A094832 -id:A094832 - cp's OEIS frontend

A094833 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 5.

1, 4, 15, 55, 199, 714, 2548, 9061, 32148, 113887, 403051, 1425471, 5039254, 17809336, 62928201, 222324436, 785402143, 2774421135, 9800231959, 34617003682, 122274355596, 431893332397, 1525507797700, 5388281150223

Offset: 1

Herbert Kociemba, Jun 13 2004

In general, a(n) = (2/m)*Sum_{r=1..m-1} sin(r*j*Pi/m)*sin(r*k*Pi/m)*(2*cos(r*Pi/m))^(2n) counts (s(0), s(1), ..., s(2n)) such that 0 < s(i) < m and |s(i)-s(i-1)| = 1 for i = 1,2,...,2n, s(0) = j, s(2n) = k.

Michael De Vlieger, Table of n, a(n) for n = 1..1825
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (6,-9,1).

Rest@ CoefficientList[Series[(-x + 2 x^2)/(-1 + 6 x - 9 x^2 + x^3), {x, 0, 24}], x] (* Michael De Vlieger, Jul 02 2021 *)

a(n) = (2/9)*Sum_{r=1..8} sin(r*Pi/3)*sin(5*r*Pi/9)*(2*cos(r*Pi/9))^(2n).
a(n) = 6a(n-1) - 9a(n-2) + a(n-3).
G.f.: (-x+2x^2)/(-1 + 6x - 9x^2 + x^3).
a(n+1) = 3*a(n) + A094832(n-1). - Philippe Deléham, Mar 20 2007
a(n) = A094829(n+1) - 2*A094829(n). - R. J. Mathar, Nov 14 2019

A216236 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=4 or if k-n>=5, 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, 0, 0, 5, 10, 10, 4, 0, 0, 5, 15, 20, 14, 0, 0, 0, 0, 20, 35, 34, 14, 0, 0, 0, 0, 20, 55, 69, 48, 0, 0, 0, 0, 0, 0, 75, 124, 117, 48, 0, 0, 0, 0, 0, 0, 75, 199, 241, 165, 0, 0, 0, 0, 0, 0, 0, 0, 274, 440, 406, 165, 0, 0, 0, 0, 0, 0, 0, 0, 274, 714, 846, 571, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 14 2013

Arithmetic hexagon of E. Lucas.

			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, ...
1, 4, 10, 20, 35, 55, 75, 75, 0, 0, 0, ...
0, 4, 14, 34, 69, 124, 199, 274, 274, 0, 0, ...
0, 0, 14, 48, 117, 241, 440, 714, 988, 988, 0, ...
...

E. Lucas, Théorie des nombres, Albert Blanchard, Paris, 1958, Tome 1, p. 89

E. Lucas, Théorie des nombres, Tome 1, Jacques Gabay, Paris, 1991, p.89

Similar sequences: A216201, A216210, A216216, A216218, A216219, A216220, A216226, A216228, A216229, A216230, A216232, A216235.

T(n+3,n) = T(n+2,n) = A094827(n).
T(n+1,n) = A094832(n).
T(n,n) = A094854(n).
T(n,n+1) = A094855(n).
T(n,n+2) = A094833(n+1).
T(n,n+3) = T(n,n+4) = A094828(n).
Sum( T(n-k,k), 0<=k<=n ) = A217733(n). - Philippe Deléham, Mar 22 2013

A217765 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=3 or if k-n >= 6, T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 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, 0, 1, 4, 6, 3, 0, 1, 5, 10, 9, 0, 0, 0, 6, 15, 19, 9, 0, 0, 0, 6, 21, 34, 28, 0, 0, 0, 0, 0, 27, 55, 62, 28, 0, 0, 0, 0, 0, 27, 82, 117, 90, 0, 0, 0, 0, 0, 0, 0, 109, 199, 207, 90, 0, 0, 0, 0, 0, 0, 0, 109, 308, 406, 297, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 24 2013

A hexagon arithmetic of E. Lucas.

