A110514 -id:A110514 - cp's OEIS frontend

A110515 Sequence array for (1 - x + x^2 + x^3)/(1 - x^4).

1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1

Offset: 0

Paul Barry, Jul 24 2005

Row sums are A106249. Diagonal sums are A110514.

			Rows begin
   1;
  -1,  1;
   1, -1,  1;
   1,  1, -1,  1;
   1,  1,  1, -1,  1;
  -1,  1,  1,  1, -1,  1;
   1, -1,  1,  1,  1,- 1,  1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Table[If[k <= n, -Sin[Pi*(n - k)/2] + Cos[Pi*(n - k)]/2 + 1/2, 0], {n,0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Aug 29 2017 *)

for(n=0,20, for(k=0,n, print1(round(if(k<=n, -sin(Pi*(n-k)/2) + cos(Pi*(n-k))/2 + 1/2, 0)), ", "))) \\ G. C. Greubel, Aug 29 2017

Riordan array ((1 - x + x^2 + x^3)/(1 - x^4), 1).
Column k has g.f. x^k*(1 - x + x^2 + x^3)/(1 - x^4).
T(n, k) = if(k <= n, -sin(Pi*(n-k)/2) + cos(Pi*(n-k))/2 + 1/2, 0).
T(n, k) = if(k <= n, Jacobi(2^(n-k), 2(n-k)+1), 0) [conjecture].

A110516 Expansion of (1-x+x^2+x^3)/(1+x-x^4-x^5).

Original entry on oeis.org

1, -2, 3, -2, 3, -4, 5, -4, 5, -6, 7, -6, 7, -8, 9, -8, 9, -10, 11, -10, 11, -12, 13, -12, 13, -14, 15, -14, 15, -16, 17, -16, 17, -18, 19, -18, 19, -20, 21, -20, 21, -22, 23, -22, 23, -24, 25, -24, 25, -26, 27, -26, 27, -28, 29, -28, 29, -30, 31, -30, 31, -32, 33, -32, 33, -34, 35, -34, 35, -36, 37, -36, 37, -38, 39, -38, 39

Offset: 0

Paul Barry, Jul 24 2005

Transform of (-1)^n by number triangle A110515. Partial sums are A110514.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,0,0,1,1)

CoefficientList[Series[(1-x+x^2+x^3)/(1+x-x^4-x^5),{x,0,80}],x] (* or *) LinearRecurrence[{-1,0,0,1,1},{1,-2,3,-2,3},80] (* Harvey P. Dale, Jul 14 2017 *)

x='x+O('x^50); Vec((1-x+x^2+x^3)/(1+x-x^4-x^5)) \\ G. C. Greubel, Aug 29 2017

G.f.: (1-x+x^2+x^3)/((1+x)(1-x^4)).
a(n) = -a(n-1) + a(n-4) + a(n-5).
a(n) = -sin(pi*n/2+pi/4)/sqrt(2) + cos(pi*n+pi/4)/sqrt(2) + (-1)^n*(2n+3)/4 + 1/4.

cp's OEIS Frontend

A110515 Sequence array for (1 - x + x^2 + x^3)/(1 - x^4).

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