178351, 1159698552323529599, 18482410314337295233, 294558866477073194129, 4694459453318833810831

Offset: 1

Paul Weisenhorn, Apr 19 2009

The sequence a(j) is A157456, the sequence n(j) is A159679, the sequence b(j) the sequence given here.

Numbers k such that 7*k^2 + 2 is a square. - Colin Barker, Mar 17 2014

Colin Barker, Table of n, a(n) for n = 1..800
K. Andersen, L. Carbone, and D. Penta, Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
Index entries for linear recurrences with constant coefficients, signature (16,-1).

Cf. A077412, A157456, A159679, A266698.

[n le 2 select 17^(n-1) else 16*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jun 03 2018

for a from 1 by 2 to 100000 do b:=sqrt((9*a*a-2)/7): if (trunc(b)=b) then
n:=(a*a-1)/7: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: end if: end do:
# Second program
seq(simplify(ChebyshevU(n-1,8) + ChebyshevU(n-2,8)), n=1..30); # G. C. Greubel, Sep 27 2022

Rest[CoefficientList[Series[x (1+x)/(1-16x+x^2),{x,0,30}],x]] (* or *) LinearRecurrence[{16,-1},{1,17},30] (* Harvey P. Dale, Dec 25 2011 *)

Vec(x*(1+x)/(1-16*x+x^2) + O(x^30)) \\ Michel Marcus, Jan 03 2016

a(n) = round((-(8-3*sqrt(7))^n*(3+sqrt(7))-(-3+sqrt(7))*(8+3*sqrt(7))^n)/(2*sqrt(7))) \\ Colin Barker, Jul 25 2016

[(lucas_number2(n,16,1)-lucas_number2(n-1,16,1))/14 for n in range(1, 20)] # Zerinvary Lajos, Nov 10 2009

The b(j) recurrence (this sequence) is b(1)=1, b(2)=17, b(t+2) = 16*b(t+1) - b(t).
From R. J. Mathar, Oct 31 2011: (Start)
G.f.: x*(1+x) / ( 1-16*x+x^2 ).
a(n) = A077412(n-1) + A077412(n-2). (End)
a(n) = 16*a(n-1) - a(n-2), with a(1)=1, a(2)=17. - Harvey P. Dale, Dec 25 2011
a(n) = ( (3-sqrt(7))*(8+3*sqrt(7))^n - (3+sqrt(7))*(8-3*sqrt(7))^n )/(2*sqrt(7)). - Colin Barker, Jul 25 2016

More terms from Zerinvary Lajos, Nov 10 2009

cp's OEIS Frontend

A041008 Numerators of continued fraction convergents to sqrt(7).

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A159678 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2-equation problem 7n(j) + 1 = a(j)a(j) and 9n(j) + 1 = b(j)b(j) with positive integer numbers.

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cp's OEIS Frontend

A041008 Numerators of continued fraction convergents to sqrt(7).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A159678 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2-equation problem 7*n(j) + 1 = a(j)*a(j) and 9*n(j) + 1 = b(j)*b(j) with positive integer numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

Extensions

A159678 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2-equation problem 7n(j) + 1 = a(j)a(j) and 9n(j) + 1 = b(j)b(j) with positive integer numbers.