cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Dec 29 2013

Keywords

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.
		

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

A234692 Decimal value of the bitmap of active segments in 7-segment display of the number n, variant 2 ("abcdefg" scheme: bits represent segments in clockwise order).

Original entry on oeis.org

63, 6, 91, 79, 102, 109, 125, 39, 127, 111, 831, 774, 859, 847, 870, 877, 893, 807, 895, 879, 11711, 11654, 11739, 11727, 11750, 11757, 11773, 11687, 11775, 11759, 10175, 10118, 10203, 10191, 10214, 10221, 10237, 10151, 10239, 10223, 13119, 13062, 13147, 13135, 13158
Offset: 0

Views

Author

M. F. Hasler, Dec 29 2013

Keywords

Comments

The bits 0-6 are assigned to the segments according to the "abcdefg" scheme (top, upper right, lower right, bottom, lower left, upper left, center), cf. section "Displaying letters" of the Wikipedia page (3rd column of the table). Other conventions are common in engineering (as well for the segment-to-bit correspondence as for the glyphs), see sequence A234691, the Wikipedia page and the comment after the Example for a(7).
For n > 9, each of the digits of the base-10 representation is coded in a separate group of 7 bits, for example, a(10) = a(1)*2^7 + a(0) = 831.
Alternatively, for n >= 10 one could define a(n) to represent a 7-segment variant of the characters A-Z and/or a-z, as in hexadecimal or base-64 encoding. In that case, one could also use a few more bits for additional segments, e.g., four half-diagonals to represent K, M, N, R, V, X, Z correctly and S distinctly from 5. But as mentioned on the Wikipedia page, a possible ambiguity of representations of alphabetic characters is not always an obstacle to common use, since whole words are usually readable nonetheless.
The Hamming weight A000120 of the terms of this sequence yields the count of lit segments, A010371(n) = A000120(a(n)) = A000120(A234691(n)). For that sequence, 5 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

			a(7) = 39 = 2^0 + 2^1 + 2^2 + 2^5, because the digit 7 is represented as
" _ " : bit 0,
"| |" : bits 5+1,
"  |" : bit 2,
and no bit 3 (bottom "_") nor 4 (lower left "|") nor 6 (central "-").
Although other glyphs do exist as well for 6, 9, 0 and maybe other digits, "7" is probably the digit where an alternate representation (without the upper left "|") is as common as the one we chose here.
		

Crossrefs

Cf. A234691 for a variant where bits 0-6 represent, in this order, the segments: top, upper left, upper right, center, lower left, lower right, bottom.
Cf. A000120 (Hamming weight), A010371 and variants A063720, A277116, A074458 and A006942: see comments.

Programs

  • PARI
    bitmap=apply(s->sum(i=1,#s=Vec(s),if(s[i]>" ",2^(i-1))), ["000000"," 11", "22 22 2", "3333  3", " 44  44", "5 55 55", "6 66666", "777  7", "8888888", "9999 99", "AAA AAA", "  bbbbb", "C  CCC "," dddd d", "E  EEEE", "F   FFF"]) \\ Could be extended to more alphabetical glyphs, see A234691.
    apply( {A234692(n)=bitmap[n%10+1]+if(n>9,self()(n\10)<<7)}, [0..99]) \\ M. F. Hasler, Jun 17 2020

Formula

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

Extensions

Extended with hex digits (AbCdEF) to n=15 by M. F. Hasler, Dec 30 2013
a(10) and a(11) corrected thanks to Kevin Ryde, M. F. Hasler, Jun 16 2020
Definition changed for consistency with A010371, etc. by M. F. Hasler, Jun 17 2020

A216261 Smallest positive number using exactly n segments on a calculator display (when '6' and '7' are represented using 6 resp. 3 segments).

