A075554 -id:A075554 - cp's OEIS frontend

A075553 Numerators in the Maclaurin series for arctan(1+x).

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

Offset: 0

Eric W. Weisstein, Sep 23 2002

Cf. A075554.

a[n_] := I ((-1 - I)^n - (-1 + I)^n)/2^Floor[1 + n/2]; Table[a[n], {n, 0, 100}]

a(n)=[0,1,-1,1,0,-1,1,-1][n%8+1] /* Michael Somos, Jul 16 2006 */

Euler transform of length 8 sequence [ -1, 1, 1, -1, 0, -1, 0, 1]. - Michael Somos, Jul 16 2006

G.f.: x(1-x+x^2)/(1+x^4) = x(1-x)(1-x^4)(1-x^6)/((1-x^2)(1-x^3)(1-x^8)). a(-n) = a(n+4) = -a(n). - Michael Somos, Jul 16 2006

A132314 a(n) = n*2^floor((n+1)/2).

Original entry on oeis.org

0, 2, 4, 12, 16, 40, 48, 112, 128, 288, 320, 704, 768, 1664, 1792, 3840, 4096, 8704, 9216, 19456, 20480, 43008, 45056, 94208, 98304, 204800, 212992, 442368, 458752, 950272, 983040, 2031616, 2097152, 4325376, 4456448, 9175040, 9437184, 19398656, 19922944, 40894464

Offset: 0

Simon Plouffe, Nov 19 2007

G. C. Greubel, Table of n, a(n) for n = 0..1000
Simon Plouffe, Illustration.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-4).

Cf. A093968, A075554.

Maple
```
seq(n*2^(floor((n+1)/2)),n=0..120);
```

LinearRecurrence[{0,4,0,-4}, {0, 2, 4, 12}, 50] (* G. C. Greubel, May 30 2016 *)

a(n) = 2*A093968(n).
From Chai Wah Wu, May 30 2016: (Start)
a(n) = 4*a(n-2) - 4*a(n-4) for n > 3.
G.f.: 2*x*(2*x^2 + 2*x + 1)/(2*x^2 - 1)^2. (End)
E.g.f.: x*(2*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x)). - G. C. Greubel, May 30 2016
Sum_{n>=1} 1/a(n) = log(2)/2 + log(1+sqrt(2))/sqrt(2). - Amiram Eldar, Feb 13 2023

cp's OEIS Frontend

A075553 Numerators in the Maclaurin series for arctan(1+x).

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