A098316 Decimal expansion of [3, 3, ...] = (3 + sqrt(13))/2.

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

Offset: 1

Author

Eric W. Weisstein, Sep 02 2004

Keywords

• nonn
• cons
• easy

Comments

For reasons following from the formula section, this constant could be called "the bronze ratio". For this, compare with A001622 and A014176.

If c is this constant and n > 0, then for n even, c^n = [A100230(n), 1, A100230(n)-1, 1, A100230(n)-1, 1, A100230(n)-1, 1, ...], for n odd, c^n = [A100230(n)+1, A100230(n)+1, A100230(n)+1, ...]. - Gerald McGarvey, Dec 15 2007

This is the shape of a 3-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 10 2011

From Vladimir Shevelev, Mar 02 2013: (Start)

An analog of Fermat theorem: for prime p, round(c^p) == 3 (mod p).

A generalization for "metallic" constants c_N = (N+sqrt(N^2+4))/2, N>=1: for prime p, round((c_N)^p) == N (mod p). (End)

This is the positive real algebraic number c of degree 2 with minimal polynomial x^3 - x - 1. The other negative root is 3 - c. - Wolfdieter Lang, Aug 29 2022

c^n = c*A006190(n) + A006190(n-1). - Gary W. Adamson, Apr 02 2024

Examples

			3.30277563...
		

Links

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

Crossrefs

Cf. A001622, A014176, A098317, A098318, A000032, A006497, A080039, A049310.

Programs

Mathematica

RealDigits[(3 + Sqrt[13])/2, 10, 100][[1]] (* G. C. Greubel, Apr 16 2017 *)

PARI

(3 + sqrt(13))/2 \\ Charles R Greathouse IV, Jul 24 2013

Formula

3 plus the constant in A085550. - R. J. Mathar, Sep 02 2008

From Hieronymus Fischer, Jan 02 2009: (Start)

Set c:=(3+sqrt(13))/2. Then the fractional part of c^n equals 1/c^n, if n odd. For even n, the fractional part of c^n is equal to 1-(1/c^n).

c:=(3+sqrt(13))/2 satisfies c-c^(-1)=floor(c)=3, hence c^n + (-c)^(-n) = round(c^n) for n>0, which follows from the general formula of A001622.

1/c=(sqrt(13)-3)/2.

See A001622 for a general formula concerning the fractional parts of powers of numbers x>1, which satisfy x-x^(-1)=floor(x).

Other examples of constants x satisfying the relation x-x^(-1)=floor(x) include A001622 (the golden ratio: where floor(x)=1) and A014176 (the silver ratio: where floor(x)=2). (End)

c=3+sum{k>=1}(-1)^(k-1)/(A006190(k)*A006190(k+1)). - Vladimir Shevelev, Feb 23 2013

A generalization for "metallic" constants c_N = (N+sqrt(N^2+4))/2, N>=1. Let {A_N(n), n>=0} be the sequence 0, 1, N, N^2+1, N^3+2*N, N^4+3*N^2+1,..., a(N) = N*a(N-1) + a(N-2). Then c_N = N + sum_{n>=1} (-1)^(n-1)/(A_N(n)*A_N(n+1)) (cf. A001622, A014176, A098316, A098317, A098318). - Vladimir Shevelev, Feb 23 2013

Equals lim_{n->oo} S(n, sqrt(13))/S(n-1, sqrt(13)), with the S-Chebyshev polynomial (see A049310). - Wolfdieter Lang, Nov 15 2023