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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A332119 a(n) = (10^(2n+1)-1)/9 + 8*10^n.

Original entry on oeis.org

9, 191, 11911, 1119111, 111191111, 11111911111, 1111119111111, 111111191111111, 11111111911111111, 1111111119111111111, 111111111191111111111, 11111111111911111111111, 1111111111119111111111111, 111111111111191111111111111, 11111111111111911111111111111, 1111111111111119111111111111111
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Comments

See A107649 = {1, 4, 26, 187, 226, 874, ...} for the indices of primes.

Links

Crossrefs

Cf. (A077795-1)/2 = A107649: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332129 .. A332189 (variants with different repeated digit 2, ..., 8).
Cf. A332112 .. A332118 (variants with different middle digit 2, ..., 8).

Programs

  • Maple
    A332119 := n -> (10^(2*n+1)-1)/9+8*10^n;
  • Mathematica
    Array[(10^(2 # + 1)-1)/9 + 8*10^# &, 15, 0]
Table[FromDigits[Join[PadRight[{},n,1],{9},PadRight[{},n,1]]],{n,0,20}] (* or *) LinearRecurrence[ {111,-1110,1000},{9,191,11911},20] (* Harvey P. Dale, Mar 30 2024 *)
  • PARI
    apply( {A332119(n)=10^(n*2+1)\9+8*10^n}, [0..15])
  • Python
    def A332119(n): return 10**(n*2+1)//9+8*10**n

Formula

a(n) = A138148(n) + 9*10^n = A002275(2n+1) + 8*10^n.
G.f.: (9 - 808*x + 700*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A138826 Concatenation of 2n-1 digits 1, n digits 0 and 2n-1 digits 1.

Original entry on oeis.org

101, 11100111, 1111100011111, 111111100001111111, 11111111100000111111111, 1111111111100000011111111111, 111111111111100000001111111111111, 11111111111111100000000111111111111111
Offset: 1

Views

Author

Omar E. Pol, Apr 06 2008

Keywords

Comments

a(n) has 5n-2 digits.
a(n) is also A147540(n) written in base 2. [Omar E. Pol, Nov 08 2008]

Examples

			n ........... a(n)
1 ........... 101
2 ......... 11100111
3 ....... 1111100011111
4 ..... 111111100001111111
5 ... 11111111100000111111111

Links

Crossrefs

Cf. A138120, A138144, A138145, A138146, A138148, A138720, A138721, A138722, A147540.

Programs

  • Mathematica
    Table[(1000^n + 10)*(100^n - 10)/900, {n, 10}] (* Paolo Xausa, Aug 08 2024 *)
  • PARI
    Vec(x*(1100000000*x^3-2000000*x^2+888910*x+101)/((x-1)*(100*x-1)*(1000*x-1)*(100000*x-1)) + O(x^100)) \\ Colin Barker, Sep 16 2013

Formula

a(n) = (10^(2n-1)-1+10^(5n-2)-10^(3n-1))/9. [R. J. Mathar, Nov 07 2008, corrected Nov 09 2008]
G.f.: x*(1100000000*x^3-2000000*x^2+888910*x+101) / ((x-1)*(100*x-1)*(1000*x-1)*(100000*x-1)). - Colin Barker, Sep 16 2013

A332121 a(n) = 2*(10^(2n+1)-1)/9 - 10^n.

Original entry on oeis.org

1, 212, 22122, 2221222, 222212222, 22222122222, 2222221222222, 222222212222222, 22222222122222222, 2222222221222222222, 222222222212222222222, 22222222222122222222222, 2222222222221222222222222, 222222222222212222222222222, 22222222222222122222222222222, 2222222222222221222222222222222
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002276 (2*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332120 .. A332129 (variants with different middle digit 0, ..., 9).
Cf. A332131 .. A332191 (variants with different repeated digit 3, ..., 9).