Original entry on oeis.org

1, 7, 4, 2, 0, 8, 10, 18, 22, 20, 28, 68, 88, 108, 188, 200, 208, 288, 688, 888, 1088, 1888, 2008, 2088, 2888, 6888, 8888, 10888, 18888, 20088, 20888, 28888, 68888, 88888, 108888, 188888, 200888, 208888, 288888, 688888, 888888, 1088888, 1888888, 2008888, 2088888, 2888888
Offset: 2

Views

Author

Reinhard Zumkeller, Mar 15 2013

Keywords

Comments

Essentially the same as A038619 and A143617. One could argue that a(3) should rather be -1 (prior to adding "positive" in the definition), which does use 3 segments on typical 7-segment displays, and is smaller than 7. Also, most pocket calculators and the Unicode standard (cf. links) use 4 rather than 3 segments to represent a '7' (as in A074458 and A010371, rather than A063720, A277116 or A006942), in which case a(3) is undefined if negative numbers are not allowed. No digit '9' will ever occur here, whether it would be represented with 6 or only 5 segments. However, digit '6' does occur, as the second smallest digit using 6 segments as does '0', which cannot occur as leading digit. If '6' is represented with 5 segments, any prefix 68 would be replaced with 80. - M. F. Hasler and Kevin Ryde, Jun 17 2020

Crossrefs

Cf. A038619 and A143617 (identical up to initial terms).

Programs

  • Haskell
    import Data.Maybe (fromJust)
    import Data.List (elemIndex)
    a216261 = fromJust . (`elemIndex` a006942_list)
    -- Reinhard Zumkeller, Mar 15 2013
    
  • Mathematica
    Drop[#, 2] &@ CoefficientList[Series[(x^2 + 6 x^3 - 3 x^4 - 2 x^5 - 2 x^6 + 8 x^7 + 2 x^8 - 2 x^9 - 56 x^10 + 28 x^11 + 28 x^12 + 60 x^13 - 60 x^14 - 28 x^17 + 28 x^18)/((1 - x) (1 - 10 x^7)), {x, 0, 50}], x] (* Michael De Vlieger, Jan 29 2016 *)
  • PARI
    Vec((x^2 +6*x^3 -3*x^4 -2*x^5 -2*x^6 +8*x^7 +2*x^8 -2*x^9 -56*x^10 +28*x^11 +28*x^12 +60*x^13 -60*x^14 -28*x^17 +28*x^18)/((1-x)*(1-10*x^7)) + O(x^50)) \\ Michel Marcus, Jan 29 2016

Formula

A006942(a(n)) = n and A006942(m) <> n for m < a(n).
a(n+7) = 10*a(n) + 8 for n > 10. This can be deduced from a(n) = min{10*a(n-A006942(r))+r, r=0..9} via strong induction. - David Radcliffe, Jan 29 2016
G.f.: (x^2 +6*x^3 -3*x^4 -2*x^5 -2*x^6 +8*x^7 +2*x^8 -2*x^9 -56*x^10 +28*x^11 +28*x^12 +60*x^13 -60*x^14 -28*x^17 +28*x^18)/((1-x)*(1-10*x^7)). - David Radcliffe, Jan 29 2016

Extensions

Name and cross-references edited by M. F. Hasler, Jun 17 2020

A357971 a(n) is the number of segments used to represent the time of n minutes past midnight in the format hh:mm on a 7-segment calculator display; version where the digits '6', '7' and '9' use 6, 4 and 6 segments, respectively.

Original entry on oeis.org

24, 20, 23, 23, 22, 23, 24, 22, 25, 24, 20, 16, 19, 19, 18, 19, 20, 18, 21, 20, 23, 19, 22, 22, 21, 22, 23, 21, 24, 23, 23, 19, 22, 22, 21, 22, 23, 21, 24, 23, 22, 18, 21, 21, 20, 21, 22, 20, 23, 22, 23, 19, 22, 22, 21, 22, 23, 21, 24, 23, 20, 16, 19, 19, 18, 19, 20
Offset: 0

Views

Author

Stefano Spezia, Oct 22 2022

Keywords

Comments

The sequence is periodic of period 1440 since the term a(1440) = a(0) = 24 corresponds to the time 00:00.

