A216212 -id:A216212 - cp's OEIS frontend

A178381 Number of paths of length n starting at initial node of the path graph P_9.

1, 1, 2, 3, 6, 10, 20, 35, 70, 125, 250, 450, 900, 1625, 3250, 5875, 11750, 21250, 42500, 76875, 153750, 278125, 556250, 1006250, 2012500, 3640625, 7281250, 13171875, 26343750, 47656250, 95312500, 172421875, 344843750

Offset: 0

Johannes W. Meijer, May 27 2010, May 29 2010

Counts all paths of length n, n>=0, starting at initial node on the path graph P_9, see the Maple program.
The a(n) represent the number of possible chess games, ignoring the fifty-move and the triple repetition rules, after n moves by White in the following position: White Ka1, Nh1, pawns a2, b6, c2, d6, f2, g3 and g4; Black Ka8, Bc8, pawns a3, b7, c3, d7, f3 and g5.
The path graphs P_(2*p) have as limit(a(n+1)/a(n), n =infinity) = 2 resp. hypergeom([(p-1)/(2*p+1),(p+2)/(2*p+1)],[1/2],3/4) and the path graphs P_(2*p+1) have as limit(a(n+1)/a(n), n =infinity) = 1+cos(Pi/(p+1)), p>=1; see the crossrefs. - Johannes W. Meijer, Jul 01 2010

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 20*x^6 + 35*x^7 + 70*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
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, arXiv:2006.06516 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Trigonometric Identities.
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-5).

This is row 9 of A094718.
a(2*n) = A147748(n) and a(2*n+1) = A081567(n).
a(4*n+2) = A109106(n) and a(4*n+3) = A179135(n).
Cf. A000007 (P_1), A000012 (P_2), A016116 (P_3), A000045 (P_4), A038754 (P_5), A028495 (P_6), A030436 (P_7), A061551 (P_8), this sequence (P_9), A336675 (P_10), A336678 (P_11), and A001405 (P_infinity).
Cf. A216212 (P_9 starting in the middle).
Cf. A033191, A179131, A179132, A128052, A179133.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x-3*x^2-2*x^3+x^4)/(1-5*x^2+5*x^4))); // G. C. Greubel, Sep 18 2018

with(GraphTheory): P:=9: G:=PathGraph(P): A:= AdjacencyMatrix(G): nmax:=36; for n from 0 to nmax do B(n):=A^n; a(n):=add(B(n)[1,k],k=1..P); od: seq(a(n),n=0..nmax);
r := j -> (-1)^(j/10) - (-1)^(1-j/10):
a := k -> add((2 + r(j))*r(j)^k, j in [1, 3, 5, 7, 9])/10:
seq(simplify(a(n)), n=0..30); # Peter Luschny, Sep 18 2020

CoefficientList[Series[(1+x-3*x^2-2*x^3+x^4)/(1-5*x^2+5*x^4), {x,0,50}], x] (* G. C. Greubel, Sep 18 2018 *)

x='x+O('x^50); Vec((1+x-3*x^2-2*x^3+x^4)/(1-5*x^2+5*x^4)) \\ G. C. Greubel, Sep 18 2018

G.f.: (1+x-3*x^2-2*x^3+x^4)/(1-5*x^2+5*x^4).
a(n) = 5*a(n-2) - 5*a(n-4) for n>=5 with a(0)=1, a(1)=1, a(2)=2, a(3)=3 and a(4)=6.
G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 + x / (1 - x / (1 - x / (1 + x / (1 + x)))))))). - Michael Somos, Feb 08 2015

A068913 Square array read by antidiagonals of number of k step walks (each step +-1 starting from 0) which are never more than n or less than -n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 2, 2, 1, 0, 4, 4, 2, 1, 0, 4, 6, 4, 2, 1, 0, 8, 12, 8, 4, 2, 1, 0, 8, 18, 14, 8, 4, 2, 1, 0, 16, 36, 28, 16, 8, 4, 2, 1, 0, 16, 54, 48, 30, 16, 8, 4, 2, 1, 0, 32, 108, 96, 60, 32, 16, 8, 4, 2, 1, 0, 32, 162, 164, 110, 62, 32, 16, 8, 4, 2, 1, 0, 64, 324, 328, 220, 124, 64, 32, 16, 8, 4, 2, 1

