A005570 Number of walks on cubic lattice. (Formerly M5038)

17, 50, 99, 164, 245, 342, 455, 584, 729, 890, 1067, 1260, 1469, 1694, 1935, 2192, 2465, 2754, 3059, 3380, 3717, 4070, 4439, 4824, 5225, 5642, 6075, 6524, 6989, 7470, 7967, 8480, 9009, 9554, 10115, 10692, 11285, 11894, 12519, 13160, 13817, 14490, 15179, 15884, 16605

Offset

1,1

Comments

Partial sums of A158057.

References

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

Links

Jeremy Gardiner, Table of n, a(n) for n = 1..999

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 (3,-3,1).

Formula

a(n) = 8*n^2 + 9*n.

G.f.: (17-x)/(1-x)^3. Simon Plouffe in his 1992 dissertation.

a(n) = 16*A000217(n) + n. - Jon Perry, Nov 05 2014

Sum_{n>=1} 1/a(n) = 80/81 +Psi(1/8)/9+gamma/9 = 0.11973.. see A001620 and A250129. - R. J. Mathar, May 30 2022

Sum_{n>=1} 1/a(n) = 80/81 - (sqrt(2)+1)*Pi/18 - log(1+sqrt(2))*sqrt(2)/9 -4*log(2)/9. - Amiram Eldar, Sep 10 2022

From Elmo R. Oliveira, Jan 28 2025: (Start)

E.g.f.: exp(x)*x*(17 + 8*x).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

Mathematica

CoefficientList[Series[(17 - x) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 05 2014 *)

Prog

(PARI) Vec((17-x)/(1-x)^3 + O(x^50)) \\ Michel Marcus, Nov 05 2014
(Magma) [8*n^2 + 9*n : n in [1..40]]; // Vincenzo Librandi, Nov 05 2014

Crossrefs

Cf. A000217, A001620, A158057, A250129.

Sequence in context: A098329 A160076 A003124 * A195037 A214660 A258598

Adjacent sequences: A005567 A005568 A005569 * A005571 A005572 A005573

Keyword

nonn,walk,easy,changed

Author

N. J. A. Sloane

Extensions

Formula and more terms from Jeffrey Shallit, Aug 15 1995

Status

approved