A005570 Number of walks on cubic lattice.
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
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).
Programs
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Magma
[8*n^2 + 9*n : n in [1..40]]; // Vincenzo Librandi, Nov 05 2014
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Mathematica
CoefficientList[Series[(17 - x) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 05 2014 *) -
PARI
Vec((17-x)/(1-x)^3 + O(x^50)) \\ Michel Marcus, Nov 05 2014
Formula
a(n) = 8*n^2 + 9*n.
G.f.: (17-x)/(1-x)^3. Simon Plouffe in his 1992 dissertation.
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)
Extensions
Formula and more terms from Jeffrey Shallit, Aug 15 1995
Comments