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A098316 Decimal expansion of [3, 3, ...] = (3 + sqrt(13))/2. 19
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 (list; constant; graph; refs; listen; history; text; internal format)
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 3-extension rectangle; see A188640 for definitions.  [From 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. [From R. J. Mathar, Sep 02 2008]

Comments 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).

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

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

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Last modified September 2 10:51 EDT 2014. Contains 246353 sequences.