A087946 -id:A087946 - cp's OEIS frontend

A214846 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 6 or if k-n >= 6, T(k,0) = T(0,k) = 1 if 0 <= k <= 5, T(n,k) = T(n-1,k) + T(n,k-1).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 0, 6, 15, 20, 15, 6, 0, 0, 6, 21, 35, 35, 21, 6, 0, 0, 0, 27, 56, 70, 56, 27, 0, 0, 0, 0, 27, 83, 126, 126, 83, 27, 0, 0, 0, 0, 0, 110, 209, 252, 209, 110, 0, 0, 0, 0, 0, 0, 110, 319, 461, 461, 319, 110, 0, 0, 0, 0, 0, 0, 0, 429, 780, 922, 780, 429, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 16 2013

An arithmetic hexagon of E. Lucas.

			Square array begins:
  1, 1,  1,   1,   1,   1,    0,    0,    0,     0,     0, ...
  1, 2,  3,   4,   5,   6,    6,    0,    0,     0,     0, ...
  1, 3,  6,  10,  15,  21,   27,   27,    0,     0,     0, ...
  1, 4, 10,  20,  35,  56,   83,  110,  110,     0,     0, ...
  1, 5, 15,  35,  70, 126,  209,  319,  429,   429,     0, ...
  1, 6, 21,  56, 126, 252,  461,  780, 1209,  1638,  1638, ...
  0, 6, 27,  83, 209, 461,  922, 1702, 2911,  4549,  6187, ...
  0, 0, 27, 110, 319, 780, 1702, 3404, 6315, 10864, 17051, ...
  ...

E. Lucas, Théorie des nombres, Gauthier-Villars, Paris 1891, Tome 1, p. 89.

Cf. similar sequences: A000007, A216218, A216216, A216210, A216219.

T(n,n) = A087944(n).
T(n,n+1) = T(n+1,n) = A087946(n).
T(n+2,n) = T(n,n+2) = A001353(n+1).
T(n+3,n) = T(n,n+3) = A216271(n).
T(n+5,n) = T(n+4,n) = T(n,n+4) = T(n,n+5) = A216263(n).
Sum_{k=0..n} T(n-k,k) = A216241(n).

A122068 Expansion of x(1-3x)(1-x)/(1-7x+14x^2-7x^3).

Original entry on oeis.org

1, 3, 10, 35, 126, 462, 1715, 6419, 24157, 91238, 345401, 1309574, 4970070, 18874261, 71705865, 272491891, 1035680954, 3936821259, 14965658694, 56893879910, 216295686467, 822315097387, 3126323230541, 11885921055638

Offset: 1

Gary W. Adamson, Oct 15 2006

G. C. Greubel, Table of n, a(n) for n = 1..1000
Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. Vol. 70, No. 1, Feb. 1997, 22-31.
R. Witula, P. Lorenc, M. Rozanski, and M. Szweda, Sums of the rational powers of roots of cubic polynomials, Zeszyty Naukowe Politechniki Slaskiej, Seria: Matematyka Stosowana z. 4, Nr. kol. 1920, 2014.
Index entries for linear recurrences with constant coefficients, signature (7,-14,7).

Cf. A087946, A081567.

Cf. A215007, A215008. - Roman Witula, May 16 2014

a:=[1,3,10];; for n in [4..30] do a[n]:=7*(a[n-1]-2*a[n-2]+a[n-3]); od; a; # G. C. Greubel, Oct 03 2019

I:=[1,3,10]; [n le 3 select I[n] else 7*(Self(n-1) -2*Self(n-2) + Self(n-3)): n in [1..30]]; // G. C. Greubel, Oct 03 2019

seq(coeff(series(x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3), x, n+1), x, n), n =1..30); # G. C. Greubel, Oct 03 2019

M = {{2,1,0,0,0,0}, {1,2,1,0,0,0}, {0,1,2,1,0,0}, {0,0,1,2,1,0}, {0,0,0, 1,2,1}, {0,0,0,0,1,2}}; v[1] = {1,1,1,1,1,1}; v[n_]:= v[n] = M.v[n-1]; Table[v[n][[1]], {n,30}]
Rest@CoefficientList[Series[x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3), {x, 0, 30}], x] (* G. C. Greubel, Oct 03 2019 *)
LinearRecurrence[{7,-14,7},{1,3,10},30] (* Harvey P. Dale, Mar 08 2020 *)

Vec(x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3)+O(x^30)) \\ Charles R Greathouse IV, Sep 27 2012

def A122068_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3)).list()
a=A122068_list(30); a[1:] # G. C. Greubel, Oct 03 2019

From Roman Witula, May 16 2014: (Start)
a(n) = (1/2)*Sum_{k=0..2}(1 - 1/sqrt(7)*cot(2^k * alpha))* (2*sin(2^k * alpha))^(2n), where alpha := 2*Pi/7.
a(n) = (A215007(n) + A215008(n+1) - 2*A215008(n))/2. (End)
a(n) = binomial(2*n-1, n-1) + Sum_{k=1..n} (-1)^k*binomial(2*n, n+7*k). - Greg Dresden, Jan 28 2023

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

A216271 Expansion of (1-x)/((1-2x)(1-4x+x^2)).

Original entry on oeis.org

1, 5, 21, 83, 319, 1209, 4549, 17051, 63783, 238337, 890077, 3322995, 12403951, 46296905, 172791861, 644886923, 2406788599, 8982333009, 33522674509, 125108627171, 466912358463, 1742541855257, 6503257159717, 24270490977915, 90578715140551, 338044386361505, 1261598863859901

Offset: 0

Philippe Deléham, Mar 16 2013

Partial sums are in A216263.

Diagonal of square array A214846.

Index entries for linear recurrences with constant coefficients, signature (6,-9,2).

Cf. A001353, A087946, A214846, A216263.

CoefficientList[Series[(1-x)/((1-2x)(1-4x+x^2)),{x,0,30}],x] (* Harvey P. Dale, Oct 05 2019 *)

a(n) = A001353(n+2) - A087946(n+1).
G.f.: (1-x)/(1-6x+9x^2-2x^3).
a(n) = 6*a(n-1) - 9*a(n-2) + 2*a(n-3), a(0) = 1, a(1) = 5, a(2) = 21.
Sum_{k=0..n} a(k) = A216263(n).

cp's OEIS Frontend

A214846 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 6 or if k-n >= 6, T(k,0) = T(0,k) = 1 if 0 <= k <= 5, T(n,k) = T(n-1,k) + T(n,k-1).

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A122068 Expansion of x*(1-3*x)*(1-x)/(1-7*x+14*x^2-7*x^3).

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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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Maple

Mathematica

PARI

Formula

A216271 Expansion of (1-x)/((1-2x)(1-4x+x^2)).

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Mathematica

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A122068 Expansion of x(1-3x)(1-x)/(1-7x+14x^2-7x^3).