A180168 -id:A180168 - cp's OEIS frontend

A026597 Expansion of (1+x)/(1-x-4*x^2).

1, 2, 6, 14, 38, 94, 246, 622, 1606, 4094, 10518, 26894, 68966, 176542, 452406, 1158574, 2968198, 7602494, 19475286, 49885262, 127786406, 327327454, 838473078, 2147782894, 5501675206, 14092806782, 36099507606, 92470734734

Offset: 0

Clark Kimberling

This sequence can generated by the following formula: a(n) = a(n-1) + 4*a(n-2) when n > 2; a[1] = 1, a[2] = 2. - Alex Vinokur (alexvn(AT)barak-online.net), Oct 21 2004
An elephant sequence, see A175654 and A175655. For the corner squares just one A[5] vector, with decimal value 325, leads to the sequence given above. For the central square this vector leads to a companion sequence that is 4 times this very same sequence with n >= -1. - Johannes W. Meijer, Aug 15 2010
Equals INVERTi transform of A180168. - Gary W. Adamson, Aug 14 2010
Start with a single cell at coordinates (0, 0), then iteratively subdivide the grid into 2 X 2 cells and remove the cells that have one '1' in their modulo 3 coordinates. a(n) is the number of cells after n iterations. Cell configuration converges to a fractal with approximate dimension 1.357. - Peter Karpov, Apr 20 2017
Also, the number of walks of length n starting at vertex 1 in the graph with 4 vertices and edges {{0,1}, {0,2}, {0,3}, {1,2}, {2,3}}. - Sean A. Irvine, Jun 02 2025

Nathaniel Johnston, Table of n, a(n) for n = 0..500
Sean A. Irvine, Walks on Graphs.
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of initial terms (n = 1..8)
Index entries for linear recurrences with constant coefficients, signature (1,4).

Cf. A006131, A006138, A007482, A026581, A026584, A122950, A175654, A175655, A180168.

[n le 2 select n else Self(n-1) + 4*Self(n-2): n in [1..41]]; // G. C. Greubel, Dec 08 2021

LinearRecurrence[{1,4},{1,2},40] (* Harvey P. Dale, Nov 28 2011 *)

[(2*i)^n*( chebyshev_U(n, -i/4) - (i/2)*chebyshev_U(n-1, -i/4) ) for n in (0..40)] # G. C. Greubel, Dec 08 2021

G.f.: (1+x)/(1-x-4*x^2).
a(n) = T(n,0) + T(n,1) + ... + T(n,2*n), T given by A026584.
a(n) = Sum_{k=0..n} binomial(floor((2*n-k-1)/2), n-k)*2^k. - Paul Barry, Feb 11 2005
a(n) = A006131(n) + A006131(n-1), n >= 1. - R. J. Mathar, Oct 20 2006
a(n) = Sum_{k=0..n} binomial(floor((2*n-k)/2),n-k)*4^floor(k/2). - Paul Barry, Feb 02 2007
Inverse binomial transform of A007482: (1, 3, 11, 39, 139, 495, ...). - Gary W. Adamson, Dec 04 2007
a(n) = Sum_{k=0..n+1} A122950(n+1,k)*3^(n+1-k). - Philippe Deléham, Jan 04 2008
a(n) = (1/2 + 3*sqrt(17)/34)*(1/2 + sqrt(17)/2)^n + (1/2 - 3*sqrt(17)/34)*(1/2 - sqrt(17)/2)^n. - Antonio Alberto Olivares, Jun 07 2011
a(n) = (2*i)^n*( chebyshevU(n, -i/4) - (i/2)*chebyshevU(n-1, -i/4) ). - G. C. Greubel, Dec 08 2021
E.g.f.: exp(x/2)*(17*cosh(sqrt(17)*x/2) + 3*sqrt(17)*sinh(sqrt(17)*x/2))/17. - Stefano Spezia, Jan 31 2023

Better name from Ralf Stephan, Jul 14 2013

A209144 Triangle of coefficients of polynomials v(n,x) jointly generated with A209143; see the Formula section.

Original entry on oeis.org

1, 3, 6, 1, 12, 5, 24, 16, 1, 48, 44, 7, 96, 112, 30, 1, 192, 272, 104, 9, 384, 640, 320, 48, 1, 768, 1472, 912, 200, 11, 1536, 3328, 2464, 720, 70, 1, 3072, 7424, 6400, 2352, 340, 13, 6144, 16384, 16128, 7168, 1400, 96, 1, 12288, 35840, 39680, 20736

Offset: 1

Clark Kimberling, Mar 06 2012

Alternating row sums: 1,3,5,7,9,11,13,15,17,...
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (3,-1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 07 2012

			First five rows:
   1;
   3;
   6,  1;
  12,  5;
  24, 16, 1;
First three polynomials v(n,x): 1, 3, 6 + x.
(3,-1, 0, 0, 0, ...) DELTA (0, 1/3, -1/3, 0, 0, ...) begins:
    1;
    3,   0;
    6,   1,   0;
   12,   5,   0, 0;
   24,  16,   1, 0, 0;
   48,  44,   7, 0, 0, 0;
   96, 112,  30, 1, 0, 0, 0;
  192, 272, 104, 9, 0, 0, 0, 0;

Cf. A209143, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209143 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209144 *)

u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
v(n,x) = u(n-1,x) + v(n-1,x) + 1,
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 07 2012: (Start)
As triangle T(n,k) with 0 <= k <= n:
T(n,k) = 2*T(n-1,k) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 0 and T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1+x)/(1-2*x-y*x^2).
Sum_{k=0..n} T(n,k)*x^k = A005408(n), A003945(n), A078057(n), A028859(n), A000244(n), A063782(n), A180168(n) for x = -1, 0, 1, 2, 3, 4, 5 respectively. (End)

A207543 Triangle read by rows, expansion of (1+yx)/(1-2y*x+y(y-1)x^2).

