cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A092936 Area of n-th triple of hexagons around a triangle.

Original entry on oeis.org

1, 9, 100, 1089, 11881, 129600, 1413721, 15421329, 168220900, 1835008569, 20016873361, 218350598400, 2381839709041, 25981886201049, 283418908502500, 3091626107326449, 33724468272088441, 367877524885646400
Offset: 1

Views

Author

Peter J. C. Moses, Apr 18 2004

Keywords

Comments

This is the unsigned member r=-9 of the family of Chebyshev sequences S_r(n) defined in A092184: ((-1)^(n+1))*a(n) = S_{-9}(n), n>=0.
a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using (1/2,1/2)-fences, red half-squares (1/2 X 1 pieces, always placed so that the shorter sides are horizontal), green half-squares, and blue half-squares. A (w,g)-fence is a tile composed of two w X 1 pieces separated by a gap of width g. a(n+1) also equals the number of tilings of an n-board using (1/4,3/4)-fences, red (1/4,1/4)-fences, green (1/4,1/4)-fences, and blue (1/4,1/4)-fences. - Michael A. Allen, Dec 30 2022

Examples

			a(5) = 10*(1089+100)-9 = 11881. From A006190, a(5) = (3*33+10)^2 = 11881.

Links

Crossrefs

Cf. A005386, A006190, A057076, A092184, A176019.

Programs

  • GAP
    a:=[1,9,100];; for n in [4..18] do a[n]:=10*(a[n-1]+a[n-2])-a[n-3]; od; a; # Muniru A Asiru, Feb 20 2018
  • Maple
    seq(fibonacci(n,3)^2,n=1..18); # Zerinvary Lajos, Apr 05 2008
  • Mathematica
    CoefficientList[Series[(1-x)*x/(1-10*x-10*x^2+x^3), {x, 0, 20}], x]
(CoefficientList[Series[x/(1-3*x-x^2), {x, 0, 20}], x])^2
Table[Round[((3+Sqrt[13])^n)^2/(13*4^n)], {n, 0, 20}]
LinearRecurrence[{10, 10, -1}, {1, 9, 100}, 18] (* Georg Fischer, Feb 22 2019 *)

Formula

a(n) = A006190(n)^2.
a(n) = 10*(a(n-1)+a(n-2)) - a(n-3).
G.f.: (1-x)*x/(1-10*x-10*x^2+x^3).
a(n) = ((3-sqrt(13))^n-(3+sqrt(13))^n)^2/(13*4^n).
a(n) = 2*(T(n, 11/2)-(-1)^n)/13 with twice the Chebyshev polynomials of the first kind evaluated at x = 11/2: 2*T(n, 11/2) = A057076(n) = ((11+3*sqrt(13))^n + (11-3*sqrt(13))^n)/2^n. - Wolfdieter Lang, Oct 18 2004
From Michael A. Allen, Dec 30 2022: (Start)
a(n+1) = 11*a(n) - a(n-1) + 2*(-1)^n.
a(n+1) = (1 + (-1)^n)/2 + 9*Sum_{k=1..n} ( k*a(n+1-k) ). (End)
Product_{n>=2} (1 + (-1)^n/a(n)) = (3 + sqrt(13))/6 (A176019) (Falcon, 2016, p. 189, eq. (3.1)). - Amiram Eldar, Dec 03 2024

A010690 Period 2: repeat (1,9).

Original entry on oeis.org

1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1, 9, 1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Digital roots of the nonzero square triangular numbers. - Ant King, Jan 21 2012
Continued fraction expansion of A176019. - R. J. Mathar, Mar 08 2012
Exp( Sum_{n >= 1} a(n-1)*x^n/n ) = 1 + x + 5*x^2 + 5*x^3 + 15*x^4 + 15*x^5 + ... is the o.g.f. for A189976 (taken with an offset of 0). - Peter Bala, Mar 13 2015
Final digit of 9^n. - Martin Renner, Jun 11 2020
Decimal expansion of 19/99. - Stefano Spezia, Feb 09 2025

Examples

			0.191919191919191919191919191919191919191...

Links

Crossrefs

Cf. A008592, A001019 (9^n), A014393, A189976.

Programs

  • Mathematica
    5+4*(-1)^# &/@Range[81] (* Ant King, Jan 21 2012 *)
  • PARI
    a(n)=1; if(n%2==1, 9, 1) \\ Felix Fröhlich, Aug 11 2014

Formula

G.f.: (1+9x)/((1-x)(1+x)). - R. J. Mathar, Nov 21 2011
a(n) = 9^n mod 10. - Martin Renner, Jun 11 2020
E.g.f.: cosh(x) + 9*sinh(x). - Stefano Spezia, Feb 09 2025
From Amiram Eldar, Jun 09 2025: (Start)
With offset 1:
Multiplicative with a(2^e) = 9, a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 1/2^(s-3)). (End)

A381485 Decimal expansion of sqrt(13)/6.

Original entry on oeis.org

6, 0, 0, 9, 2, 5, 2, 1, 2, 5, 7, 7, 3, 3, 1, 5, 4, 8, 8, 5, 3, 2, 0, 3, 5, 4, 4, 5, 7, 8, 4, 1, 5, 9, 9, 1, 0, 4, 1, 8, 8, 2, 7, 6, 2, 3, 0, 7, 5, 4, 1, 0, 3, 5, 4, 5, 1, 7, 4, 2, 1, 7, 6, 0, 3, 7, 8, 6, 1, 1, 5, 8, 0, 4, 8, 8, 3, 5, 0, 7, 4, 2, 0, 0, 7, 6, 9, 8, 4, 7, 0, 0, 3, 0, 8, 1, 7, 8, 6, 2, 7, 8, 9, 1, 9
Offset: 0

Views

Author

Amiram Eldar, Feb 24 2025

Keywords

Comments

The greatest possible minimum distance between 6 points in a unit square.
The solution was found by Ronald L. Graham and reported by Schaer (1965).

Examples

			0.60092521257733154885320354457841599104188276230754...

References

  • Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved Problems in Geometry, Springer, 1991, Section D1, p. 108.
  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

Links

Crossrefs

Solutions for k points: A002193 (k = 2), A120683 (k = 3), 1 (k = 4), A010503 (k = 5), this constant (k = 6), A379338 (k = 7), A101263 (k = 8), A020761 (k = 9), A281065 (k = 10).
Cf. A010470, A142464, A176019, A295330, A344069.

Programs

  • Mathematica
    RealDigits[Sqrt[13] / 6, 10, 120][[1]]
  • PARI
    list(len) = digits(floor(10^len*quadgen(52)/6));

Formula

Equals A010470 / 6 = A295330 / 3 = A344069 / 2 = A176019 - 1/2 = sqrt(A142464).
Minimal polynomial: 36*x^2 - 13.
Showing 1-3 of 3 results.

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