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A316599 The nonnegative integers sorted by increasing width of bounding box when printing their decimal representation using the Arial font, and in case of ties, sorted by increasing value. 2
1, 7, 0, 3, 9, 8, 2, 6, 5, 4, 11, 71, 31, 51, 91, 81, 61, 21, 41, 12, 14, 10, 13, 16, 17, 18, 19, 15, 72, 74, 70, 32, 52, 92, 73, 76, 77, 82, 78, 79, 34, 54, 62, 94, 30, 50, 84, 90, 80, 33, 36, 37, 53, 56, 57, 75, 93, 96, 97, 64, 83, 86, 87, 38, 39, 58, 59, 60, 98, 99, 88, 89, 22, 63, 66, 67, 35, 55, 68, 69 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The linked xkcd webcomic number 2016 (subtitled "OEIS keeps rejecting my submissions") shows as its second example "SUB[44]: Integers in increasing order of width when printed in Helvetica". To make a sequence involving a font well-defined, the definition can be interpreted using information provided in the corresponding Adobe Font Metrics File.  However since Helvetica is a proprietary font, the almost-identical font metrics of the Arial regular font are used here instead. Also, the vague description "width when printed" is interpreted as the size of the bounding box in the printing direction. This takes into account the actual left and right margin of the characters, which is smaller than the fixed width of 556/1000 size units for all digits, as well as the effect of the special kerning data for the character pair "11" (which is printed closer together by -74/1000 size units than other digit pairs).

The sequence is a permutation of the nonnegative integers.

LINKS

Hugo Pfoertner, Table of n, a(n) for n = 0..100000

Adobe, Adobe Font Metrics File Format Specification, Version 4.1, 1998

Randall Munroe, OEIS Submissions, xkcd Web Comic #2016, Jul 06 2018.

Hugo Pfoertner, Example of printed numbers, showing only samples of larger numbers.

EXAMPLE

All dimensions are given in units of 1/1000 of the font size. The relevant dimensions are the constant width of all numerical digits of w=556 units and the lower and upper x-limits llx and urx of the bounding box:

       llx urx

  zero  41 509

  one  108 373

  two   30 504

  three 41 511

  four  12 508

  five  41 517

  six   37 511

  seven 47 511

  eight 40 513

  nine  41 513

Additionally an addend of kpx("1","1") = -74 is applied for each occurrence of this pair.

.

a(0) = 1 because the character "1" has the smallest width of bounding box: 373 - 108 = 265.

a(1) = 7, because 7 is the digit with next smallest bounding box width: 511 - 47 = 464.

a(6) = 2, a(7) = 6 both have a bounding box width of 474 units, but 2 < 6.

The width of the bounding box for a(10) = 11 is calculated as 2*w - llx(1) - (w-urx(1)) + kpx("1","1") = 2*556 - 108  - (556-373) - 74 = 747.

The next term a(11) = 71 has bounding box width 2*w - llx(7) - (w-urx(1)) = 2*556 - 47 - (556-373) = 882.

a(29368) = 111111 is a first notable anomaly, because its bounding box width of 2675 lies between those of a(29367) = 49115, with bounding box width 2655, and a(29369) = 70002, with bounding box width 2681.

CROSSREFS

Cf. A001477, A316600.

Sequence in context: A126583 A220359 A021591 * A077185 A020830 A200122

Adjacent sequences:  A316596 A316597 A316598 * A316600 A316601 A316602

KEYWORD

nonn,base,dumb,look

AUTHOR

Hugo Pfoertner, Jul 08 2018

STATUS

approved

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Last modified April 25 16:59 EDT 2019. Contains 322461 sequences. (Running on oeis4.)