Programs

  • Maple
    A332121 := n -> 2*(10^(2*n+1)-1)/9-10^n;
  • Mathematica
    Array[2 (10^(2 # + 1)-1)/9 - 10^# &, 15, 0]
  • PARI
    apply( {A332121(n)=10^(n*2+1)\9*2-10^n}, [0..15])
  • Python
    def A332121(n): return 10**(n*2+1)//9*2-10**n

Formula

a(n) = 2*A138148(n) + 1*10^n = A002276(2n+1) - 10^n.
G.f.: (1 + 101*x - 300*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332113 a(n) = (10^(2n+1)-1)/9 + 2*10^n.

Original entry on oeis.org

3, 131, 11311, 1113111, 111131111, 11111311111, 1111113111111, 111111131111111, 11111111311111111, 1111111113111111111, 111111111131111111111, 11111111111311111111111, 1111111111113111111111111, 111111111111131111111111111, 11111111111111311111111111111, 1111111111111113111111111111111
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Comments

See A107123 = {0, 1, 2, 19, 97, 9818, ...} for the indices of primes.

Links

Crossrefs

Cf. (A077779-1)/2 = A107123: indices of primes; A331864 & A331865 (non-palindromic variants).
Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332123 .. A332193 (variants with different repeated digit 2, ..., 9).
Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

Programs

  • Maple
    A332113 := n -> (10^(2*n+1)-1)/9+2*10^n;
  • Mathematica
    Array[(10^(2 # + 1)-1)/9 + 2*10^# &, 15, 0]
  • PARI
    apply( {A332113(n)=10^(n*2+1)\9+2*10^n}, [0..15])
  • Python
    def A332113(n): return 10**(n*2+1)//9+2*10**n

Formula

a(n) = A138148(n) + 3*10^n = A002275(2n+1) + 2*10^n.
G.f.: (3 - 202*x + 100*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332129 a(n) = 2*(10^(2n+1)-1)/9 + 7*10^n.

Original entry on oeis.org

9, 292, 22922, 2229222, 222292222, 22222922222, 2222229222222, 222222292222222, 22222222922222222, 2222222229222222222, 222222222292222222222, 22222222222922222222222, 2222222222229222222222222, 222222222222292222222222222, 22222222222222922222222222222, 2222222222222229222222222222222
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002276 (2*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332119 .. A332189 (variants with different repeated digit 1, ..., 8).
Cf. A332120 .. A332128 (variants with different middle digit 0, ..., 8).

Programs

  • Maple
    A332129 := n -> 2*(10^(2*n+1)-1)/9+7*10^n;
  • Mathematica
    Array[2 (10^(2 # + 1)-1)/9 + 7*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{9,292,22922},20] (* Harvey P. Dale, Jun 25 2020 *)
  • PARI
    apply( {A332129(n)=10^(n*2+1)\9*2+7*10^n}, [0..15])
  • Python
    def A332129(n): return 10**(n*2+1)//9*2+7*10**n

Formula

a(n) = 2*A138148(n) + 9*10^n = A002276(2n+1) + 7*10^n.
G.f.: (9 - 707*x + 500*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332180 a(n) = 8*(10^(2n+1)-1)/9 - 8*10^n.

Original entry on oeis.org

0, 808, 88088, 8880888, 888808888, 88888088888, 8888880888888, 888888808888888, 88888888088888888, 8888888880888888888, 888888888808888888888, 88888888888088888888888, 8888888888880888888888888, 888888888888808888888888888, 88888888888888088888888888888, 8888888888888880888888888888888
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002282 (8*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332120 .. A332190 (variants with different repeated digit 2, ..., 9).
Cf. A332181 .. A332189 (variants with different middle digit 1, ..., 9).
Subsequence of A006072 (numbers with mirror symmetry about middle), A153806 (strobogrammatic cyclops numbers), and A204095 (numbers whose decimal digits are in {0,8}).

