A331529 -id:A331529 - cp's OEIS frontend

A006942 Number of segments used to represent n on calculator display, variant 5: digits '6', '7' and '9' use 6, 3 and 6 segments, respectively.

6, 2, 5, 5, 4, 5, 6, 3, 7, 6, 8, 4, 7, 7, 6, 7, 8, 5, 9, 8, 11, 7, 10, 10, 9, 10, 11, 8, 12, 11, 11, 7, 10, 10, 9, 10, 11, 8, 12, 11, 10, 6, 9, 9, 8, 9, 10, 7, 11, 10, 11, 7, 10, 10, 9, 10, 11, 8, 12, 11, 12, 8, 11, 11, 10, 11, 12, 9, 13, 12, 9, 5, 8, 8, 7, 8, 9

Offset: 0

N. J. A. Sloane

a(A216261(n)) = n and a(m) <> n for m < A216261(n). - Reinhard Zumkeller, Mar 15 2013
If we mark with * resp. ' the graphical representations which use more resp. less segments, we have the following variants:
  A063720 (6', 7', 9'), A277116 (6*, 7', 9'), A074458 (6*, 7*, 9'),
  _____________ this: A006942 (6*, 7', 9*), A010371 (6*, 7*, 9*).
  Sequences A234691, A234692 and variants make precise which segments are lit in each digit. These are related through the Hamming weight function A000120, e.g., A010371(n) = A000120(A234691(n)) = A000120(A234692(n)). - M. F. Hasler, Jun 17 2020

			As depicted below, zero uses 6 segments, so a(0)=6.
   _     _  _       _   _   _   _   _
  | | |  _| _| |_| |_  |_    | |_| |_|
  |_| | |_  _|   |  _| |_|   | |_|  _|
.
[Edited by _Jon E. Schoenfield_, Jul 30 2017]

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 65.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
Netnews group rec.puzzles, Frequently Asked Questions (FAQ) file, Instructions problem series.31, and Series Section solution series.31.
Index entries for sequences related to calculator display

Cf. A216261 (least inverse), A165244 (sorted digits), A302552 (primes), A328330 (iterations), A331529 (histogram).
Variants are A010371, A063720, A074458, A277116, see comments.
See also A234691, A234692, A000120.

a006942 n = a006942_list !! n
a006942_list = [6,2,5,5,4,5,6,3,7,6] ++ f 10 where
   f x = (a006942 x' + a006942 d) : f (x + 1)
         where (x',d) = divMod x 10
-- Reinhard Zumkeller, Mar 15 2013

A006942 := proc(n) local d,dig,j,s: if(n=0)then return 6:fi: dig:=[6,2,5,5,4,5,6,3,7,6]: d:=convert(n,base,10): s:=0: for j from 1 to nops(d) do s:=s+dig[d[j]+1]: od: return s: end: seq(A006942(n),n=0..100); # Nathaniel Johnston, May 08 2011

MapIndexed[ (f[First[#2] - 1] = #1)& , {6, 2, 5, 5, 4, 5, 6, 3, 7, 6}]; a[n_] := Plus @@ f /@ IntegerDigits[n]; Table[a[n], {n, 0, 76}] (* Jean-François Alcover, Sep 25 2012 *)
a[n_] := Plus @@ (IntegerDigits@ n /. {0 -> 6, 1 -> 2, 2 -> 5, 3 -> 5, 7 -> 3, 8 -> 7, 9 -> 6}); Array[a, 77, 0] (* Robert G. Wilson v, Jun 20 2018 *)

a(n)=if(n==0, return(6)); my(d=digits(n),v=vector(10)); for(i=1,#d, v[d[i]+1]++); v*[6, 2, 5, 5, 4, 5, 6, 3, 7, 6]~ \\ Charles R Greathouse IV, Feb 05 2018

def a(n): return sum([6, 2, 5, 5, 4, 5, 6, 3, 7, 6][int(d)] for d in str(n))
print([a(n) for n in range(77)]) # Michael S. Branicky, Jun 02 2021

a(n) = a(floor(n/10)) + a(n mod 10) for n > 9. - Reinhard Zumkeller, Mar 15 2013

a(n) = A010371(n) - A102679(n) + A102681(n) (subtract the number of digits 7 in n) = A277116(n) + A102683(n) (add number of digits 9 in n); and in particular, A063720(n) <= A277116(n) <= a(n) = A010371(n). - M. F. Hasler, Jun 17 2020

More terms from Matthew Conroy, Sep 13 2001

A331530 a(n) is the number of nonnegative integers that can be represented in a 7-segment display by using only n segments (version A010371).

