A005571 Number of walks on cubic lattice. (Formerly M5352)

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

0,1

References

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

Links

Table of n, a(n) for n=0..35.

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).

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

Mathematica

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

Prog

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

Crossrefs

Sequence in context: A251078 A153676 A060316 * A233877 A067987 A167586

Adjacent sequences: A005568 A005569 A005570 * A005572 A005573 A005574

Keyword

nonn,walk,easy

Author

N. J. A. Sloane

Status

approved