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A132050 Denominator of 2*n*A000111(n-1)/A000111(n): approximations of Pi using Euler (up/down) numbers. 6
1, 1, 1, 5, 8, 61, 136, 1385, 3968, 50521, 176896, 2702765, 260096, 199360981, 951878656, 19391512145, 104932671488, 2404879675441, 14544442556416, 74074237647505, 2475749026562048, 69348874393137901, 507711943253426176 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

The rationals r(n)=2*n*e(n-1)/e(n), where e(n)=A000111(n), approximate Pi as n -> oo. - M. F. Hasler, Apr 03 2013

Numerators are given in A132049.

See the Delahaye reference and a link by W. Lang given in A132049.

From Paul Curtz, Mar 17 2013: (Start)

Apply the Akiyama-Tanigawa transform (or algorithm) to A046978(n+2)/A016116(n+1):

1,        1/2,      0,   -1/4,  -1/4,  -1/8,      0, 1/16, 1/16;

1/2,        1,    3/4,      0,  -5/8,  -3/4,  -7/16,    0;  = Balmer0(n)

-1/2,     1/2,    9/4,    5/2,   5/8, -15/8, -49/16;

-1,      -7/2,   -3/4,   15/2,  25/2,  57/8;

5/2,    -11/2,  -99/4,    -20, 215/8;

8,       77/2,  -57/4, -375/2;

-61/2,  211/2, 2079/4;

-136, -1657/2;

1385/2;

The first column is PIEULER(n) = 1, 1/2, -1/2, -1, 5/2, 8, -61/2, -136, 1385/2,... = c(n)/d(n). Abs c(n+1)=1,1,1,5,8,61,... =a(n) with offset=1.

For numerators of Balmer0(n) see A076109, A000265 and A061037(n-1) (End).

Other completely unrelated rational approximations of Pi are given by A063674/A063673 and other references there. - M. F. Hasler, Apr 03 2013

LINKS

Table of n, a(n) for n=1..23.

FORMULA

a(n)=denominator(r(n)) with the rationals r(n):=2*n*e(n-1)/e(n) where e(n):=A000111(n).

EXAMPLE

Rationals r(n): [2, 4, 3, 16/5, 25/8, 192/61, 427/136, 4352/1385, 12465/3968, 158720/50521,...].

MATHEMATICA

e[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n + 1)*(2^(n + 1) - 1)*BernoulliB[n + 1])/(n + 1)]]; r[n_] := 2*n*(e[n - 1]/e[n]); a[n_] := Denominator[r[n]]; Table[a[n], {n, 1, 23}] (* Jean-Fran├žois Alcover, Mar 26 2013 *)

CROSSREFS

Cf. triangle A008281 (main diagonal give zig-zag numbers A000111).

Sequence in context: A117474 A165716 A068478 * A250067 A213239 A123819

Adjacent sequences:  A132047 A132048 A132049 * A132051 A132052 A132053

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang Sep 14 2007

EXTENSIONS

Definition made more explicit, and initial terms a(1)=a(2)=1 added by M. F. Hasler, Apr 03 2013

STATUS

approved

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Last modified December 7 23:55 EST 2016. Contains 278902 sequences.