Offset: 0

Henry Bottomley, Mar 06 2002

			Rows start:
  1,  0,  0,  0,  0, ...
  1,  2,  2,  4,  4, ...
  1,  2,  4,  6, 12, ...
  1,  2,  4,  8, 14, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Cf. early rows: A000007, A016116 (without initial term), A068911, A068912, A216212, A216241, A235701.

Central and lower diagonals are A000079, higher diagonals include A000918, A028399.

T[n_,0]=1; T[n_,k_]:=2^k/(n+1) Sum[(-1)^r Cos[(Pi (2r-1))/(2 (n+1))]^k Cot[(Pi (1-2r))/(4 (n+1))],{r,1,n+1}]; Table[T[r,n-r],{n,0,20},{r,0,n}]//Round//Flatten (* Herbert Kociemba, Sep 23 2020 *)

Starting with T(n, 0) = 1, if (k-n) is negative or even then T(n, k) = 2*T(n, k-1), otherwise T(n, k) = 2*T(n, k-1) - A061897(n+1, (k-n-1)/2). So for n>=k, T(n, k) = 2^k. [Corrected by Sean A. Irvine, Mar 23 2024]

T(n,0) = 1, T(n,k) = (2^k/(n+1))*Sum_{r=1..n+1} (-1)^r*cos((Pi*(2*r-1))/(2*(n+1)))^k*cot((Pi*(1-2*r))/(4*(n+1))). - Herbert Kociemba, Sep 23 2020

A216219 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=5 or if k-n>=5, T(4,0) = T(3,0) = T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 0, 5, 10, 10, 5, 0, 0, 5, 15, 20, 15, 5, 0, 0, 0, 20, 35, 35, 20, 0, 0, 0, 0, 20, 55, 70, 55, 20, 0, 0, 0, 0, 0, 75, 125, 125, 75, 0, 0, 0, 0, 0, 0, 75, 200, 250, 200

Offset: 0

Philippe Deléham, Mar 13 2013

			Square array begins:
1, 1,  1,  1,   1,   0,   0,   0,    0,    0, 0, ...
1, 2,  3,  4,   5,   5,   0,   0,    0,    0, 0, ...
1, 3,  6, 10,  15,  20,  20,   0,    0,    0, 0, ...
1, 4, 10, 20,  35,  55,  75,  75,    0,    0, 0, ...
1, 5, 15, 35,  70, 125, 200, 275,  275,    0, 0, ...
0, 5, 20, 55, 125, 250, 450, 725, 1000, 1000, 0, ...
0, 0, 20, 75, 200, 450, 900, ...

Cf. A030191, A039717, A068913, A081567, A147748, A216210, A216212, A216218

T(n,n) = A147748(n).
T(n+1,n) = T(n,n+1) = A081567(n).
T(n+2,n) = T(n,n+2) = A039717(n+1).
T(n+3,n) = T(n+4,n) = T(n,n+3) = T(n,n+4) = A030191(n).
Sum_{k, 0<=k<=n} T(n-k,k) = A068913(4,n) = A216212(n).

A217777 Expansion of (1+x)(1+2x)(1-x)/(1-5x^2+5*x^4).

Original entry on oeis.org

1, 2, 4, 8, 15, 30, 55, 110, 200, 400, 725, 1450, 2625, 5250, 9500, 19000, 34375, 68750, 124375, 248750, 450000, 900000, 1628125, 3256250, 5890625, 11781250, 21312500, 42625000, 77109375, 154218750, 278984375, 557968750, 1009375000, 2018750000, 3651953125

Offset: 0

Philippe Deléham, Mar 24 2013

Index entries for linear recurrences with constant coefficients, signature (0, 5, 0, -5).

