A005555 -id:A005555 - cp's OEIS frontend

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

N. J. A. Sloane

Partial sums of A158057.

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

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).

Cf. A000217, A001620, A158057, A250129.

[8*n^2 + 9*n : n in [1..40]]; // Vincenzo Librandi, Nov 05 2014

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

Vec((17-x)/(1-x)^3 + O(x^50)) \\ Michel Marcus, Nov 05 2014

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)

Formula and more terms from Jeffrey Shallit, Aug 15 1995

A005567 Number of walks on square lattice.

Original entry on oeis.org

10, 70, 308, 1092, 3414, 9834, 26752, 69784, 176306, 434382, 1048812, 2490636, 5833006, 13500754, 30933368, 70255008, 158335434, 354419190, 788529700, 1744831060, 3841983110, 8422163130, 18387829488

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

R. K. Guy, Letter to N. J. A. Sloane, May 1990
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6, (see Figure 6).
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

A005567:=2*(5-10*z+4*z**2)/(2*z-1)**3/(z-1)**3; # conjectured by Simon Plouffe in his 1992 dissertation

a(n) = 26 + 11*n + n^2 + (-16 + 24*n + 8*n^2)*2^n; \\ Michel Marcus, Oct 13 2014

a(n) = 26 + 11*n + n^2 + (-16 + 24*n + 8*n^2)*2^n. - Fitted by John W. Layman

A005569 Number of walks on square lattice.

Original entry on oeis.org

4, 34, 308, 3024, 31680, 349206, 4008004, 47530912, 579058896, 7215393640, 91644262864, 1183274479040, 15497363512800, 205519758825150, 2755739674246500, 37314861293793600, 509727100259638800

Offset: 1

N. J. A. Sloane

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

R. K. Guy, Letter to N. J. A. Sloane, May 1990
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

a(n)=4*if(n<2,n==1,a(n-1)*n*(2*n-1)*(2*n+1)*(5*n+7)/((n-1)*(n+2)*(n+3)*(5*n+2)))

G.f.: 4_F_3 ( [ 2, 17/5, 5/2, 3/2 ]; [ 4, 5, 12/5 ];16 x ).

a(n)(n-1)(n+2)(n+3)(5n+2) = a(n-1)4n(2n-1)(2n+1)(5n+7), n>1. - Michael Somos, Mar 04 2003

A005571 Number of walks on cubic lattice.

Original entry on oeis.org

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

N. J. A. Sloane

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

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).

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

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

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

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A005570 Number of walks on cubic lattice.

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A005567 Number of walks on square lattice.

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A005569 Number of walks on square lattice.

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A005571 Number of walks on cubic lattice.

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