A090550 -id:A090550 - cp's OEIS frontend

A004187 a(n) = 7*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.

0, 1, 7, 48, 329, 2255, 15456, 105937, 726103, 4976784, 34111385, 233802911, 1602508992, 10983760033, 75283811239, 516002918640, 3536736619241, 24241153416047, 166151337293088, 1138818207635569, 7805576116155895, 53500214605455696, 366695926122033977

Offset: 0

N. J. A. Sloane, R. K. Guy

Define the sequence T(a_0,a_1) by a_{n+2} is the greatest integer such that a_{n+2}/a_{n+1}= 0 . A004187 (with initial 0 omitted) is T(1,7).
This is a divisibility sequence.
For n>=2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 7's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
a(n) and b(n) := A056854(n) are the proper and improper nonnegative solutions of the Pell equation b(n)^2 - 5*(3*a(n))^2 = +4. see the cross-reference to A056854 below. - Wolfdieter Lang, Jun 26 2013
For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,2,3,4,5,6}. - Milan Janjic, Jan 25 2015
The digital root is A253298, which shares its digital root with A253368. - Peter M. Chema, Jul 04 2016
Lim_{n->oo} a(n+1)/a(n) = 2 + 3*phi = 1+ A090550 = 6.854101...  - Wolfdieter Lang, Nov 16 2023

			a(2) = 7*a(1) - a(0) = 7*7 - 1 = 48. - _Michael B. Porter_, Jul 04 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.
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.
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3 , example 12
D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993
Zvonko Cerin, Some alternating sums of Lucas numbers, Centr. Eur. J. Math. vol 3 no 1 (2005) 1-13.
R. Flórez, R. A. Higuita, and A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=7, q=-1.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Tanya Khovanova, Recursive Sequences
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eq.(44), lhs, m=9.
Index entries for sequences related to Chebyshev polynomials.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (7,-1).
Index entries for Pisot sequences

Cf. A000027, A001906, A001353, A004254, A001109, A049685, A033888. a(n)=sqrt((A056854(n)^2 - 4)/45).

Second column of array A028412.

[Fibonacci(4*n)/3 : n in [0..30]]; // Vincenzo Librandi, Jun 07 2011

/* By definition: */ [n le 2 select n-1 else 7*Self(n-1)-Self(n-2): n in [1..23]]; // Bruno Berselli, Dec 24 2012

seq(combinat:-fibonacci(4*n)/3, n = 0 .. 30); # Robert Israel, Jan 26 2015

LinearRecurrence[{7,-1},{0,1},30] (* Harvey P. Dale, Jul 13 2011 *)
CoefficientList[Series[x/(1 - 7*x + x^2), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 23 2012 *)

a[0]:0$ a[1]:1$ a[n]:=7*a[n-1] - a[n-2]$ A004187(n):=a[n]$ makelist(A004187(n),n,0,30); /* Martin Ettl, Nov 11 2012 */