Original entry on oeis.org

0, 0, 1, 0, 3, 3, 8, 7, 16, 23, 48, 70, 125, 192, 345, 561, 972, 1578, 2683, 4436, 7537, 12536, 21114, 35163, 59123, 98837, 166006, 277650, 465619, 779296, 1306674, 2188248, 3667717, 6142653, 10293460, 17242678, 28892956, 48402553, 81099234, 135863965, 227636213

Offset: 0

Stefano Spezia, Jan 19 2020

The nonnegative integers are displayed as in A010371, where a 7 is depicted by 4 segments.

Given the set S = {2, 4, 5, 6, 7}, the function f defined in S as f(4) = 2, f(5) = f(6) = 3 and f(2) = f(7) = 1, a(n) is equal to the difference between the number b(n) of S-restricted f-weighted integer compositions of n with that of n-6, i.e., b(n-6). The latter one provides the number of all those excluded cases where a nonnegative integer is displayed with leading zeros. b(n) is calculated as the sum of polynomial coefficients or extended binomial coefficients (see Equation 3 in Eger) where the index of summation is positive and it covers the numbers of possible digits that can be displayed by n segments (see first formula).

			a(6) = 8 since 0, 6, 9, 14, 17, 41, 71, 111 are displayed by 6 segments.
   __       __      __
  |  |     |__     |__|     |  |__|
  |__|     |__|     __|     |     |
  (0)      (6)      (9)       (14)
     __                   __
  | |  |     |__|  |     |  |  |    |  |  |
  |    |        |  |        |  |    |  |  |
   (17)        (41)        (71)      (111)

Colin Barker, Table of n, a(n) for n = 0..1000
Steffen Eger, Restricted Weighted Integer Compositions and Extended Binomial Coefficients, Journal of Integer Sequences, Vol. 16, Article 13.1.3, (2013).
Index entries for linear recurrences with constant coefficients, signature (0,1,0,2,3,3,1).
Index entries for sequences related to calculator display
Index entries for sequences related to compositions

Cf. A002426, A004526, A010371, A331529, A343314, A343315.

P[x_]:=x^2+2x^4+3x^5+3x^6+x^7; b[n_]:=Coefficient[Sum[P[x]^k,{k,Max[1,Ceiling[n/7]],Floor[n/2]}],x,n];a[n_]:=b[n]-b[n-6]; Array[a,41,0]

concat([0,0], Vec(x^2*(1 - x)*(1 + x)*(1 - x + x^2)*(1 + x + x^2)*(1 + 2*x^2 + 3*x^3 + 3*x^4 + x^5) / (1 - x^2 - 2*x^4 - 3*x^5 - 3*x^6 - x^7) + O(x^41))) \\ Colin Barker, Jan 20 2020

a(n) = b(n) - b(n-6), where b(n) = [x^n] Sum_{k=max(1,ceiling(n/7))..floor(n/2)} P(x)^k with P(x) = x^2 + 2*x^4 + 3*x^5 + 3*x^6 + x^7.
From Colin Barker, Jan 20 2020: (Start)
G.f.: x^2*(1 - x)*(1 + x)*(1 - x + x^2)*(1 + x + x^2)*(1 + 2*x^2 + 3*x^3 + 3*x^4 + x^5) / (1 - x^2 - 2*x^4 - 3*x^5 - 3*x^6 - x^7).
a(n) = a(n-2) + 2*a(n-4) + 3*a(n-5) + 3*a(n-6) + a(n-7) for n>13.
(End)

A343314 a(n) is the number of nonnegative integers that can be represented in a 7-segment display by using only n segments (version A063720).

Original entry on oeis.org

0, 0, 1, 1, 2, 7, 5, 16, 19, 39, 77, 103, 226, 334, 636, 1106, 1827, 3386, 5568, 10059, 17281, 29890, 52771, 90283, 159191, 274976, 479035, 835476, 1447278, 2528496, 4386143, 7640592, 13293308, 23106132, 40245277, 69946521, 121762316, 211791205, 368418674, 641125867

Offset: 0

Stefano Spezia, Apr 11 2021

The nonnegative integers are displayed as in A063720.

