A063210 -id:A063210 - cp's OEIS frontend

A153860 Triangle by columns: leftmost column = (1, 0, 1, -1, 1, -1, 1, ...); columns >1 = (1, 1, 0, 0, 0, ...).

1, 0, 1, 1, 1, 1, -1, 0, 1, 1, 1, 0, 0, 1, 1, -1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, -1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Gary W. Adamson, Jan 03 2009

As an infinite lower triangular matrix M; M * [1,2,3,...] = A063210: (1, 2, 6, 6, 10, 10, 14, 14, ...).
M * [1, 3, 5, 7, ...] = A047471, {1,3} mod 8.
Eigensequence of the triangle = A066983 starting (1, 1, 3, 3, 7, 9, 17, 25, ...).
Binomial transform of the triangle = A153861.
Row sums = A153284: (1, 1, 3, 1, 3, 1, 3, 1, ...).

			First few rows of the triangle:
   1;
   0, 1;
   1, 1, 1;
  -1, 0, 1, 1;
   1, 0, 0, 1, 1;
  -1, 0, 0, 0, 1, 1;
   1, 0, 0, 0, 0, 1, 1;
  -1, 0, 0, 0, 0, 0, 1, 1;
   1, 0, 0, 0, 0, 0, 0, 1, 1;
  ...

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

Cf. A153861 (binomial transform), A153284 (row sums), A063210, A047471, A066983.

a153860 n k = a153860_tabl !! (n-1) !! (k-1)
a153860_row n = a153860_tabl !! (n-1)
a153860_tabl = [1] : [0, 1] : iterate (\(x:xs) -> -x : 0 : xs) [1, 1, 1]
-- Reinhard Zumkeller, Dec 16 2013

Triangle by columns: leftmost column = (1, 0, 1, -1, 1, ...); columns > 1 = (1, 1, 0, 0, 0, ...).

A168276 a(n) = 2*n - (-1)^n - 1.

Original entry on oeis.org

2, 2, 6, 6, 10, 10, 14, 14, 18, 18, 22, 22, 26, 26, 30, 30, 34, 34, 38, 38, 42, 42, 46, 46, 50, 50, 54, 54, 58, 58, 62, 62, 66, 66, 70, 70, 74, 74, 78, 78, 82, 82, 86, 86, 90, 90, 94, 94, 98, 98, 102, 102, 106, 106, 110, 110, 114, 114, 118, 118, 122, 122, 126, 126, 130, 130

Offset: 1

Vincenzo Librandi, Nov 22 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A039722, A168277.

Cf. A063210. - R. J. Mathar, Nov 25 2009

[2*n-1-(-1)^n: n in [1..70]]; // Vincenzo Librandi, Sep 16 2013

CoefficientList[Series[2 (1 + x^2) / ((1 + x) (1 - x)^2), {x, 0, 80}], x] (* Vincenzo Librandi, Sep 16 2013 *)
Table[2 n - 1 - (-1)^n, {n, 70}] (* Bruno Berselli, Sep 17 2013 *)
LinearRecurrence[{1,1,-1},{2,2,6},70] (* Harvey P. Dale, Oct 22 2014 *)

a(n) = 4*n - a(n-1) - 4, with n>1, a(1)=2.
from R. J. Mathar, Nov 25 2009: (Start)
a(n) = 2*n - (-1)^n - 1.
a(n) = 2*A109613(n-1).
G.f.: 2*x*(1 + x^2)/((1+x)*(1-x)^2). (End)
a(n) = a(n-1) + a(n-2) - a(n-3). - Vincenzo Librandi, Sep 16 2013
a(n) = A168277(n) + 1. - Vincenzo Librandi, Sep 17 2013
E.g.f.: (-1 + 2*exp(x) + (2*x -1)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 16 2016
Sum_{n>=1} 1/a(n)^2 = Pi^2/16. - Amiram Eldar, Aug 21 2022

Previous definition replaced with closed-form expression by Bruno Berselli, Sep 17 2013

cp's OEIS Frontend

A153860 Triangle by columns: leftmost column = (1, 0, 1, -1, 1, -1, 1, ...); columns >1 = (1, 1, 0, 0, 0, ...).

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