A336675 -id:A336675 - 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

A061551 Number of paths along a corridor width 8, starting from one side.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 20, 35, 69, 124, 241, 440, 846, 1560, 2977, 5525, 10490, 19551, 36994, 69142, 130532, 244419, 460737, 863788, 1626629, 3052100, 5743674, 10782928, 20283121, 38092457, 71632290, 134560491, 252989326, 475313762

Offset: 0

Henry Bottomley, May 16 2001

Counts all paths of length n starting at initial node on the path graph P_8. - Paul Barry, May 11 2004
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, pawns a2, b6, d2, d6 and g2; Black Ka8, Bc8, pawns a3, b7, d3, d7 and g3. - Johannes W. Meijer, May 29 2010
Define the 4 X 4 tridiagonal unit-primitive matrix (see [Jeffery]) M = A_{9,1} = [0,1,0,0; 1,0,1,0; 0,1,0,1; 0,0,1,1]; then a(n)=[M^n](4,4). - _L. Edson Jeffery, Mar 18 2011
a(n) is the length of n-th word derived by certain iterated substitutions on four letters {1,2,3,4} as follows. Define the substitution rules 1 -> {2}, 2 -> {1,3}, 3 -> {2,4}, 4 -> {3,4}, in which "," denotes concatenation, so 1 -> 2, 2 -> 13, 3 -> 24, 4 -> 34. Let w(k) be the k-th word formed by applying the substitution rules to each letter (digit) in word w(k-1), k>0, putting w(0) = 1. Then, for n=0,1,..., {w(n)} = {1, 2, 13, 224, 131334, 2242242434, 13133413133413342434, ...} in which {length(w(n))} = {1,1,2,3,6,10,...} = A061551. The maps 1 -> 2, etc., are given by the above matrix A_{9,1} by taking i -> {j : [A_{9,1}](i,j) <> 0}, i, j in {1,2,3,4}. Moreover, the entry in row 1 and column j of [A{9,1}]^n gives the relative frequency of the letter j in the n-th word w(n). Finally, the sum of the first-row entries of [A_{9,1}]^n again gives a(n), so obviously a(n) = sum of relative frequencies of each j in word w(n). - L. Edson Jeffery, Feb 06 2012
Range of row n of the circular Pascal array of order 9. - Shaun V. Ault, Jun 05 2014
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=8. - Herbert Kociemba, Sep 17 2020

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 20*x^6 + 35*x^7 + 69*x^8 + ....

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Shaun V. Ault and Charles Kicey, Counting paths in corridors using circular Pascal arrays, Discrete Mathematics (2014).
Shaun V. Ault and Charles Kicey, Counting paths in corridors using circular Pascal arrays, arXiv:1407.2197 [math.CO], (8-July-2014)
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Johann Cigler, Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials, arXiv:2212.02118 [math.NT], 2022.
Nachum Dershowitz, Between Broadway and the Hudson, arXiv:2006.06516 [math.CO], 2020.
L. E. Jeffery, Unit-primitive matrices
Marc A. A. Van Leeuwen, Some simple bijections involving lattice walks and ballot sequences, arXiv:1010.4847 [math.CO], (23-October-2010)
Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1).

Narrower corridors effectively produce A000007, A000012, A016116, A000045, A038754, A028495, A030436, A061551, A178381, A336675, A336678.
An infinitely wide corridor (i.e., just one wall) would produce A001405.
Equivalently, the above mentioned corridor numbers are exactly the ranges of the circular Pascal array of order d = 2, 3, 4, 5, 6, 7, 8, respectively, and this is true for any natural number d greater than or equal to 2.
a(n) = A094718(8, n).
Cf. A030436 and A178381.

a[0]:=1: a[1]:=1: a[2]:=2: a[3]:=3: a[4]:=6: a[5]:=10: a[6]:=20: a[7]:=35: for n from 8 to 33 do a[n]:=7*a[n-2]-15*a[n-4]+10*a[n-6]-a[n-8] od: seq(a[n],n=0..33); # Emeric Deutsch, Aug 14 2006
with(GraphTheory): P:=8: G:=PathGraph(P): A:= AdjacencyMatrix(G): nmax:=33; 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); # Johannes W. Meijer, May 29 2010
X := j -> (-1)^(j/9) - (-1)^(1-j/9):
a := k -> add((2 + X(j))*X(j)^k, j in [1, 3, 5, 7])/9:
seq(simplify(a(n)), n=0..30); # Peter Luschny, Sep 17 2020

LinearRecurrence[{1,3,-2,-1},{1,1,2,3},40] (* Harvey P. Dale, Dec 19 2011 *)
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,8],{n,0,40}]//Round (* Herbert Kociemba, Sep 17 2020 *)

a(n) = sum(b(n,i)) where b(n,0) = b(n,9) = 0, b(0,1)=1, b(0, n)=0 if n!=1 and b(n,i) = b(n-1,i) + b(n+1,i) if 0 < n < 9.
From Emeric Deutsch, Aug 14 2006: (Start)
G.f.: (1-2*x^2)/((1-x)*(1-3*x^2-x^3)).
a(n) = 7*a(n-2) - 15*a(n-4) + 10*a(n-6) - a(n-8). (End)
a(2*n) = A094854(n) and a(2*n+1) = A094855(n). - Johannes W. Meijer, May 29 2010
a(n) = a(n-1) + 3*a(n-2) - 2*a(n-3) - a(n-4), for n > 3, with {a(k)}={1,1,2,3}, k=0,1,2,3. - L. Edson Jeffery, Mar 18 2011
a(n) = A187498(3*n + 2). - L. Edson Jeffery, Mar 18 2011
a(n) = A205573(3,n). - L. Edson Jeffery, Feb 06 2012
G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 + x / (1 - x / (1 - x / (1 + x))))))). - Michael Somos, Feb 08 2015
a(n) = 2^n/9*Sum_{r=1..8} (1-(-1)^r)cos(Pi*r/9)^n*(1+cos(Pi*r/9)). - Herbert Kociemba, Sep 17 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.

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=11. - 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 (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

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

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A061551 Number of paths along a corridor width 8, starting from one side.

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

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