numlib::fibonacci(4*n)/3 $ n = 0..25; // Zerinvary Lajos, May 09 2008

a(n)=fibonacci(4*n)/3 \\ Charles R Greathouse IV, Mar 09 2012

concat(0, Vec(x/(1-7*x+x^2) + O(x^99))) \\ Altug Alkan, Jul 03 2016

[lucas_number1(n,7,1) for n in range(27)] # Zerinvary Lajos, Jun 25 2008

[fibonacci(4*n)/3 for n in range(0, 21)] # Zerinvary Lajos, May 15 2009

G.f.: x/(1-7*x+x^2).
a(n) = F(4*n)/3 = A033888(n)/3, where F=A000045 (the Fibonacci sequence).
a(n) = S(2*n-1, sqrt(9))/sqrt(9) = S(n-1, 7); S(n, x) := U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310.
a(n) = Sum_{i = 0..n-1} C(2*n-1-i, i)*5^(n-i-1). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
[A049685(n-1), a(n)] = [1,5; 1,6]^n * [1,0]. - Gary W. Adamson, Mar 21 2008
a(n) = A167816(4*n). - Reinhard Zumkeller, Nov 13 2009
a(n) = (((7+sqrt(45))/2)^n-((7-sqrt(45))/2)^n)/sqrt(45). - Noureddine Chair, Aug 31 2011
a(n+1) = Sum_{k = 0..n} A101950(n,k)*6^k. - Philippe Deléham, Feb 10 2012
a(n) = (A081072(n)/3)-1. - Martin Ettl, Nov 11 2012
From Peter Bala, Dec 23 2012: (Start)
Product {n >= 1} (1 + 1/a(n)) = (1/5)*(5 + 3*sqrt(5)).
Product {n >= 2} (1 - 1/a(n)) = (1/14)*(5 + 3*sqrt(5)). (End)
From Peter Bala, Apr 02 2015: (Start)
Sum_{n >= 1} a(n)*x^(2*n) = -A(x)*A(-x), where A(x) = Sum_{n >= 1} Fibonacci(2*n)* x^n.
1 + 5*Sum_{n >= 1} a(n)*x^(2*n) = F(x)*F(-x) = G(x)*G(-x), where F(x) = 1 + A(x) and G(x) = 1 + 5*A(x).
1 + Sum_{n >= 1} a(n)*x^(2*n) = H(x)*H(-x) = I(x)*I(-x), where H(x) = 1 + Sum_{n >= 1} Fibonacci(2*n + 3)*x^n and I(x) = 1 + x + x*Sum_{n >= 1} Fibonacci(2*n - 1)*x^n. (End)
E.g.f.: 2*exp(7*x/2)*sinh(3*sqrt(5)*x/2)/(3*sqrt(5)). - Ilya Gutkovskiy, Jul 03 2016
a(n) = Sum_{k = 0..n-1} (-1)^(n+k+1)*9^k*binomial(n+k, 2*k+1). - Peter Bala, Jul 17 2023
a(n) = Sum_{k = 0..floor(n/2)} (-1)^k*7^(n-2*k)*binomial(n-k, k). - Greg Dresden, Aug 03 2024
From Peter Bala, Jul 22 2025: (Start)
The following products telescope:
Product {n >= 2} (1 + (-1)^n/a(n)) = (3/14)*(3 + sqrt(5)).
Product {n >= 1} (1 - (-1)^n/a(n)) = (1/3)*(3 + sqrt(5)).
Product_{n >= 1} (a(2*n) + 1)/(a(2*n) - 1) = (3/5)*sqrt(5). (End)

Entry improved by comments from Michael Somos and Wolfdieter Lang, Aug 02 2000

A090388 Decimal expansion of 1 + sqrt(3).

Original entry on oeis.org

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

Offset: 1

Felix Tubiana, Feb 05 2004

1 + sqrt(3) is the length of the minimal Steiner network that connects the four vertices of a unit square. - Lekraj Beedassy, May 02 2008
This is the case n = 12 in the identity (Gamma(1/n)/Gamma(3/n))*(Gamma((n-1)/n)/Gamma((n-3)/n)) = 1 + 2*cos(2*Pi/n). - Bruno Berselli, Dec 14 2012
Equals n + n/(n + n/(n + n/(n + ...))) for n = 2. - Stanislav Sykora, Jan 23 2014
A non-optimal solution to the problem of finding the length of shortest fence that protects privacy of a square garden [Kawohl]. Cf. A256965. - N. J. A. Sloane, Apr 14 2015
Perimeter of a 30-60-90 triangle with longest leg equal to 1. - Wesley Ivan Hurt, Apr 09 2016
Length of the second shortest diagonal in a regular 12-gon with unit side. - Mohammed Yaseen, Dec 13 2020
Surface area of a square pyramid (Johnson solid J_1) with unit edges. - Paolo Xausa, Aug 04 2025

			2.7320508075688772...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Bernd Kawohl, Some nonconvex shape optimization problems, in: Optimal Shape Design, Eds. A.Cellina u. A. Ornelas, Springer Lecture Notes in Math.1740 (2000), p. 7-46.
Ian Stewart, Pursuing Polygonal Privacy, Mathematical Recreations Column, Scientific American, 284 (No. 2, 2001), 88-89.
Wikipedia, Square pyramid.
Index entries for algebraic numbers, degree 2

Cf. n + n/(n + n/(n + ...)): A090458 (n = 3), A090488 (n = 4), A090550 (n = 5), A092294 (n = 6), A092290 (n = 7), A090654 (n = 8), A090655 (n = 9), A090656 (n = 10). - Stanislav Sykora, Jan 23 2014

Cf., also A256965.

