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Search: a143617 -id:a143617
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A010371
Number of segments used to represent n on a 7-segment calculator display; version where '6', '7' and '9' use 6, 4 and 6 segments, respectively.
+10
21
6, 2, 5, 5, 4, 5, 6, 4, 7, 6, 8, 4, 7, 7, 6, 7, 8, 6, 9, 8, 11, 7, 10, 10, 9, 10, 11, 9, 12, 11, 11, 7, 10, 10, 9, 10, 11, 9, 12, 11, 10, 6, 9, 9, 8, 9, 10, 8, 11, 10, 11, 7, 10, 10, 9, 10, 11, 9, 12, 11, 12, 8, 11, 11, 10, 11, 12, 10, 13, 12, 10, 6, 9, 9, 8, 9, 10, 8, 11, 10, 13, 9, 12, 12
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Except for 1 and 3 every positive integer occurs; A143616 and A143617 give record values and where they occur. - Reinhard Zumkeller, Aug 27 2008
The difference between this sequence and A006942 lies in the representation chosen for the digit 7,
_ _
| | |
| (here), vs. | in A006942.
If we mark with ' the "sans serif" graphical representation which uses one segment less and with * the "heavier" version, we have the following variants:
A063720 (6', 7', 9'), A277116 (6*, 7', 9'), A074458 (6*, 7*, 9'),
_____________________ A006942 (6*, 7', 9*), A010371 (6*, 7*, 9*) = this.
Sequences A234691, A234692 and variants make precise which segments are lit in each digit. They are related through the Hamming weight A000120, see formula. The sequence could be extended to negative arguments with a(-n) = a(n)+1. - M. F. Hasler, Jun 17 2020
LINKS
R. Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to calculator display
FORMULA
For n > 9, a(n) = a(floor(n/10)) + a(n mod 10). - Reinhard Zumkeller, Aug 27 2008
a(n) = A000120(A234691(n)) = A000120(A234692(n))
= A006942(n) + A102679(n) - A102681(n) (add number of digits 7)
= A074458(n) + A102683(n) (add number of digits 9). - M. F. Hasler, Jun 17 2020
EXAMPLE
LCD Display (cf. Casio scientific calculator fx-3600P):
_ _ _ _ _ _ _ _
| | | _| _| |_| |_ |_ | | |_| |_|
|_| | |_ _| | _| |_| | |_| _|
MATHEMATICA
MapIndexed[(f[#2[[1]]-1] = #1)&, {6, 2, 5, 5, 4, 5, 6, 4, 7, 6}]; a[n_] := Total[f /@ IntegerDigits[n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 08 2017 *)
PROG
(Haskell)
a010371 n = a010371_list !! n
a010371_list = [6, 2, 5, 5, 4, 5, 6, 4, 7, 6] ++ f 10 where
f x = (a010371 x' + a010371 d) : f (x + 1)
where (x', d) = divMod x 10
-- Reinhard Zumkeller, Mar 15 2013
(PARI) apply( {A010371(n)=digits(6255456476)[n%10+1]+if(n>9, self()(n\10))}, [0..99]) \\ M. F. Hasler, Jun 17 2020
CROSSREFS
Segment variations: A006942, A063720, A074458, A277116.
Cf. A000120, A234691, A234692.
KEYWORD
nonn,base,easy,nice,look
AUTHOR
Olivier.Gagneux(AT)roche.com
EXTENSIONS
Corrected and extended by Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 27 1999
Edited name, comments, cross-references. - M. F. Hasler, Jun 17 2020
STATUS
approved
A216261
Smallest positive number using exactly n segments on a calculator display (when '6' and '7' are represented using 6 resp. 3 segments).
+10
9
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
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
LINKS
Table of n, a(n) for n=2..47.
Unicode, Symbols for Legacy Computing, The Unicode Standard, Version 13.0, 2020.
Index entries for sequences related to calculator display
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,10,-10).
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
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 *)
PROG
(Haskell)
import Data.Maybe (fromJust)
import Data.List (elemIndex)
a216261 = fromJust . (`elemIndex` a006942_list)
-- Reinhard Zumkeller, Mar 15 2013
(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
CROSSREFS
Cf. A006942, A010371, A063720, A074458, A277116; A234691, A234692.
Cf. A038619 and A143617 (identical up to initial terms).
KEYWORD
nonn,base
AUTHOR
Reinhard Zumkeller, Mar 15 2013
EXTENSIONS
Name and cross-references edited by M. F. Hasler, Jun 17 2020
STATUS
approved
A038619
Smallest positive number that needs more lines when shown on a 7-segment display (digital clock) than any previous term.
+10
4
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
LINKS
Table of n, a(n) for n=1..45.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,10,-10).
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
EXAMPLE
Digits 0, 1, 2, ..., 9 use 6, 2, 5, 5, 4, 5, 6, 3, 7, 6 lines / segments.
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 *)
PROG
(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
CROSSREFS
Cf. A143617, A216261; A234691, A234692; A277116, A074458, A006942, A010371, A063720.
KEYWORD
nonn,base,easy,nice
AUTHOR
Jan Kristian Haugland
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
STATUS
approved
A143616
Record values in A010371.
+10
2
6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
a(n)=A010371(A143617(n)) and A010371(m)<a(n) for m < A143617(n);
{1,2,3,4,5,10} = complement(range of this sequence).
LINKS
Table of n, a(n) for n=1..71.
Index entries for linear recurrences with constant coefficients, signature (2,-1).
FORMULA
a(n) = n + 6 for n > 4. [Charles R Greathouse IV, Oct 26 2011]
PROG
(PARI) a(n)=n+5+(n>4) \\ Charles R Greathouse IV, Oct 26 2011
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Aug 27 2008
STATUS
approved
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