Given the set S = {2, 3, 4, 5, 6, 7}, the function f defined in S as f(5) = 5 and f(s) = 1 elsewhere, a(n) is equal to the difference between the number b(n) of S-restricted f-weighted integer compositions of n with that of n-6, i.e., b(n-6). The latter one provides the number of all those excluded cases where a nonnegative integer is displayed with leading zeros. b(n) is calculated as the sum of polynomial coefficients or extended binomial coefficients (see Equation 3 in Eger) where the index of summation is positive and it covers the numbers of possible digits that can be displayed by n segments (see third formula).

			a(6) = 5 since 0, 14, 41, 77 and 111 are displayed by 6 segments.
    __                                   __   __
   |  |      | |__|      |__|    |         |    |      |    |    |
   |__|      |    |         |    |         |    |      |    |    |
    (0)       (14)          (41)            (77)          (111)

Steffen Eger, Restricted Weighted Integer Compositions and Extended Binomial Coefficients, Journal of Integer Sequences, Vol. 16, Article 13.1.3, (2013).
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,5,1,1).
Index entries for sequences related to calculator display
Index entries for sequences related to compositions

Cf. A002426, A004526, A063720, A216261, A331529, A331530, A343315.

P[x_]:=x^2+x^3+x^4+5x^5+x^6+x^7; b[n_]:=Coefficient[Sum[P[x]^k, {k, Max[1, Ceiling[n/7]], Floor[n/2]}], x, n]; a[n_]:=b[n]-b[n-6]; Array[a, 40, 0]

G.f.: x^2*(1 - x)*(1 + x)*(1 - x + x^2)*(1 + x + x^2)*(1 + x + x^2 + 5*x^3 + x^4 + x^5)/(1 - x^2 - x^3 - x^4 - 5*x^5 - x^6 - x^7).
a(n) = a(n-2) + a(n-3) + a(n-4) + 5*a(n-5) + a(n-6) + a(n-7) for n > 13.
a(n) = b(n) - b(n-6), where b(n) = [x^n] Sum_{k=max(1,ceiling(n/7))..floor(n/2)} P(x)^k with P(x) = x^2 + x^3 + x^4 + 5*x^5 + x^6 + x^7.

A343315 a(n) is the number of nonnegative integers that can be represented in a 7-segment display by using only n segments (version A277116).

Original entry on oeis.org

0, 0, 1, 1, 2, 6, 6, 14, 19, 36, 67, 103, 194, 315, 560, 971, 1651, 2895, 4933, 8581, 14798, 25515, 44165, 76067, 131563, 227034, 392032, 677152, 1168742, 2018769, 3485255, 6018422, 10392472, 17943750, 30985861, 53501944, 92385050, 159523542, 275451221, 475633952

Offset: 0

Stefano Spezia, Apr 11 2021

The nonnegative integers are displayed as in A277116.

Given the set S = {2, 3, 4, 5, 6, 7}, the function f defined in S as f(5) = 4, f(6) = 2 and f(s) = 1 elsewhere, a(n) is equal to the difference between the number b(n) of S-restricted f-weighted integer compositions of n with that of n-6, i.e., b(n-6). The latter one provides the number of all those excluded cases where a nonnegative integer is displayed with leading zeros. b(n) is calculated as the sum of polynomial coefficients or extended binomial coefficients (see Equation 3 in Eger) where the index of summation is positive and it covers the numbers of possible digits that can be displayed by n segments (see third formula).

			a(5) = 6 since 2, 3, 5, 9, 17 and 71 are displayed by 5 segments.
   __        __        __        __          __       __
   __|       __|      |__       |__|      |    |        |    |
  |__        __|       __|         |      |    |        |    |
   (2)       (3)       (5)       (9)       (17)          (71)

Steffen Eger, Restricted Weighted Integer Compositions and Extended Binomial Coefficients, Journal of Integer Sequences, Vol. 16, Article 13.1.3, (2013).
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,4,2,1).
Index entries for sequences related to calculator display
Index entries for sequences related to compositions

Cf. A002426, A004526, A216261, A277116, A331529, A331530, A343314.

P[x_]:=x^2+x^3+x^4+4x^5+2x^6+x^7; b[n_]:=Coefficient[Sum[P[x]^k, {k, Max[1, Ceiling[n/7]], Floor[n/2]}], x, n]; a[n_]:=b[n]-b[n-6]; Array[a, 40, 0]

G.f.: x^2*(1 - x)*(1 + x)*(1 - x + x^2)*(1 + x + x^2)*(1 + x + x^2 + 4*x^3 + 2*x^4 + x^5)/(1 - x^2 - x^3 - x^4 - 4*x^5 - 2*x^6 - x^7).
a(n) = a(n-2) + a(n-3) + a(n-4) + 4*a(n-5) + 2*a(n-6) + a(n-7) for n > 13.
a(n) = b(n) - b(n-6), where b(n) = [x^n] Sum_{k=max(1,ceiling(n/7))..floor(n/2)} P(x)^k with P(x) = x^2 + x^3 + x^4 + 4*x^5 + 2*x^6 + x^7.