			Square array begins:
1, 1, 1, 1, 1, 1, 0, 0, 0, ... row n=0
1, 2, 3, 4, 5, 6, 6, 0, 0, ... row n=1
1, 3, 6, 10, 15, 21, 27, 27, 0, 0, ... row n=2
0, 3, 9, 19, 34, 55, 82, 109, 109, 0, 0, ... row n=3
0, 0, 9, 28, 62, 117, 199, 308, 417, 417, 0, 0, ... row n=4
0, 0, 0, 28, 90, 207, 406, 714, 1131, 1548, 1548, 0, 0, ... row n=5
...
Square array, read by rows, with 0 omitted:
1, 1, 1, 1, 1, 1
1, 2, 3, 4, 5, 6, 6
1, 3, 6, 10, 15, 21, 27, 27
3, 9, 19, 34, 55, 82, 109, 109
9, 28, 62, 117, 199, 308, 417, 417
28, 90, 207, 406, 714, 1131, 1548, 1548
90, 297, 703, 1417, 2548, 4096, 5644, 5644
297, 1000, 2417, 4965, 9061, 14705, 20349, 20349
1000, 3417, 8382, 17443, 32148, 52497, 72846, 72846
3417, 11799, 29242, 61390, 113887, 186733, 259579, 259579
11799, 41041, 102431, 216318, 403051, 662630, 922209, 922209
...

Cf. Similar sequences: A216201, A216210, A216216, A216218, ...

T(n,n+4) = T(n,n+5) = A094829(n+2).
T(n,n+3) = A094834(n+1).
T(n,n+2) = A094833(n+1).
T(n,n+1) = A094832(n).
T(n,n) = A094831(n).
T(n+1,n) = T(n+2,n) = A094826(n).
sum(T(n-k,k), 0<=k<=n) = A065455(n).

A188022 Expansion of x(1+x) / (1-3x^2-x^3).

Original entry on oeis.org

0, 1, 1, 3, 4, 10, 15, 34, 55, 117, 199, 406, 714, 1417, 2548, 4965, 9061, 17443, 32148, 61390, 113887, 216318, 403051, 762841, 1425471, 2691574, 5039254, 9500193, 17809336, 33539833, 62928201, 118428835, 222324436, 418214706, 785402143, 1476968554

Offset: 0

L. Edson Jeffery, Mar 18 2011

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](3,4)=[M^n](4,3).

Michael De Vlieger, Table of n, a(n) for n = 0..3650
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
L. E. Jeffery, Unit-primitive matrices
Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
Index entries for linear recurrences with constant coefficients, signature (0,3,1).

Cf. A094832 (bisection), A094833 (bisection).

Cf. A052931, A187498.

LinearRecurrence[{0, 3, 1}, {0, 1, 1}, 36] (* or *)
CoefficientList[Series[x (1 + x)/(1 - 3 x^2 - x^3), {x, 0, 35}], x] (* Michael De Vlieger, Mar 10 2020 *)

a(n) = 3*a(n-2)+a(n-3).
a(n) = A187498(3*n+1).
a(n) = A052931(n-2)+A052931(n-1). - R. J. Mathar, Mar 22 2011

cp's OEIS Frontend

A094833 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 5.

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A216236 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=4 or if k-n>=5, 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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A217765 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=3 or if k-n >= 6, T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A188022 Expansion of x(1+x) / (1-3x^2-x^3).

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cp's OEIS Frontend

A094833 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 5.

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A216236 Square array T, read by antidiagonals: T(n,k) = 0 if n-k>=4 or if k-n>=5, 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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A217765 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=3 or if k-n >= 6, T(2,0) = T(1,0) = T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).

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A188022 Expansion of x*(1+x) / (1-3*x^2-x^3).

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A188022 Expansion of x(1+x) / (1-3x^2-x^3).