Programs

  • Maple
    A332180 := n -> 8*((10^(2*n+1)-1)/9-10^n);
  • Mathematica
    Array[8 ((10^(2 # + 1)-1)/9 - 10^#) &, 15, 0]
  • PARI
    apply( {A332180(n)=(10^(n*2+1)\9-10^n)*8}, [0..15])
  • Python
    def A332180(n): return (10**(n*2+1)//9-10**n)*8

Formula

a(n) = 8*A138148(n) = A002282(2n+1) - 8*10^n.
G.f.: 8*x*(101 - 200*x)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
E.g.f.: 8*exp(x)*(10*exp(99*x) - 9*exp(9*x) - 1)/9. - Stefano Spezia, Jul 13 2024

A332130 a(n) = (10^(2n+1)-1)/3 - 3*10^n.

Original entry on oeis.org

0, 303, 33033, 3330333, 333303333, 33333033333, 3333330333333, 333333303333333, 33333333033333333, 3333333330333333333, 333333333303333333333, 33333333333033333333333, 3333333333330333333333333, 333333333333303333333333333, 33333333333333033333333333333, 3333333333333330333333333333333
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002277 (3*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332120 .. A332190 (variants with different repeated digit 2, ..., 9).
Cf. A332131 .. A332139 (variants with different middle digit 1, ..., 9).

Programs

  • Maple
    A332130 := n -> (10^(2*n+1)-1)/3-3*10^n;
  • Mathematica
    Array[ ((10^(2 # + 1)-1)/3 - 3*10^#) &, 15, 0]
  • PARI
    apply( {A332130(n)=10^(n*2+1)\3-3*10^n}, [0..15])
  • Python
    def A332130(n): return 10**(n*2+1)//3-3*10**n

Formula

a(n) = 3*A138148(n) = A002277(2n+1) - 3*10^n.
G.f.: 3*x*(101 - 200*x)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
E.g.f.: exp(x)*(10*exp(99*x) - 9*exp(9*x) - 1)/3. - Stefano Spezia, Jul 13 2024

A153806 Strobogrammatic cyclops numbers.

Original entry on oeis.org

0, 101, 609, 808, 906, 11011, 16091, 18081, 19061, 61019, 66099, 68089, 69069, 81018, 86098, 88088, 89068, 91016, 96096, 98086, 99066, 1110111, 1160911, 1180811, 1190611, 1610191, 1660991, 1680891, 1690691, 1810181, 1860981
Offset: 1

Views

Author

Omar E. Pol, Jan 15 2009

Keywords

Comments

Intersection of A000787 and A134808.

Examples

			1680891 is a member because it is the same upside down (A000787) and also a cyclops number (A134808).

Links

Crossrefs

Cf. A000787, A007597, A134808, A134809, A138131, A138148, A153807.

Programs

  • Mathematica
    Select[Range[10^7], And[OddQ@ Length@#, Part[#, Ceiling[Length[#]/2]] == 0, Times @@ Boole@ Map[MemberQ[{0, 1, 6, 8, 9}, #] &, Union@ #] == 1, Count[#, 0] == 1, (Take[#, Floor[Length[#]/2]] /. {6 -> 9, 9 -> 6}) ==
Reverse@ Take[#, -Floor[Length[#]/2]]] &@ IntegerDigits@ # &] (* Michael De Vlieger, Jul 05 2016 *)
  • Python
    import sys
f = open('b153806.txt', 'w')
i = 1
n = 0
a = [""]
r = [""]  #reversed strobogrammatically
while True:
    for x,y in zip(a,r):
        f.write(str(i)+" "+x+"0"+y+"\n")
        i += 1
        if i>20000:
            f.close()
            sys.exit()
    a = sum([[x+"1",x+"6",x+"8",x+"9"] for x in a],[])
    r = sum([["1"+x,"9"+x,"8"+x,"6"+x] for x in r],[])
# Kenny Lau, Jul 05 2016

Extensions

Extended beyond 11011 by R. J. Mathar, Jan 17 2009

A160717 Cyclops triangular numbers.

