A234691 Decimal value of the bitmap of active segments in 7-segment display of the number n, variant 1: bits 0-6 refer to segments from top to bottom, left to right.

119, 36, 93, 109, 46, 107, 123, 39, 127, 111, 4727, 4644, 4701, 4717, 4654, 4715, 4731, 4647, 4735, 4719, 12023, 11940, 11997, 12013, 11950, 12011, 12027, 11943, 12031, 12015, 14071, 13988, 14045, 14061, 13998, 14059, 14075, 13991, 14079, 14063, 6007, 5924, 5981, 5997

Offset: 0

Author

M. F. Hasler, Dec 29 2013

Keywords

• nonn
• base

Comments

The bits 0-6 are assigned to the segments as follows: top (bit 0), upper left, upper right, middle, lower left, lower right, bottom (bit 6). Other conventions are common in engineering (as well for the segment-to-bit correspondence as for the glyphs), see sequence A234692 (the so-called ABCDEF coding), Wikipedia and the Example for a(7).

For n > 9, the decimal representation of n is encoded using a group of 7 bits for each digit, e.g., a(10) = a(1)*2^7 + a(0).

The sequence could have been extended differently to indices > 9, for example, by coding 7-segment representations of upper- and/or lowercase letters, as used for hexadecimal and base-64 representation.

The Hamming weight A000120 of the terms of this sequence yields the count of lit segments, A010371(n) = A000120(A234691(n)) = A000120(A234692(n)). For that sequence, 4 other variants are in the OEIS, depending on the number of segments used to represent digits 6, 7 and 9: A063720 (6', 7', 9'), A277116 (7', 9'), A074458 (9') and A006942 (7'), where x' means that the "sans serif" variant (one segment less than here) is used for digit x. - M. F. Hasler, Jun 17 2020

Examples

			The digits are assumed to be rendered as follows:
   _     _  _       _   _   _   _   _
  | | |  _| _| |_| |_  |_  | | |_| |_|
  |_| | |_  _|   |  _| |_|   | |_|  _|
For example, a(7) = 39 = 2^0 + 2^1 + 2^2 + 2^5, because the nonzero bits are
   _  : bit 0,
  | | : bits 1+2,
    | : bit 5,
while bits 3 (central '-'), 4 (lower left '|') and 6 (bottom '_') remain 0.
For the digit '7', the "sans serif" representation (without the upper left '|') is also very common; it would correspond to a(7) = 37. Similarly, digits 6 and 9 are sometimes represented without the top resp. bottom bar, see comments.
		

Links

• Wikipedia, Seven-segment display.

Crossrefs

Cf. A234692 for the "abcdefg" coding variant (clockwise from top and ending with bit 6 for the central "-").

Cf. A010371 (number of segments lit), and variants A063720, A277116, A074458: see comments.

Cf. A000120 (Hamming weight).

Programs

PARI

/* Bitmaps computed from 7-character strings. Extended beyond 9 with characters most frequently used to code hexadecimal digits A-F, chessboard coordinates a-h, and other letters. Alternate glyphs used more or less frequently: sans serif " 6 6666", "7 7  7 ", "9999 9 "; "a aaaaa", B same as 8, "eeeee e", g same as 9, H same as X, "i   i i" or "     i ", "J J JJJ" or "J J  JJ", S same as 5, "TT  T  ", " YYY Y ". Note: Z below is the same as 2! */
bitmaps = { apply( s2b = s->sum(i=1,#s=Vec(s), if(s[i]>" ",2^(i-1))), ["000 000", "  1  1 ", "2 222 2", "3 33 33", " 444 4 ", "55 5 55", "66 6666", "777  7 ", "8888888", "9999 99", "AAAAAA ", " b bbbb", "CC   CC", "  ddddd",
  "EE EEEE", "FF FF  ", "GG  GGG", " h hhh ", " I  I  ", "  J  JJ", "KK KKK ",
  " L  L L", "MMM MM ", "   nnn ", "   oooo", "PPPPP  ", "qqqq q ", "   rr  ", "   s ss", " t tt t", "    uuu", " VV VVV", " XXXXX ", " YYY YY", "Z ZZZ Z"])} \\ Extended to letters by M. F. Hasler, Jun 16 2020
apply( {A234691(n)=bitmaps[n%10+1]+if(n>9,self()(n\10)<<7)}, [0..99]) \\ Extended to n > 9 by M. F. Hasler, Jun 17 2020

Formula

a(n) = a(n mod 10) + a(floor(n/10))*2^7 for n > 9. - M. F. Hasler, Jun 17 2020

Extensions

a(6) corrected thanks to Kevin Ryde, M. F. Hasler, Jun 16 2020

Edited and extended to n > 9 by M. F. Hasler, Jun 17 2020