Digits:=100: evalf((1+sqrt(3))); # Wesley Ivan Hurt, Apr 09 2016

RealDigits[1 + Sqrt[3], 10, 100][[1]] (* Alonso del Arte, Feb 23 2014 *)

1 + sqrt(3) \\ Michel Marcus, Apr 10 2016

Equals 1 + A002194. - R. J. Mathar, Oct 16 2015

Equals A019973 -1 . - R. J. Mathar, May 25 2023

Better definition from Rick L. Shepherd, Jul 02 2004

A090488 Decimal expansion of 2 + 2*sqrt(2).

Original entry on oeis.org

4, 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

Offset: 1

Felix Tubiana, Feb 05 2004

Side length of smallest square containing five circles of radius 1. - Charles R Greathouse IV, Apr 05 2011
Equals n + n/(n +n/(n +n/(n +....))) for n = 4.  See also A090388. - Stanislav Sykora, Jan 23 2014
Also the area of a regular octagon with unit edge length. - Stanislav Sykora, Apr 12 2015
The positive solution to x^2 - 4*x - 4 = 0. The negative solution is -1 * A163960 = -0.82842... . - Michal Paulovic, Dec 12 2023

			4.828427124746190097603377448419396157139343750...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Octagon
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 2

Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090655 (n=9), A090656 (n=10). - Stanislav Sykora, Jan 23 2014
Cf. A002193, A010466, A014176, A086178, A156035, A157258, A163960, A365823.
Cf. Areas of other regular polygons: A120011, A102771, A104956, A178817, A256853, A178816, A256854, A178809.

RealDigits[2+2Sqrt[2],10,120][[1]] (* Harvey P. Dale, Mar 11 2015 *)

2*(1 + sqrt(2)) \\ G. C. Greubel, Jul 03 2017

Equals 1 + A086178 = 2*A014176. - R. J. Mathar, Sep 03 2007
From Michal Paulovic, Dec 12 2023: (Start)
Equals A010466 + 2.
Equals A156035 - 1.
Equals A157258 - 5.
Equals A163960 + 4.
Equals A365823 - 2.
Equals [4; 1, 4, ...] (periodic continued fraction expansion).
Equals sqrt(4 + 4 * sqrt(4 + 4 * sqrt(4 + 4 * sqrt(4 + 4 * ...)))). (End)

Better definition from Rick L. Shepherd, Jul 02 2004

A090458 Decimal expansion of (3 + sqrt(21))/2.

Original entry on oeis.org

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

Offset: 1

Felix Tubiana, Feb 05 2004

Decimal expansion of the solution to n/x = x-n for n-3. n/x = x-n with n=1 gives the Golden Ratio = 1.6180339887...
n/x = x-n ==> x^2 - n*x - n = 0 ==> x = (n + sqrt(n^2 + 4*n)) / 2 (Positive Root) n = 3: x = (3 + sqrt(21))/2 = 3.79128784747792...
x=3.7912878474... is the shape of a rectangle whose geometric partition (as at A188635) consists of 3 squares, then 1 square, then 3 squares, etc., matching the continued fraction of x, which is [3,1,3,1,3,1,3,1,3,1,...]. (See the Mathematica program below.) - Clark Kimberling, May 05 2011
x appears to be the limit for n to infinity of the ratio of the number of even numbers that take n steps to reach 1 to the number of odd numbers that take n steps to reach 1 in the Collatz iteration. As A005186(n-1) is the number of even numbers that take n steps to reach 1, this means x = lim A005186(n-1)/A176866(n). - Markus Sigg, Oct 20 2020
From Wolfdieter Lang, Sep 02 2022: (Start)
This integer in the quadratic number field Q(sqrt(21)) equals the (real) cube root of 27 + 6*sqrt(21) = 54.4954541... . See Euler, Elements of Algebra, Article 748 or Algebra (in German) p. 306, Kapitel 12, 187.
Subtracting 3 from the present number gives the (real) cube root of
  -27 +  6*sqrt(21) = 0.4954541... . (End)

			3.79128784747792...