A339700 a(n) is the n-th nonnegative number to light exactly n segments when displayed on a calculator.

Original entry on oeis.org

71, 77, 47, 61, 70, 52, 62, 99, 136, 190, 246, 263, 306, 589, 882, 1085, 1838, 2059, 2308, 2869, 5886, 8689, 10800, 18098, 20268, 20896, 28608, 58880, 86886, 106898, 180889, 200858, 208698, 283888, 588868, 868880, 1068889, 1808886, 2008086, 2086868, 2809888, 5888808, 8688868, 10688886, 18088880

Offset: 5

Graham Holmes, Dec 13 2020

a(n) is undefined for n<5, as there are no numbers with 1 segment, 1 with 2 segments, 1 with 3 segments, and 2 with 4 segments. If 0 is excluded as a valid input - so the series would refer to "the n-th positive number" - then a(6) would be 111 rather than 77.

			For n=7, 47 is the 7th positive number to light 7 segments, after 8, 12, 13, 15, 21, and 31.

Graham Holmes, Table of n, a(n) for n = 5..82
Index entries for sequences related to calculator display

Cf. A006942 (segments lit), A216261, A331529.

s=[6,2,5,5,4,5,6,3,7,6];p=[];a=[];for(i=2;i<=100;i++)p[i]=0;for(i=1;i<=1000000;i++){d=i;n=0;do{x=d%10;n+=s[x];d=(d-x)/10;}while(d>0)p[n]++;if(p[n]==n)a[n]=i;}for(c=2;c<=40;c++)document.write(c+": "+a[c]+"
");

A350177 a(n) is the number of nonnegative integers that can be represented by lighting only n segments on a 9-segment display, used by the Russian postal service.

Original entry on oeis.org

0, 0, 0, 2, 3, 2, 5, 13, 17, 22, 47, 86, 127, 211, 387, 645, 1044, 1794, 3086, 5135, 8608, 14674, 24805, 41631, 70322, 119069, 200768, 338429, 571845, 965823, 1629253, 2749904, 4643876, 7838862, 13229487, 22333638, 37704236, 63642469, 107427241, 181351098, 306133271

Offset: 0

Stefano Spezia, Dec 18 2021

The nonnegative integers are displayed as in A350131.

Given the set S = {3, 4, 5, 6, 7}, the function f defined in S as f(3) = f(5) = 2, f(4) = 3, and f(6) = f(7) = 1, a(n) is equal to the difference between the number b(n) of S-restricted f-weighted integer compositions of n with that of n-6, i.e., b(n-6). The latter one provides the number of all those excluded cases where a nonnegative integer is displayed with leading zeros. b(n) is calculated as the sum of polynomial coefficients or extended binomial coefficients (see Equation 3 in Eger) where the index of summation is positive and it covers the numbers of possible digits that can be displayed by n segments (see first formula).

			a(6) = 5 since 0, 11, 17, 71 and 77 are displayed by 6 segments.
   _                 _    _        _  _
  | |    /| /|    /| /    /  /|    /  /
  |_|     |  |     | |    |   |    |  |
  (0)     (11)    (17)     (71)    (77)

Steffen Eger, Restricted Weighted Integer Compositions and Extended Binomial Coefficients, Journal of Integer Sequences, Vol. 16, Article 13.1.3, (2013).
Wikipedia, Postal code template.
Wikipedia, Postal codes in Russia.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,3,2,1,1).
Index entries for sequences related to calculator display
Index entries for sequences related to compositions

Cf. A002426, A004526, A331529, A331530, A343314, A343315, A350131.

P[x_]:=2x^3+3x^4+2x^5+x^6+x^7; b[n_]:=Coefficient[Sum[P[x]^k, {k, Max[1, Ceiling[n/7]], Floor[n/2]}], x, n]; a[n_]:=b[n]-b[n-6]; Array[a, 41, 0]

a(n) = b(n) - b(n-6), where b(n) = [x^n] Sum_{k=max(1,ceiling(n/7))..floor(n/2)} P(x)^k with P(x) = 2*x^3 + 3*x^4 + 2*x^5 + x^6 + x^7.
G.f.: x^3*(1 - x)*(1 + x)*(1 - x + x^2)*(1 + x + x^2)*(2 + 3*x + 2*x^2 + x^3 + x^4)/(1 - 2*x^3 - 3*x^4 - 2*x^5 - x^6 - x^7).
a(n) = 2*a(n-3) + 3*a(n-4) + 2*a(n-5) + a(n-6) + a(n-7) for n > 13.