Crossrefs

Programs

  • Mathematica
    a055643[n_]:=FromDigits@ Apply[Join, PadLeft[#, 2] & /@ IntegerDigits@ IntegerDigits[n, 60]]; a010371[n_] := Plus @@ (IntegerDigits@ n /. {0 -> 6, 1 -> 2, 2 -> 5, 3 -> 5, 7 -> 4, 8 -> 7, 9 -> 6}); a[n_]:=a010371[a055643[n]]+6(4-Ceiling[Log10[a055643[n]+1]]); Join[{24},Array[a,66]]

Formula

a(n) = A010371(A055643(n)) + 6*(4 - ceiling(log10(A055643(n) + 1))) for n > 0.
a(n) <= 26.

A038619 Smallest positive number that needs more lines when shown on a 7-segment display (digital clock) than any previous term.

Original entry on oeis.org

1, 2, 6, 8, 10, 18, 20, 28, 68, 88, 108, 188, 200, 208, 288, 688, 888, 1088, 1888, 2008, 2088, 2888, 6888, 8888, 10888, 18888, 20088, 20888, 28888, 68888, 88888, 108888, 188888, 200888, 208888, 288888, 688888, 888888, 1088888, 1888888, 2008888, 2088888, 2888888, 6888888, 8888888
Offset: 1

Views

Author

Keywords

Comments

For n > 1, a(n) uses n + 3 segments to be displayed, when a digit '6' uses 6 segments (as in A234691, A234692 and A277116, A074458, A006942, A010371, but not in A063720). Sequence A143617 is the same but starts with 0, 8, ... and A216261 has additional terms 7 & 4 before 2 and 22 before 20. - M. F. Hasler, Jun 23 2020

Examples

			Digits 0, 1, 2, ..., 9 use 6, 2, 5, 5, 4, 5, 6, 3, 7, 6 lines / segments.
		

Crossrefs

Programs

  • Mathematica
    Block[{f, s}, MapIndexed[(f[#2[[1]] - 1] = #1) &, {6, 2, 5, 5, 4, 5, 6, 3, 7, 6}]; s = Array[Total[f /@ IntegerDigits[#]] &, 10^7]; Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* or *)
    Nest[Append[#1, If[#2 > 13, 10 #1[[-7]] + 8, 10 #1[[-6]] + Boole[#2 != 13] 8]] & @@ {#, Length@ # + 1} &, {1, 2, 6, 8, 10, 18, 20}, 36] (* Michael De Vlieger, Jun 23 2020 *)
    LinearRecurrence[{1,0,0,0,0,0,10,-10},{1,2,6,8,10,18,20,28,68,88,108,188,200,208},50] (* Harvey P. Dale, Aug 11 2025 *)
  • PARI
    apply( {A038619(n)=if(n>7, self()(n-6-(n>13))*10+(n!=13)*8, [1,2,6, 8,10,18,20][n])}, [1..33]) \\ M. F. Hasler, Jun 23 2020

Formula

For n >= 3, the terms with n digits are given by: 108*A + B, 188*A + B, 200*A + B, 208*A + B, 288*A + B, 688*A + B, 888*A + B where A = 10^(n-3), B = 8*(A - 1)/9.
From M. F. Hasler, Jun 23 2020: (Start)
a(n) = 10*a(n-7) + 8 for n > 13 (and with a(n-6) for 7 < n < 13).
G.f.: (1 + x + 4*x^2 + 2*x^3 + 2*x^4 + 8*x^5 + 2*x^6 - 2*x^7 + 30*x^8 - 20*x^9 + 60*x^11 - 68*x^12 - 12*x^13)/((1 - x)*(1 - x^10)).
(End)
a(n) = a(n-1) + 10*a(n-7) - 10*a(n-8), for n >= 15. - Wesley Ivan Hurt, Jun 29 2020

Extensions

Edited and offset corrected to 1 by M. F. Hasler, Jun 23 2020
More terms from Michael De Vlieger, Jun 23 2020
More terms from M. F. Hasler, Jun 23 2020

A074459 Number of segments which change from display of a number n to the next number n+1 on a 7-segment display: version where '6', '7', '9' use 6, 4, resp. 5 segments.