Leonhard Euler, Vollständige Anleitung zur Algebra, (1770), Reclam, Leipzig, 1883, p.306, Kapitel 12, 187.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Leonhard Euler, Elements of Algebra, p. 244, Article 748.
Markus Sigg, Heuristic argument for the asymptotic value of the even/odd ratio of number of steps in the Collatz iteration, 2020.
Index entries for algebraic numbers, degree 2

Of the same type as this: A090388 (n=2), A090488 (n=4), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090655 (n=9), A090656 (n=10).
Equals 3*A176014 (constant).
Cf. A356034.

FromContinuedFraction[{3,1,{3,1}}]
ContinuedFraction[%,20]
RealDigits[N[%%,120]] (*A090458*)
N[%%%,40]
RealDigits[(3 + Sqrt[21])/2, 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

solve(x=3,4,x^2-3*x-3) \\ Charles R Greathouse IV, Oct 04 2011

(3+sqrt(21))/2 \\ Charles R Greathouse IV, Oct 04 2011

Equals (27 + 6*sqrt(21))^(1/3). - Wolfdieter Lang, Sep 01 2022

Additional comments from Rick L. Shepherd, Jul 02 2004

A090654 Decimal expansion of 4 + 2*sqrt(6).

Original entry on oeis.org

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

Offset: 1

Felix Tubiana, Feb 05 2004

Equals n +n/(n +n/(n +n/(n +....))) for n = 8. See also A090388. - Stanislav Sykora, Jan 23 2014

			8.898979485566356196394568149...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 2

Cf. A010480, A176213.
Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090488 (n=4), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090655 (n=9), A090656 (n=10). - Stanislav Sykora, Jan 23 2014
Essentially the same as A010480.

RealDigits[4 + 2*Sqrt[6], 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

4 + 2*sqrt(6) \\ G. C. Greubel, Jul 03 2017

A092290 Decimal expansion of solution to n/x = x-n for n = 7.

Original entry on oeis.org

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

Offset: 1

Felix Tubiana, Feb 05 2004

n/x = x-n with n=1 gives the Golden Ratio = 1.6180339887...

Equals n +n/(n +n/(n +n/(n +....))) for n = 7. See also A090388. - Stanislav Sykora, Jan 23 2014

Chai Wah Wu, Table of n, a(n) for n = 1..10001
Index entries for algebraic numbers, degree 2.

Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090488 (n=4), A090550 (n=5), A092294 (n=6), A090654 (n=8), A090655 (n=9), A090656 (n=10). - Stanislav Sykora, Jan 23 2014

RealDigits[(7+Sqrt[77])/2, 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

(7+sqrt(77))/2 \\ Charles R Greathouse IV, Oct 18 2012

n/x = x-n ==> x^2 - n*x - n = 0 ==> x = (n + sqrt(n^2 + 4*n)) / 2 (Positive Root) n = 7: x = (7 + sqrt(77))/2 = 7.88748219369606...

A092294 Decimal expansion of 3 + sqrt(15).

Original entry on oeis.org

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

Offset: 1

Felix Tubiana, Feb 05 2004

Equals n +n/(n +n/(n +n/(n +....))) for n = 6. See also A090388. - Stanislav Sykora, Jan 23 2014

			6.87298334620741688...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 2

Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090488 (n=4), A090550 (n=5), A092290 (n=7), A090654 (n=8), A090655 (n=9), A090656 (n=10). - Stanislav Sykora, Jan 23 2014

RealDigits[3+Sqrt[15],10,120][[1]] (* Harvey P. Dale, Aug 29 2012 *)

sqrt(15)+3 \\ Charles R Greathouse IV, Apr 25 2016

Equals A010472 plus 3. - R. J. Mathar, Sep 08 2008

Equals 1/A176016 + 6. - Hugo Pfoertner, Mar 19 2024

A090655 Decimal expansion of solution to n/x = x-n for n = 9.

