A308321 Decimal expansion of 2^(-9/4); exact width of the A4 paper size measured in meters according to the ISO 216 standard.

2, 1, 0, 2, 2, 4, 1, 0, 3, 8, 1, 3, 4, 2, 8, 6, 3, 5, 7, 5, 7, 7, 8, 1, 3, 6, 9, 0, 5, 8, 3, 0, 3, 7, 2, 3, 7, 6, 0, 0, 0, 8, 5, 6, 5, 5, 8, 9, 1, 9, 6, 1, 2, 7, 7, 0, 3, 3, 0, 6, 5, 2, 1, 4, 9, 3, 7, 3, 1, 1, 8, 8, 7, 3, 8, 4, 7, 5, 5, 5, 9, 9, 5, 3, 5, 8, 1, 0

Offset

0,1

Comments

Also exact length of the A5 paper size measured in meters.

According to the ISO 216 standard, the A0 paper size is defined to have an area of 1 square meter where the ratio of the length to the width is sqrt(2), so the length is 2^(1/4) m and the width is 2^(-1/4) m. For each n >= 0, the length of the size A(n+1) is equal to the width of the size A(n) and the width of the size A(n+1) is equal to half of the length of the size A(n), so the area of the size A(n+1) is half of that of A(n). Equivalently, the length of the A(n) size is 2^(-n/2 + 1/4) m and the width is 2^(-n/2 - 1/4) m. For the A4 size, the exact length and width are 2^(-7/4) m = 297.301... mm (A308320) and 2^(-9/4) m = 210.224... mm, and the actual length and width are 297 mm and 210 mm.

Links

Table of n, a(n) for n=0..87.

Prepressure, A4.

Wikipedia, ISO 216.

Example

The exact lengths and widths (rounded to the nearest 1/10 mm) and areas of the A-series are as follows:
.
  size |       exact length      |       exact width      | exact area (mm^2)
   A0  | 2^(  1/4) m = 1189.2 mm | 2^(- 1/4) m = 840.9 mm |  1000000
   A1  | 2^(- 1/4) m =  840.9 mm | 2^(- 3/4) m = 594.6 mm |   500000
   A2  | 2^(- 3/4) m =  594.6 mm | 2^(- 5/4) m = 420.4 mm |   250000
   A3  | 2^(- 5/4) m =  420.4 mm | 2^(- 7/4) m = 297.3 mm |   125000
   A4  | 2^(- 7/4) m =  297.3 mm | 2^(- 9/4) m = 210.2 mm |    62500
   A5  | 2^(- 9/4) m =  210.2 mm | 2^(-11/4) m = 148.7 mm |    31250
   A6  | 2^(-11/4) m =  148.7 mm | 2^(-13/4) m = 105.1 mm |    15625
   A7  | 2^(-13/4) m =  105.1 mm | 2^(-15/4) m =  74.3 mm |     7812.5
   A8  | 2^(-15/4) m =   74.3 mm | 2^(-17/4) m =  52.6 mm |     3906.25
   A9  | 2^(-17/4) m =   52.6 mm | 2^(-19/4) m =  37.2 mm |     1953.125
   A10 | 2^(-19/4) m =   37.2 mm | 2^(-21/4) m =  26.3 mm |      976.5625
.
And the actual lengths, widths and areas (note that the actual areas are always smaller than the exact areas) are as follows:
.
  size | actual length (mm) | actual width (mm) | actual area (mm^2)
   A0  |        1189        |        841        |  999949 (99.9949%)
   A1  |         841        |        594        |  499554 (99.9108%)
   A2  |         594        |        420        |  249480 (99.7920%)
   A3  |         420        |        297        |  124740 (99.7920%)
   A4  |         297        |        210        |   62370 (99.7920%)
   A5  |         210        |        148        |   31080 (99.4560%)
   A6  |         148        |        105        |   15540 (99.4560%)
   A7  |         105        |         74        |    7770 (99.4560%)
   A8  |          74        |         52        |    3848 (98.5088%)
   A9  |          52        |         37        |    1924 (98.5088%)
   A10 |          37        |         26        |     962 (98.5088%)

Mathematica

RealDigits[2^(-9/4), 10, 88][[1]] (* James C. McMahon, Feb 26 2024 *)

Prog

(PARI) default(realprecision, 100); 2^(-9/4)

Crossrefs

Cf. A010767 (2^(1/4)), A228497 (2^(-1/4)), A308320 (2^(-7/4)).

Sequence in context: A128095 A316658 A189962 * A097854 A161515 A145580

Adjacent sequences: A308318 A308319 A308320 * A308322 A308323 A308324

Keyword

nonn,cons

Author

Jianing Song, May 20 2019

Extensions

Edited by Jon E. Schoenfield, Feb 25 2024

Status

approved