A030436 -id:A030436 - cp's OEIS frontend

A217257 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 1 or if k-n >= 7, T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,3) = T(0,4) = T(0,5) = T(0,6) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 5, 0, 0, 0, 1, 5, 9, 5, 0, 0, 0, 0, 6, 14, 14, 0, 0, 0, 0, 0, 6, 20, 28, 14, 0, 0, 0, 0, 0, 0, 26, 48, 42, 0, 0, 0, 0, 0, 0, 0, 26, 74, 90, 42, 0, 0, 0, 0, 0, 0, 0, 0, 100, 164, 132, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 264, 296, 132, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 364, 560, 428, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 17 2013

A hexagon arithmetic of E. Lucas.

			Square array begins:
1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... row n=0
0, 1, 2, 3, 4, 5, 6, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... row n=1
0, 0, 2, 5, 9, 14, 20, 26, 26, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... row n=2
0, 0, 0, 5, 14, 28, 48, 74, 100, 100, 0, 0, 0, 0, 0, 0, 0, ... row n=3
0, 0, 0, 0, 14, 42, 90, 162, 264, 364, 364, 0, 0, 0, 0, 0, ... row n=4
0, 0, 0, 0, 0, 42, 132, 296, 560, 924, 1288, 1288, 0, 0, 0, ... row n=5
...

E. Lucas, Théorie des nombres, A. Blanchard, Paris, 1958, p.89

E. Lucas, Théorie des nombres, Tome 1, Jacques Gabay, Paris, p. 89

Cf. similar sequences: A216230, A216228, A216226, A216238, A216054.

T(n,n) = A024175(n).
T(n,n+1) = A024175(n+1).
T(n,n+2) = A094803(n+1).
T(n,n+3) = A007070(n).
T(n,n+4) = A094806(n+2).
T(n,n+5) = T(n,n+6) = A094811(n+2).
Sum_{k, 0<=k<=n} T(n-k,k) = A030436(n).

a(69) = 0 deleted by Georg Fischer, Oct 16 2021

A336675 Number of paths of length n starting at initial node of the path graph P_10.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 251, 460, 911, 1690, 3327, 6225, 12190, 22950, 44744, 84626, 164407, 312019, 604487, 1150208, 2223504, 4239225, 8181175, 15621426, 30108147, 57556155, 110820165, 212037241, 407946421, 781074572, 1501844193, 2877011660, 5529362694

Offset: 0

Nachum Dershowitz, Jul 30 2020

Also the number of paths along a corridor width 10, starting from one side.

In general, a(n,m) = (2^n/(m+1))*Sum_{r=1..m} (1-(-1)^r)*cos(Pi*r/(m+1))^n*(1+cos(Pi*r/(m+1))) gives the number of paths of length n starting at the initial node on the path graph P_m. Here we have m=10. - Herbert Kociemba, Sep 14 2020

Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-3,1).

This is row 10 of A094718. Bisections give A224514 (even part), A216710 (odd part).

Cf. A000004 (row 0), A000007 (row 1), A000012 (row 2), A016116 (row 3), A000045 (row 4), A038754 (row 5), A028495 (row 6), A030436 (row 7), A061551 (row 8), A178381 (row 9), this sequence (row 10), A336678 (row 11), A001405 (limit).

X := j -> (-1)^(j/11) - (-1)^(1-j/11):
a := k -> add((2 + X(j))*X(j)^k, j in [1, 3, 5, 7, 9])/11:
seq(simplify(a(n)), n=0..30); # Peter Luschny, Sep 17 2020

a[n_,m_]:=2^(n+1)/(m+1) Module[{x=(Pi r)/(m+1)},Sum[Cos[x]^n (1+Cos[x]),{r,1,m,2}]]
Table[a[n,10],{n,0,40}]//Round (* Herbert Kociemba, Sep 14 2020 *)

my(x='x+O('x^44)); Vec((1 - 3*x^2 + x^4)/(1 - x - 4*x^2 + 3*x^3 + 3*x^4 - x^5)) \\ Joerg Arndt, Jul 31 2020

From Stefano Spezia, Jul 30 2020: (Start)
G.f.: (1 - 3*x^2 + x^4)/(1 - x - 4*x^2 + 3*x^3 + 3*x^4 - x^5).
a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 3*a(n-4) + a(n-5) for n > 4. (End)
a(n) = (2^n/11)*Sum_{r=1..10} (1-(-1)^r)*cos(Pi*r/11)^n*(1+cos(Pi*r/11)). - Herbert Kociemba, Sep 14 2020