Original entry on oeis.org

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

Offset: 1

Felix Tubiana, Feb 05 2004

n/x = x-n with n=1 gives the Golden Ratio = 1.6180339887...

Equals n +n/(n +n/(n +n/(n +....))) for n = 9. See also A090388. - Stanislav Sykora, Jan 23 2014

			9.90832691319598...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 2.

Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090488 (n=4), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090656 (n=10). - Stanislav Sykora, Jan 23 2014

RealDigits[(3/2)*(3+Sqrt[13]), 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

(3/2)*(3 + sqrt(13)) \\ G. C. Greubel, Jul 03 2017

n/x = x-n ==> x^2 - n*x - n = 0 ==> x = (n + sqrt(n^2 + 4*n)) / 2 (Positive Root) n = 9: x = (9 + sqrt(117))/2 = 9.90832691319598...

Equals (3/2)*(3 + sqrt(13)). - G. C. Greubel, Jul 03 2017

A090656 Decimal expansion of 5 + sqrt(35).

Original entry on oeis.org

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

Offset: 2

Felix Tubiana, Feb 05 2004

Equals n+n/(n+n/(n+n/(n+....))) for n = 10. See also A090388. - Stanislav Sykora, Jan 23 2014

			10.9160797830996160...

Chai Wah Wu, Table of n, a(n) for n = 2..10003
Index entries for algebraic numbers, degree 2

Equals A010490 plus 5. - R. J. Mathar, Sep 08 2008
Cf. A161321.
Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090488 (n=4), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090655 (n=9). - Stanislav Sykora, Jan 23 2014

RealDigits[5+Sqrt[35],10,120][[1]] (* Harvey P. Dale, Jun 17 2014 *)

sqrt(35)+5 \\ Charles R Greathouse IV, Apr 25 2016

A132429 Period 4: repeat [3, 1, -1, -3].

Original entry on oeis.org

3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1, -1, -3, 3, 1

Offset: 0

Paul Curtz, Nov 13 2007

Nonsimple continued fraction expansion of (7 + 3*sqrt(5))/2 = 6.85410196624... = 1 + A090550. - R. J. Mathar, Mar 08 2012

Pisano period lengths: 1, 1, 4, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ... . - R. J. Mathar, Aug 10 2012

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

Cf. A084101 (1, 3, 3, 1), A090550.

a132429 = (3 -) . (* 2) . (`mod` 4)
a132429_list = cycle [3, 1, -1, -3]  -- Reinhard Zumkeller, Aug 15 2015

&cat [[3, 1, -1, -3]^^30]; // Wesley Ivan Hurt, Jul 10 2016

A132429:=n->3 - 2 * (n mod 4); seq(A132429(n), n=0..100); # Wesley Ivan Hurt, Apr 18 2014

PadRight[{}, 104, {3,1,-1,-3}] (* Harvey P. Dale, Nov 12 2011 *)

a(n)=3-2*(n%4) \\ Jaume Oliver Lafont, Aug 28 2009

def A132429(n): return 3 - 2*(n & 3) # Chai Wah Wu, May 25 2022

G.f.: (3 + 4*x + 3*x^2)/((1+x)*(1+x^2)). - Jaume Oliver Lafont, Aug 30 2009
a(n) = (-1)^n + 2(-1)^((2n + (-1)^n - 1)/4). - Brad Clardy, Mar 10 2013
a(n) = 3 - 2*(n mod 4). - Joerg Arndt, Mar 10 2013
a(n) = (-1)^n + 2(-1)^floor(n/2). - Wesley Ivan Hurt, Apr 17 2014
From Wesley Ivan Hurt, Jul 10 2016: (Start)
a(n) + a(n-1) + a(n-2) + a(n-3) = 0 for n>2, a(n) = a(n-4) for n>3.
a(n) = 2*cos(n*Pi/2) + cos(n*Pi) + 2*sin(n*Pi/2). (End)

cp's OEIS Frontend

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