

A098316


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


21



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
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OFFSET

1,1


COMMENTS

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 3extension 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)


LINKS

Table of n, a(n) for n=1..102.


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 cc^(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 xx^(1)=floor(x).
Other examples of constants x satisfying the relation xx^(1)=floor(x) include A001622 (the golden ratio: where floor(x)=1) and A014176 (the silver ratio: where floor(x)=2).
c=3+sum{k>=1}(1)^(k1)/(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(N1) + a(N2). Then c_N = N + sum_{n>=1} (1)^(n1)/(A_N(n)*A_N(n+1)) (cf. A001622, A014176, A098316, A098317, A098318).  Vladimir Shevelev, Feb 23 2013


EXAMPLE

3.30277563...


PROG

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


CROSSREFS

Cf. A001622, A014176, A098317, A098318, A000032, A006497, A080039.
Sequence in context: A010607 A118522 A179119 * A160165 A084055 A084103
Adjacent sequences: A098313 A098314 A098315 * A098317 A098318 A098319


KEYWORD

nonn,cons,easy


AUTHOR

Eric W. Weisstein, Sep 02 2004


EXTENSIONS

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


STATUS

approved