A336678 Number of paths of length n starting at initial node of the path graph P_11.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 461, 922, 1702, 3404, 6315, 12630, 23494, 46988, 87533, 175066, 326382, 652764, 1217483, 2434966, 4542526, 9085052, 16950573, 33901146, 63255670, 126511340, 236063915, 472127830, 880983606, 1761967212, 3287837741

Offset: 0

Nachum Dershowitz, Jul 30 2020

Also the number of paths along a corridor width 11, starting from one side.

Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-9,0,2).

This is row 11 of A094718. Bisections give A087944 (even part), A087946 (odd part).
Cf. A000004 (row 0), A000007 (row 1), A000012 (row 2), A016116 (row 3), A000045 (row 4), A038754 (row 5), A028495 (row 6), A030436 (row 7), A061551 (row 8),
A178381 (row 9), A336675 (row 10), this sequence (row 11), A001405 (limit).

X := j -> (-1)^(j/12) - (-1)^(1-j/12):
a := k -> add((2 + X(j))*X(j)^k, j in [1, 3, 5, 7, 9, 11])/12:
seq(simplify(a(n)), n=0..30); # Peter Luschny, Sep 17 2020

LinearRecurrence[{0, 6, 0, -9, 0, 2}, {1, 1, 2, 3, 6, 10}, 40] (* Harvey P. Dale, Sep 08 2020 *)
a[n_,m_]:=2^(n+1)/(m+1) Module[{x=(Pi r)/(m+1)},Sum[Cos[x]^n (1+Cos[x]),{r,1,m,2}]]
Table[a[n,11], {n,0,40}]//Round (* Herbert Kociemba, Sep 14 2020 *)

my(x='x+O('x^44)); Vec(-(x^5+3*x^4-3*x^3-4*x^2+x+1)/((2*x^2-1)*(x^4-4*x^2+1))) \\ Joerg Arndt, Jul 31 2020

G.f.: -(x^5+3*x^4-3*x^3-4*x^2+x+1)/((2*x^2-1)*(x^4-4*x^2+1)).

a(n) = (2^n/12)*Sum_{r=1..11} (1-(-1)^r)*cos(Pi*r/12)^n*(1+cos(Pi*r/12)). - Herbert Kociemba, Sep 14 2020

A030435 Expansion of g.f. (1 + x - 2x^2 - x^3)/(1/2 - 2x^2 + x^4).

Original entry on oeis.org

2, 2, 4, 6, 12, 20, 40, 68, 136, 232, 464, 792, 1584, 2704, 5408, 9232, 18464, 31520, 63040, 107616, 215232, 367424, 734848, 1254464, 2508928, 4283008, 8566016, 14623104, 29246208, 49926400, 99852800, 170459392, 340918784, 581984768, 1163969536, 1987020288

Offset: 0

N. J. A. Sloane

Also (starting 4,6,...) the number of zig-zag paths from top to bottom of a rectangle of width 7, whose color is that of the top right corner. - Joseph Myers, Dec 24 2008

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-2).

Twice A030436.

CoefficientList[Series[(1+x-2*x^2-x^3)/(1/2-2*x^2+x^4),{x,0,40}],x] (* Harvey P. Dale, Oct 05 2020 *)

Vec((1+x-2*x^2-x^3)/(1/2-2*x^2+x^4)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

E.g.f.: cosh(r*x) + cosh(s*x) + (r*sinh(r*x) + s*sinh(s*x))/2, where r = sqrt(2 - sqrt(2)) and s = sqrt(2 + sqrt(2)). - Stefano Spezia, Jun 14 2023

cp's OEIS Frontend

A217257 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 1 or if k-n >= 7, T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,3) = T(0,4) = T(0,5) = T(0,6) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

A336675 Number of paths of length n starting at initial node of the path graph P_10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A336678 Number of paths of length n starting at initial node of the path graph P_11.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A030435 Expansion of g.f. (1 + x - 2*x^2 - x^3)/(1/2 - 2*x^2 + x^4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A030435 Expansion of g.f. (1 + x - 2x^2 - x^3)/(1/2 - 2x^2 + x^4).