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a(0)=1, a(n)=3 for n > 0.
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%I #52 Aug 06 2024 21:35:54

%S 1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,

%T 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,

%U 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3

%N a(0)=1, a(n)=3 for n > 0.

%C Continued fraction for (sqrt(13) - 1)/2 = A223139.

%C Decimal expansion of 4/30. - _Alonso del Arte_, Aug 16 2012

%C 4/3 is the volume of the regular octahedron inscribed in the unit-radius sphere. - _Amiram Eldar_, Jun 02 2023

%D Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Springer, 2013, pp. 95-96, 224.

%H Jun Yan, <a href="https://arxiv.org/abs/2404.07958">Results on pattern avoidance in parking functions</a>, arXiv:2404.07958 [math.CO], 2024. See p. 4.

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).

%F a(n) = 3 - 2*0^n.

%F G.f.: (1 + 2*x)/(1 - x).

%F Sum_{n >= 0} a(n)*10^(-n) = 4/3.

%F From _Amiram Eldar_, Jun 05 2021: (Start)

%F 4/3 = Product_{k>=1} (1 + 1/2^(2^k)).

%F 4/3 = Sum_{k>=0} binomial(2*k,k)/((k+2)*4^k). (End)

%F Sum_{k>0} 3*k/4^k = 4/3 [Nicole Oresme]. - _Stefano Spezia_, Jun 27 2024

%F K_{n>=3} n/(n-2) = 4/3 (see Clawson at p. 224). - _Stefano Spezia_, Jul 01 2024

%F E.g.f.: 3*exp(x) - 2. - _Elmo R. Oliveira_, Aug 05 2024

%t RealDigits[4/3, 10, 105][[1]] (* _Alonso del Arte_, Aug 16 2012 *)

%t PadRight[{1},120,3] (* _Harvey P. Dale_, Jul 21 2023 *)

%o (PARI) a(n)=(n>=0)+2*(n>0) \\ _Jaume Oliver Lafont_, Mar 26 2009

%Y Cf. A118273 (cube), A339259 (regular icosahedron), A363437 (regular tetrahedron), A363438 (regular dodecahedron).

%Y Cf. A223139.

%K nonn,cofr,easy,cons

%O 0,2

%A _Philippe Deléham_, Sep 20 2006