Original entry on oeis.org

1, 0, 3, 0, 1, 5, 0, 0, 5, 7, 0, 0, 1, 14, 9, 0, 0, 0, 7, 30, 11, 0, 0, 0, 1, 27, 55, 13, 0, 0, 0, 0, 9, 77, 91, 15, 0, 0, 0, 0, 1, 44, 182, 140, 17, 0, 0, 0, 0, 0, 11, 156, 378, 204, 19, 0, 0, 0, 0, 0, 1, 65, 450, 714, 285, 21, 0

Offset: 0

Philippe Deléham, Feb 24 2012

Previous name was: "A scaled version of triangle A082985."

Triangle, read by rows, given by (0, 1/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -4/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

			Triangle begins :
1
0, 3
0, 1, 5
0, 0, 5, 7
0, 0, 1, 14, 9
0, 0, 0, 7, 30, 11
0, 0, 0, 1, 27, 55, 13
0, 0, 0, 0, 9, 77, 91, 15
0, 0, 0, 0, 1, 44, 182, 140, 17
0, 0, 0, 0, 0, 11, 156, 378, 204, 19
0, 0, 0, 0, 0, 1, 65, 450, 714, 285, 21
0, 0, 0, 0, 0, 0, 13, 275, 1122, 1254, 385, 23

R. Zielinski, Induction and Analogy in a Problem of Finite Sums arXiv:1608.04006 [math.GM], 2016.

Cf. A082985 which is another version of this triangle.

Cf. Diagonals : A005408, A000330, A005585, A050486, A054333, A057788. Cf. A119900.

s := (1+y*x)/(1-2*y*x+y*(y-1)*x^2): t := series(s,x,12):
seq(print(seq(coeff(coeff(t,x,n),y,m),m=0..n)),n=0..11); # Peter Luschny, Aug 17 2016

T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 3.
G.f.: (1+y*x)/(1-2*y*x+y*(y-1)*x^2).
Sum_{i, i>=0} T(n+i,n) = A000204(2*n+1).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A078069(n), A000007(n), A003945(n), A111566(n) for x = -1, 0, 1, 2 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A090131(n), A005408(n), A003945(n), A078057(n), A028859(n), A000244(n), A063782(n), A180168(n) for x = -1, 0, 1, 2, 3, 4, 5, 6 respectively.
From Peter Bala, Aug 17 2016: (Start)
Let S(k,n) = Sum_{i = 1..n} i^k. Calculations in Zielinski 2016 suggest the following identity holds involving the p-th row elements of this triangle:
Sum_{k = 0..p} T(p,k)*S(2*k,n) = 1/2*(2*n + 1)*(n*(n + 1))^p.
For example, for row 6 we find S(6,n) + 27*S(8,n) + 55*S(10,n) + 13*S(12,n) = 1/2*(2*n + 1)*(n*(n + 1))^6.
There appears to be a similar result for the odd power sums S(2*k + 1,n) involving A119900. (End)

New name using a formula of the author from Peter Luschny, Aug 17 2016

A176812 Expansion of 3(1+x)/(1-2x-5*x^2).

Original entry on oeis.org

3, 9, 33, 111, 387, 1329, 4593, 15831, 54627, 188409, 649953, 2241951, 7733667, 26677089, 92022513, 317430471, 1094973507, 3777099369, 13029066273, 44943629391, 155032590147, 534783327249, 1844729605233, 6363375846711

Offset: 0

Roger L. Bagula, Apr 26 2010

Binomial transform of A026532 after dropping A026532(0). [From R. J. Mathar, Apr 27 2010]

Index entries for linear recurrences with constant coefficients, signature (2,5).

a[n_] = 2^n*(((3 + Sqrt[ 6])/2)*((1 + Sqrt[6])/2)^n + ((3 - Sqrt[6])/2)*((1 - Sqrt[6])/2)^n); Table[FullSimplify[a[n]], {n, 0, 30}]
CoefficientList[Series[(-3(1+x))/(5x^2+2x-1),{x,0,40}],x]  (* Harvey P. Dale, Feb 24 2011 *)

Binet form: a(n)=2^n*(((3 + Sqrt[6])/2)*((1 + Sqrt[6])/2)^n + ((3 - Sqrt[6])/2)*((1 - Sqrt[6])/2)^n) = 3*A180168(n).

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A026597 Expansion of (1+x)/(1-x-4*x^2).

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A209144 Triangle of coefficients of polynomials v(n,x) jointly generated with A209143; see the Formula section.

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A207543 Triangle read by rows, expansion of (1+y*x)/(1-2*y*x+y*(y-1)*x^2).

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A176812 Expansion of 3*(1+x)/(1-2*x-5*x^2).

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A207543 Triangle read by rows, expansion of (1+yx)/(1-2y*x+y(y-1)x^2).

A176812 Expansion of 3(1+x)/(1-2x-5*x^2).