A350437 a(n) is the number of integers that can be represented in a 7-segment display by using only n segments (version A006942).

Original entry on oeis.org

0, 0, 1, 2, 3, 7, 12, 18, 31, 52, 92, 158, 269, 460, 786, 1350, 2317, 3969, 6798, 11643, 19952, 34197, 58601, 100410, 172042, 294791, 505143, 865589, 1483206, 2541480, 4354847, 7462119, 12786520, 21909974, 37543133, 64330800, 110232005, 188884671, 323657539, 554593317

Offset: 0

Stefano Spezia, Dec 31 2021

The integers are displayed as in A006942, where a 7 is depicted by 3 segments. The negative integers are depicted by using 1 segment more for the minus sign.
Since the integer 0 is depicted by 6 segments, in order to avoid considering -0 in the case n = 7, a(7) is obtained by decreasing of a unit the result of the sum A331529(7) + A331529(6) = 12 + 7 = 19, i.e., a(7) = 19 - 1 = 18.
The same sequence is obtained when 7 and 9 are depicted respectively by 4 and 5 segments (A074458).

			a(7) = 18 since -111, -77, -41, -14, -9, -6, 8, 12, 13, 15, 21, 31, 47, 51, 74, 117, 171 and 711 are displayed by 7 segments.
segments.
                       __   __                                      __
   __   |  |  |     __   |    |    __ |__|  |    __   | |__|    __ |__|
        |  |  |          |    |          |  |         |    |        __|
        (-111)          (-77)         (-41)          (-14)        (-9)
       __      __        __         __         __      __        __
   __ |__     |__|    |  __|     |  __|     | |__      __|  |    __|  |
      |__|    |__|    | |__      |  __|     |  __|    |__   |    __|  |
     (-6)      (8)     (12)       (13)       (15)       (21)      (31)
        __      __        __               __       __        __
  |__|    |    |__   |      | |__|    |  |   |    |   |  |      |  |  |
     |    |     __|  |      |    |    |  |   |    |   |  |      |  |  |
     (47)        (51)        (74)       (117)       (171)        (711)

Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,3,3,1).
Index entries for sequences related to calculator display
Index entries for sequences related to compositions

Cf. A006942, A074458, A331529.

P[x_]:=x^2+x^3+x^4+3x^5+3x^6+x^7; c[n_]:=Coefficient[Sum[P[x]^k, {k, Max[1, Ceiling[n/7]], Floor[n/2]}], x, n]; b[n_]:=c[n]-c[n-6]; (* A331529 *)
a[n_]:=If[n!=7,b[n]+b[n-1],18]; Array[a,40,0]

a(7) = 18, otherwise a(n) = A331529(n) + A331529(n-1).
G.f.: x^2*(1 + 2*x + 2*x^2 + 4*x^3 + 6*x^4 + 3*x^5 - x^7 - x^8 - 3*x^9 - 3*x^10 - x^11)/(1 - x^2 - x^3 - x^4 - 3*x^5 - 3*x^6 - x^7).
a(n) = a(n-2) + a(n-3) + a(n-4) + 3*a(n-5) + 3*a(n-6) + a(n-7) for n > 13.

cp's OEIS Frontend

A006942 Number of segments used to represent n on calculator display, variant 5: digits '6', '7' and '9' use 6, 3 and 6 segments, respectively.

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A331530 a(n) is the number of nonnegative integers that can be represented in a 7-segment display by using only n segments (version A010371).

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A343314 a(n) is the number of nonnegative integers that can be represented in a 7-segment display by using only n segments (version A063720).

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A343315 a(n) is the number of nonnegative integers that can be represented in a 7-segment display by using only n segments (version A277116).

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A339700 a(n) is the n-th nonnegative number to light exactly n segments when displayed on a calculator.

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A350177 a(n) is the number of nonnegative integers that can be represented by lighting only n segments on a 9-segment display, used by the Russian postal service.

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A350437 a(n) is the number of integers that can be represented in a 7-segment display by using only n segments (version A006942).

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