A134931 -id:A134931 - cp's OEIS frontend

A120463 Expansion of x(1+x+2x^3) / ((x-1)(1+x)(3*x^2-1)).

0, 1, 1, 4, 6, 13, 21, 40, 66, 121, 201, 364, 606, 1093, 1821, 3280, 5466, 9841, 16401, 29524, 49206, 88573, 147621, 265720, 442866, 797161, 1328601, 2391484, 3985806, 7174453, 11957421, 21523360, 35872266, 64570081, 107616801, 193710244

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jun 30 2006

Top element of the vector obtained by multiplying the n-th power of the 5 X 5 matrix [[0, 1, 0, 0, 0], [1, 0, 1, 0, 0], [0, 1, 0, 1, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 0]] by the column vector [0, 1, 1, 2, 3].

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-3).

Cf. A003462, A134931.

LinearRecurrence[{0,4,0,-3},{0,1,1,4,6},40] (* Harvey P. Dale, Dec 15 2018 *)

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

a(2n+2) = A134931(n). a(2n+1) = A003462(n+1). - R. J. Mathar, Nov 07 2011

A166124 Triangle, read by rows, given by [0,1/2,1/2,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 2, 0, 1, 2, 0, 1, 1, 2, 0, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 0

Philippe Deléham, Oct 07 2009

			Triangle begins :
1 ;
0,2 ;
0,1,2 ;
0,1,1,2 ;
0,1,1,1,2 ;
0,1,1,1,1,2 ;
0,1,1,1,1,1,2 ; ...

Sum_{k, 0<=k<=n} T(n,k)*x^(n-k)= A166122(n), A166114(n), A084222(n), A084247(n), A000034(n), A040000(n), A000027(n+1), A000079(n), A007051(n), A047849(n), A047850(n), A047851(n), A047852(n), A047853(n), A047854(n), A047855(n), A047856(n) for x= -5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^k= A000007(n), A000027(n+1), A033484(n), A134931(n), A083597(n) for x= 0,1,2,3,4 respectively.
T(n,k)= A166065(n,k)/2^(n-k).
G.f.: (1-x+x*y)/(1-x-x*y+x^2*y). - Philippe Deléham, Nov 09 2013
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 09 2013

A208324 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -2, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 4, 3, 10, 8, 4, 18, 28, 16, 5, 28, 64, 72, 32, 6, 40, 120, 200, 176, 64, 7, 54, 200, 440, 576, 416, 128, 8, 70, 308, 840, 1456, 1568, 960, 256, 9, 88, 448, 1456, 3136, 4480, 4096, 2176, 512, 10, 108, 624, 2352, 6048, 10752, 13056, 10368, 4864

Offset: 0

Philippe Deléham, Feb 25 2012

Row sums are A134931(n).
Diagonal sums are A140253(n).
Compare this sequence with A207627.
Column k is divisible by 2^k.

			Triangle begins :
1
2, 4
3, 10, 8
4, 18, 28, 16
5, 28, 64, 72, 32
6, 40, 120, 200, 176, 64
7, 54, 200, 440, 576, 416, 128
8, 70, 308, 840, 1456, 1568, 960, 256
9, 88, 448, 1456, 3136, 4480, 4096, 2176, 512
10, 108, 624, 2352, 6048, 10752, 13056, 10368, 4864, 1024

Cf. A207627

T(n,0) = n+1.
T(n,1) = 2*T(n,0) + T(n-1,1).
T(n,k) = 2*T(n-1,k-1) + T(n-1,k) for k>1.
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1) with T(0,0) = 1, T(1,0) = 2, T(1,1) = 4.
G.f.: (1+2*y*x)/(1-2*(1+y)*x+(1+2*y)*x^2).

A220946 Expansion of (1+2x+2x^2-x^3)/((1-x)(1+x)(1-3x^2)).

Original entry on oeis.org

1, 2, 6, 7, 21, 22, 66, 67, 201, 202, 606, 607, 1821, 1822, 5466, 5467, 16401, 16402, 49206, 49207, 147621, 147622, 442866, 442867, 1328601, 1328602, 3985806, 3985807, 11957421, 11957422, 35872266, 35872267, 107616801, 107616802, 322850406, 322850407

Offset: 0

Philippe Deléham, Apr 14 2013

Index entries for linear recurrences with constant coefficients, signature (0, 4, 0, -3).

Cf. A060816, A134931.

LinearRecurrence[{0, 4, 0, -3}, {1, 2, 6, 7}, 40] (* T. D. Noe, Apr 17 2013 *)

a(n) = a(n-1)+1 if n odd, a(n) = a(n-1)*3 if n even.
a(2n) = A134931(n), a(2n+1) = A060816(n+1).
a(n) = 4*a(n-2) - 3*a(n-4) with a(0)=1, a(1)=2, a(2)=6, a(3)=7.

cp's OEIS Frontend

A120463 Expansion of x(1+x+2x^3) / ((x-1)(1+x)(3*x^2-1)).

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A166124 Triangle, read by rows, given by [0,1/2,1/2,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A208324 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -2, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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Formula

A220946 Expansion of (1+2x+2x^2-x^3)/((1-x)(1+x)(1-3x^2)).

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cp's OEIS Frontend

A120463 Expansion of x*(1+x+2*x^3) / ((x-1)*(1+x)*(3*x^2-1)).

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Author

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Formula

A166124 Triangle, read by rows, given by [0,1/2,1/2,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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Examples

Formula

A208324 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -2, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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Formula

A220946 Expansion of (1+2*x+2*x^2-x^3)/((1-x)*(1+x)*(1-3x^2)).

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A120463 Expansion of x(1+x+2x^3) / ((x-1)(1+x)(3*x^2-1)).

A220946 Expansion of (1+2x+2x^2-x^3)/((1-x)(1+x)(1-3x^2)).