A005571 Number of walks on cubic lattice.

76, 288, 700, 1376, 2380, 3776, 5628, 8000, 10956, 14560, 18876, 23968, 29900, 36736, 44540, 53376, 63308, 74400, 86716, 100320, 115276, 131648, 149500, 168896, 189900, 212576, 236988, 263200, 291276, 321280, 353276, 387328, 423500, 461856, 502460, 545376

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• walk
• easy

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
• Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6 (see Figure 7).
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Programs

Mathematica

a[n_] := 4 (n + 1) (n + 3) (8 n + 19)/3; Array[a, 30, 0] (* Amiram Eldar, Sep 10 2022 *)

PARI

vector(40, n, n--; 4*(n+1)*(n+3)*(8*n+19)/3) \\ Michel Marcus, Oct 13 2014

Formula

G.f.: 4*(19-4*x+x^2)/(x-1)^4. - Simon Plouffe in his 1992 dissertation

a(n) = 4(n+1)(n+3)(8n+19)/3.

Sum_{n>=0} 1/a(n) = 499/1936 + (6*log(1+sqrt(2))*sqrt(2) - 3*(sqrt(2)-1)*Pi - 24*log(2))/55. - Amiram Eldar, Sep 10 2022