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A316599
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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.
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2
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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
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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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.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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