Original entry on oeis.org

4, 5, 2, 3, 3, 1, 4, 3, 2, 5, 4, 5, 2, 3, 3, 1, 4, 3, 2, 8, 4, 5, 2, 3, 3, 1, 4, 3, 2, 5, 4, 5, 2, 3, 3, 1, 4, 3, 2, 6, 4, 5, 2, 3, 3, 1, 4, 3, 2, 6, 4, 5, 2, 3, 3, 1, 4, 3, 2, 4, 4, 5, 2, 3, 3, 1, 4, 3, 2, 7, 4, 5, 2, 3, 3, 1, 4, 3, 2, 6, 4, 5, 2, 3, 3, 1, 4, 3, 2, 5
Offset: 0

Views

Author

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 22 2002

Keywords

Comments

The glyph variants used here are the same as in A074458 (other variants are described in A006942, A010371, A063720 and A277116). - M. F. Hasler, Jun 17 2020

Examples

			Consider the representations of digits '0', '1' and '2' given below.
To change from 0 to 1, we need to delete 4 segments, from 1 to 2, we need to delete 1 segment and add 4 segments, so 5 segments in total are needed to be changed.
From _M. F. Hasler_, Jun 23 2020: (Start)
We consider the following 7-segment representations of the digits 0 - 9:
   _       _   _       _   _   _   _   _
  | |   |  _|  _| |_| |_  |_  | | |_| |_|
  |_|   | |_   _|   |  _| |_|   | |_|   | .
To switch from displaying number 9 to displaying number 10, one has to activate 2 segments for the additional digit '1', and change 3 segments from the representation of '9' to get that of '0', whence a(9) = 2 + 3 = 5.
To switch from 19 to 20 one has a(19) = a(1) + 3 = 8. (End)
		

Crossrefs

Cf. A074458.

Programs

  • PARI
    apply( {A074459(n)=if(n%10<9,digits(452331432)[n%10+1],n>9,3+self()(n\10),5)}, [0..99]) \\ M. F. Hasler, Jun 23 2020

Formula

For n % 10 < 9, a(n) = a(n % 9), where % is the modulo (remainder) operator. - M. F. Hasler, Jun 23 2020

Extensions

Edited and terms for n > 8 added by M. F. Hasler, Jun 23 2020

A350131 Number of segments lit to display the number n on a 9-segment display used by the Russian postal service.

Original entry on oeis.org

6, 3, 4, 4, 4, 5, 5, 3, 7, 5, 9, 6, 7, 7, 7, 8, 8, 6, 10, 8, 10, 7, 8, 8, 8, 9, 9, 7, 11, 9, 10, 7, 8, 8, 8, 9, 9, 7, 11, 9, 10, 7, 8, 8, 8, 9, 9, 7, 11, 9, 11, 8, 9, 9, 9, 10, 10, 8, 12, 10, 11, 8, 9, 9, 9, 10, 10, 8, 12, 10, 9, 6, 7, 7, 7, 8, 8, 6, 10, 8
Offset: 0

Views

Author

Bartlomiej Malarz, Dec 16 2021

Keywords

Examples

			The Russian postal service uses a special template for entering postal codes, which makes automatic sorting of their parcels faster. It has nine segments, and digits looks like this (please check the first link in the Links section for a more readable, graphical example):
._       _   _         _        _   _    _
| |  /|   |  /_  |_|  |_   /_   /  |_|  |_|
|_|   |  /_   /    |   _|  |_|  |  |_|   /
The template for a single digit contains nine segments: four vertical, three horizontal and two diagonal:
Vertical      Horizontal      Diagonal
                   _
   | |             _              /
   | |             _              /
Using a combination of vertical, horizontal and/or diagonal segments, it is possible to create digits:
.
                  number of segments
         =====================================
  digit  total  vertical  horizontal  diagonal
  -----  -----  --------  ----------  --------
    0      6        4          2          0
    1      3        2          0          1
    2      4        1          2          1
    3      4        0          2          2
    4      4        3          1          0
    5      5        2          3          0
    6      5        2          2          1
    7      3        1          1          1
    8      7        4          3          0
    9      5        2          2          1
		

Crossrefs

Cf. A350177 (histogram).

