login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A157142 Signed denominators of Leibniz series for Pi/4 12
1, -3, 5, -7, 9, -11, 13, -15, 17, -19, 21, -23, 25, -27, 29, -31, 33, -35, 37, -39, 41, -43, 45, -47, 49, -51, 53, -55, 57, -59, 61, -63, 65, -67, 69, -71, 73, -75, 77, -79, 81, -83, 85, -87, 89, -91, 93, -95, 97, -99, 101, -103, 105, -107, 109, -111, 113, -115 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Numerators are all 1.

a(n) is also the determinant of the n X n matrix with 1's on the diagonal and 2's elsewhere (cf. A000354). - Jody Nagel (SejeongY(AT)aol.com), May 01 2010

LINKS

Table of n, a(n) for n=0..57.

X. Gourdon and P. Sebah, Archimedes' constant

Mathpages, How Leibniz might have anticipated Euler

Wikipedia, Leibniz formula for Pi

FORMULA

Euler transform of length 2 sequence [ -3, 2]. - Michael Somos, Mar 26 2011

a(n) = b(2*n + 1) where b(n) is completely multiplicative with b(2) = 0, b(p) = p if p == 1 (mod 4), b(p) = -p if p == 3 (mod 4). - Michael Somos, Mar 26 2011

With offset 1 this sequence is the exponential reversion of A005264. - Michael Somos, Mar 26 2011

a(-1 - n) = a(n). a(n + 1) + a(n - 1) = -2 * a(n). - Michael Somos, Mar 26 2011

E.g.f.: (1 - 2*x) * exp(-x). - Michael Somos, Mar 26 2011

a(n) = A005408(n) * A033999(n).

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

a(0)=1, a(1)=-3, a(n)=-2a(n-1)-a(n-2) for n>=2

Sum_{n=0..inf} 1/a(n) = Pi/4

EXAMPLE

1 - 3*x + 5*x^2 - 7*x^3 + 9*x^4 - 11*x^5 + 13*x^6 - 15*x^7 + 17*x^8 + ...

PROG

(PARI) {a(n) = (2*n + 1) * (-1)^n}

CROSSREFS

Cf. A005264, A005408, A033999.

Cf. A157327. [From Jaume Oliver Lafont, Mar 03 2009]

Sequence in context: A081874 A165747 A053229 * A247328 A004273 A005408

Adjacent sequences:  A157139 A157140 A157141 * A157143 A157144 A157145

KEYWORD

frac,sign

AUTHOR

Jaume Oliver Lafont, Feb 24 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 8 20:44 EST 2016. Contains 278950 sequences.