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%I #68 Apr 04 2024 10:34:05
%S 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,
%T 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,
%U 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
%N Decimal expansion of [3, 3, ...] = (3 + sqrt(13))/2.
%C For reasons following from the formula section, this constant could be called "the bronze ratio". For this, compare with A001622 and A014176.
%C 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
%C This is the shape of a 3-extension rectangle; see A188640 for definitions. - _Clark Kimberling_, Apr 10 2011
%C From _Vladimir Shevelev_, Mar 02 2013: (Start)
%C An analog of Fermat theorem: for prime p, round(c^p) == 3 (mod p).
%C 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)
%C 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 c^n = c*A006190(n) + A006190(n-1). - _Gary W. Adamson_, Apr 02 2024
%H G. C. Greubel, <a href="/A098316/b098316.txt">Table of n, a(n) for n = 1..10000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Metallic_mean">Metallic mean</a>
%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>
%F 3 plus the constant in A085550. - _R. J. Mathar_, Sep 02 2008
%F From _Hieronymus Fischer_, Jan 02 2009: (Start)
%F 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).
%F 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.
%F 1/c=(sqrt(13)-3)/2.
%F See A001622 for a general formula concerning the fractional parts of powers of numbers x>1, which satisfy x-x^(-1)=floor(x).
%F 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)
%F c=3+sum{k>=1}(-1)^(k-1)/(A006190(k)*A006190(k+1)). - _Vladimir Shevelev_, Feb 23 2013
%F 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
%F 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
%e 3.30277563...
%t RealDigits[(3 + Sqrt[13])/2, 10, 100][[1]] (* _G. C. Greubel_, Apr 16 2017 *)
%o (PARI) (3 + sqrt(13))/2 \\ _Charles R Greathouse IV_, Jul 24 2013
%Y Cf. A001622, A014176, A098317, A098318, A000032, A006497, A080039, A049310.
%K nonn,cons,easy
%O 1,1
%A _Eric W. Weisstein_, Sep 02 2004
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