Programs

  • Mathematica
    Table[Total[IntegerDigits[n]/.{0->6, 1->3, 2->4, 3->4, 6->5, 7->3, 8->7, 9->5}], {n, 0, 79}] (* Stefano Spezia, Dec 17 2021 *)
  • PHP
     $segments[$n], str_split($i)));
        echo $usedSegments . "\n";
    } // Simplified by Bartlomiej Malarz, Oct 30 2024
    
  • Python
    segments = [6, 3, 4, 4, 4, 5, 5, 3, 7, 5]
    def a(n): return sum(segments[int(d)] for d in str(n))
    print([a(n) for n in range(80)]) # Michael S. Branicky, Dec 22 2021

A386910 Number of iterations of seven segments count x -> A063720(x) to go from n to a fixed point.

Original entry on oeis.org

2, 2, 1, 1, 0, 0, 1, 2, 3, 1, 4, 1, 3, 3, 2, 3, 3, 1, 2, 3, 2, 3, 5, 5, 2, 5, 5, 4, 4, 5, 2, 3, 5, 5, 2, 5, 5, 4, 4, 5, 5, 2, 2, 2, 4, 2, 2, 3, 2, 2, 2, 3, 5, 5, 2, 5, 5, 4, 4, 5, 2, 3, 5, 5, 2, 5, 5, 4, 4, 5, 2, 1, 4, 4, 3, 4, 4, 2, 5, 4, 4, 2, 4, 4, 2, 4, 4
Offset: 0

Views

Author

Marco RipĂ , Aug 07 2025

Keywords

Comments

A063720 is a strictly decreasing function A063720(x) < x whenever x >= 10 and all single digit x reach a fixed point A063720(x) = x with x in {4, 5}.
This sequence is unbounded and the first occurrence of a(n) = k is at n = A338255(k + 1) for any k >= 3.

Examples

			For n = 12, the a(12) = 3 steps are 12 -> 7 -> 3 -> 5 segments, and 5 is a fixed point A063720(5) = 5.
		

Crossrefs

Cf. A006942, A010371, A074458, A277116 (segments variation).

A123587 Number of bars that must change on a standard digital clock when the minutes change from n-1 to n.

Original entry on oeis.org

5, 4, 5, 2, 3, 3, 1, 5, 4, 1, 6, 4, 5, 2, 3, 3, 1, 5, 4, 1, 7, 4, 5, 2, 3, 3, 1, 5, 4, 1, 4, 4, 5, 2, 3, 3, 1, 5, 4, 1, 5, 4, 5, 2, 3, 3, 1, 5, 4, 1, 5, 4, 5, 2, 3, 3, 1, 5, 4, 1, 5, 4, 5, 2, 3, 3, 1, 5, 4, 1, 6, 4, 5, 2, 3, 3, 1, 5, 4, 1, 7, 4, 5, 2, 3, 3, 1, 5, 4, 1, 4, 4, 5, 2, 3, 3, 1, 5, 4, 1, 5, 4, 5, 2, 3, 3, 1, 5, 4, 1, 5, 4, 5, 2, 3, 3, 1, 5, 4, 1
Offset: 0

Views

Author

Danny Rorabaugh, Nov 13 2006

Keywords

Comments

Cyclic with period 60.

Examples

			a(20)=7 because when the clock changes from 19 to 20 minutes, 1 bar turns off and 4 bars turn on in the tens digit for the "0" to become a "1" and 1 bar turns off and 1 turns on in the units digit for the "9" to become a "0". 1 + 4 + 1 + 1 = 7.
		

Crossrefs

A165244 The numbers commonly displayed with 7 segments in electric clocks, in ascending order of number of segments lit.

Original entry on oeis.org

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

Views

Author

Kurt Eisemann (keiseman(AT)mail.sdsu.edu), Sep 09 2009

Keywords

Crossrefs

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