Original entry on oeis.org

0, 105, 406, 703, 903, 11026, 13041, 14028, 15051, 27028, 36046, 41041, 43071, 46056, 61075, 66066, 75078, 77028, 83028, 85078, 93096, 1110795, 1130256, 1160526, 1180416, 1250571, 1290421, 1330896, 1350546, 1360425, 1380291
Offset: 1

Views

Author

Omar E. Pol, Jun 08 2009

Keywords

Comments

Triangular numbers (A000217) that are also cyclops numbers (A134808).

Examples

			105 is in the sequence since it is both a triangular number (105 = 1 + 2 + ... + 14) and a Cyclops number (number of digits is odd, and the only zero is the middle digit). - _Michael B. Porter_, Jul 08 2016

Links

Crossrefs

Cf. A000217, A134808, A134809, A138131, A138148, A153806, A160711, A160712.

Programs

  • Maple
    count:= 1: A[1]:= 0:
for d from 1 to 3 do
  for x from 0 to 9^d-1 do
    L:= convert(x+9^d,base,9);
    X:= add((L[i]+1)*10^(i-1),i=1..d);
    for y from 0 to 9^d-1 do
      L:= convert(y+9^d,base,9);
      Y:= add((L[i]+1)*10^(i-1),i=1..d);
      Z:= Y + 10^(d+1)*X;
      if issqr(1+8*Z) then
        count:= count+1;
        A[count]:= Z;
      fi
od od od:
seq(A[i],i=1..count); # Robert Israel, Jul 08 2016
  • Mathematica
    cyclopsQ[n_] := Block[{id=IntegerDigits@n,lg=Floor[Log[10,n]+1]}, Count[id,0]==1 && OddQ@lg && id[[(lg+1)/2]]==0]; lst = {0}; Do[t = n (n + 1)/2; If[ cyclopsQ@t, AppendTo[lst, t]], {n, 0, 1670}]; lst (* Robert G. Wilson v, Jun 09 2009 *)
cyclpsQ[n_]:=With[{len=IntegerLength[n]},OddQ[len]&&DigitCount[n,10,0]==1&&IntegerDigits[n][[(len+1)/2]]==0]; Join[{0},Select[ Accumulate[ Range[2000]],cyclpsQ]] (* Harvey P. Dale, Nov 05 2024 *)

Extensions

More terms from Robert G. Wilson v, Jun 09 2009
Offset and b-file changed by N. J. A. Sloane, Jul 27 2016

A332139 a(n) = (10^(2*n+1)-1)/3 + 6*10^n.

Original entry on oeis.org

9, 393, 33933, 3339333, 333393333, 33333933333, 3333339333333, 333333393333333, 33333333933333333, 3333333339333333333, 333333333393333333333, 33333333333933333333333, 3333333333339333333333333, 333333333333393333333333333, 33333333333333933333333333333, 3333333333333339333333333333333
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002277 (3*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332129 .. A332189 (variants with different repeated digit 2, ..., 8).
Cf. A332130 .. A332138 (variants with different middle digit 0, ..., 8).

Programs

  • Maple
    A332139 := n -> (10^(2*n+1)-1)/3+6*10^n;
  • Mathematica
    Array[ (10^(2 # + 1)-1)/3 + 6*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{9,393,33933},20] (* Harvey P. Dale, Sep 17 2020 *)
  • PARI
    apply( {A332139(n)=10^(n*2+1)\3+6*10^n}, [0..15])
  • Python
    def A332139(n): return 10**(n*2+1)//3+6*10**n

Formula

a(n) = 3*A138148(n) + 9*10^n = A002277(2n+1) + 6*10^n = 3*A332113(n).
G.f.: (9 - 606*x + 300*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
Previous Showing 11-20 of 102 results. Next

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