A122553 a(0)=1, a(n)=3 for n > 0.

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, 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, 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

Offset: 0

Author

Philippe Deléham, Sep 20 2006

Keywords

• nonn
• cofr
• easy
• cons

Comments

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

Decimal expansion of 4/30. - Alonso del Arte, Aug 16 2012

4/3 is the volume of the regular octahedron inscribed in the unit-radius sphere. - Amiram Eldar, Jun 02 2023

References

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

Links

• Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 4.
• Index entries for linear recurrences with constant coefficients, signature (1).

Crossrefs

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

Cf. A223139.

Programs

Mathematica

RealDigits[4/3, 10, 105][[1]] (* Alonso del Arte, Aug 16 2012 *)
PadRight[{1},120,3] (* Harvey P. Dale, Jul 21 2023 *)

PARI

a(n)=(n>=0)+2*(n>0) \\ Jaume Oliver Lafont, Mar 26 2009

Formula

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

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

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

From Amiram Eldar, Jun 05 2021: (Start)

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

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

Sum_{k>0} 3*k/4^k = 4/3 [Nicole Oresme]. - Stefano Spezia, Jun 27 2024

K_{n>=3} n/(n-2) = 4/3 (see Clawson at p. 224). - Stefano Spezia, Jul 01 2024

E.g.f.: 3*exp(x) - 2. - Elmo R. Oliveira, Aug 05 2024