Cf. A217770

Vec((1+x)*(1+2*x)*(1-x)/(1-5*x^2+5*x^4)+O(x^66)) /* Joerg Arndt, Mar 29 2013 */

a(n) = A216212(n+1)/2.
a(2n) = A039717(n+1), a(2n+1) = 2*a(2n) = 2*A039717(n+1).
a(n) = sum(A217770(n-k,k), 0<=k<=n).
a(n) = 5*a(n-2) - 5*a(n-4) for n>=4, a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 8.
G.f.: (1+x)*(1+2*x)*(1-x)/(1-5*x^2+5*x^4).

Corrected name (g.f.), Joerg Arndt, Mar 29 2013

A216241 Number of n-step walks (each step +-1 starting from 0) which are never more than 5 or less than -5.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 62, 124, 236, 472, 890, 1780, 3340, 6680, 12502, 25004, 46732, 93464, 174554, 349108, 651740, 1303480, 2432918, 4865836, 9080956, 18161912, 33892954, 67785908, 126494956, 252989912, 472095062, 944190124, 1761901676, 3523803352, 6575544410, 13151088820

Offset: 0

Philippe Deléham, Mar 15 2013

Cf. Rows of A068913: A000007, A016116 (without initial term), A068911, A068912, A214846, A216212.

nn=35;CoefficientList[Series[(1+2x)(1-x^2)^2/(1-6x^2+9x^4-2x^6),{x,0,nn}],x] (* Geoffrey Critzer, Jan 14 2014 *)

a(n) = A068913(5,n).
a(n) = 6*a(n-2) - 9*a(n-4) + 2*a(n-6).
a(n) = 2^n for n < 6.
G.f.: ((1-x)^2*(1+x)^2*(1+2*x)) / ((1-2*x^2)*(1-4*x^2+x^4)).
a(2*n+1) = 2*a(2*n).
a(n) = Sum_{k=0..n} A214846(n-k, k). - Philippe Deléham, Mar 25 2013

a(34) corrected by Sean A. Irvine, May 19 2019

A235701 Number of n step walks (each step +/- 1 starting from 0) which are never more than 6 or less than -6.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 126, 252, 490, 980, 1890, 3780, 7252, 14504, 27734, 55468, 105840, 211680, 403368, 806736, 1535954, 3071908, 5845406, 11690812, 22238062, 44476124, 84582428, 169164856, 321661970, 643323940, 1223146232, 2446292464, 4650833040

Offset: 0

Geoffrey Critzer, Jan 14 2014

Index entries for linear recurrences with constant coefficients, signature (0,7,0,-14,0,7)

Cf. A068911, A068912, A216212, A216241.

Row n=6 of A068913.

nn=30;r=Solve[{s==1 + x a + x b, a==x s + x c, b==x s +x d, c==x a +x e, d==x b + x f, e==x c+x g, f==x d+x h,g==x e+x i, h==x f+x j,i==x g+x k,j==x h+x l,k==x i,l==x j, z==x k + x l }, {s,a,b,c,d,e,f,g,h,i,j,k,l,z}]; CoefficientList[Series[s+a+b+c+d+e+f+g+h+i+j+k+l/.r,{x,0,nn}],x]

G.f.: (1 + x - 2*x^2 - x^3)^2/(1 - 7*x^2 + 14*x^4 - 7*x^6).

cp's OEIS Frontend

A178381 Number of paths of length n starting at initial node of the path graph P_9.

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A068913 Square array read by antidiagonals of number of k step walks (each step +-1 starting from 0) which are never more than n or less than -n.

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A216219 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=5 or if k-n>=5, T(4,0) = T(3,0) = T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A217777 Expansion of (1+x)*(1+2*x)*(1-x)/(1-5*x^2+5*x^4).

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A216241 Number of n-step walks (each step +-1 starting from 0) which are never more than 5 or less than -5.

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A235701 Number of n step walks (each step +/- 1 starting from 0) which are never more than 6 or less than -6.

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A217777 Expansion of (1+x)(1+2x)(1-x)/(1-5x^2+5*x^4).