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.

Showing 1-6 of 6 results.

A332197 a(n) = 10^(2n+1) - 1 - 2*10^n.

Original entry on oeis.org

7, 979, 99799, 9997999, 999979999, 99999799999, 9999997999999, 999999979999999, 99999999799999999, 9999999997999999999, 999999999979999999999, 99999999999799999999999, 9999999999997999999999999, 999999999999979999999999999, 99999999999999799999999999999, 9999999999999997999999999999999
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Comments

According to Kamada, n = 118 and n = 145126 are the only known indices of primes (the so-called palindromic near-repdigit or wing primes).

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only).
Cf. A332190 .. A332196, A181965 (variants with different middle digit 0, ..., 8).
Cf. A332117 .. A332187 (variants with different repeated digit 1, ..., 9).

Programs

  • Maple
    A332197 := n -> 10^(n*2+1)-1-2*10^n;
  • Mathematica
    Array[ 10^(2 # + 1) -1 -2*10^# &, 15, 0]
Table[FromDigits[Join[PadRight[{},n,9],{7},PadRight[{},n,9]]],{n,0,20}] (* or *) LinearRecurrence[{111,-1110,1000},{7,979,99799},20] (* Harvey P. Dale, Mar 03 2023 *)
  • PARI
    apply( {A332197(n)=10^(n*2+1)-1-2*10^n}, [0..15])
  • Python
    def A332197(n): return 10**(n*2+1)-1-2*10^n

Formula

a(n) = 9*A138148(n) + 7*10^n.
G.f.: (7 + 202*x - 1100*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.

A332116 a(n) = (10^(2n+1)-1)/9 + 5*10^n.

Original entry on oeis.org

6, 161, 11611, 1116111, 111161111, 11111611111, 1111116111111, 111111161111111, 11111111611111111, 1111111116111111111, 111111111161111111111, 11111111111611111111111, 1111111111116111111111111, 111111111111161111111111111, 11111111111111611111111111111, 1111111111111116111111111111111
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Comments

See A107126 = {10, 14, 40, 59, 160, 412, ...} for the indices of primes.

Links

Crossrefs

Cf. (A077706-1)/2 = A107126: 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. A332126 .. A332196 (variants with different repeated digit 2, ..., 9).
Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

Programs

  • Maple
    A332116 := n -> (10^(2*n+1)-1)/9+5*10^n;
  • Mathematica
    Array[(10^(2 # + 1)-1)/9 + 5*10^# &, 15, 0]
  • PARI
    apply( {A332116(n)=10^(n*2+1)\9+5*10^n}, [0..15])
  • Python
    def A332116(n): return 10**(n*2+1)//9+5*10**n

Formula

a(n) = A138148(n) + 6*10^n = A002275(2n+1) + 5*10^n.
G.f.: (6 - 505*x + 400*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.
E.g.f.: exp(x)*(10*exp(99*x) + 45*exp(9*x) - 1)/9. - Stefano Spezia, Jul 13 2024

A332126 a(n) = 2*(10^(2n+1)-1)/9 + 4*10^n.

Original entry on oeis.org

6, 262, 22622, 2226222, 222262222, 22222622222, 2222226222222, 222222262222222, 22222222622222222, 2222222226222222222, 222222222262222222222, 22222222222622222222222, 2222222222226222222222222, 222222222222262222222222222, 22222222222222622222222222222, 2222222222222226222222222222222
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. A332116 .. A332196 (variants with different repeated digit 1, ..., 9).
Cf. A332120 .. A332129 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332126 := n -> 2*(10^(2*n+1)-1)/9+4*10^n;
  • Mathematica
    Array[2 (10^(2 # + 1)-1)/9 + 4*10^# &, 15, 0]
Table[FromDigits[Join[PadRight[{},n,2],{6},PadRight[{},n,2]]],{n,0,20}] (* or *) LinearRecurrence[{111,-1110,1000},{6,262,22622},20] (* Harvey P. Dale, Oct 17 2021 *)
  • PARI
    apply( {A332126(n)=10^(n*2+1)\9*2+4*10^n}, [0..15])
  • Python
    def A332126(n): return 10**(n*2+1)//9*2+4*10**n

Formula

a(n) = 2*A138148(n) + 6*10^n = A002276(2n+1) + 4*10^n = 2*A332113(n).
G.f.: (6 - 404*x + 200*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.
E.g.f.: 2*exp(x)*(10*exp(99*x) + 18*exp(9*x) - 1)/9. - Stefano Spezia, Jul 13 2024

A332136 a(n) = 3*(10^(2n+1)-1)/9 + 3*10^n.

Original entry on oeis.org

6, 363, 33633, 3336333, 333363333, 33333633333, 3333336333333, 333333363333333, 33333333633333333, 3333333336333333333, 333333333363333333333, 33333333333633333333333, 3333333333336333333333333, 333333333333363333333333333, 33333333333333633333333333333, 3333333333333336333333333333333
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. A332126 .. A332196 (variants with different repeated digit 2, ..., 9).
Cf. A332130 .. A332139 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332136 := n -> (10^(2*n+1)-1)/3+3*10^n);
  • Mathematica
    Array[ (10^(2 # + 1)-1)/3 + 3*10^# &, 15, 0]
  • PARI
    apply( {A332136(n)=10^(n*2+1)\3+3*10^n}, [0..15])
  • Python
    def A332136(n): return 10**(n*2+1)//3+3*10**n

Formula

a(n) = 3*A138148(n) + 6*10^n = A002277(2n+1) + 3*10^n = 3*A332112(n).
G.f.: (6 - 303*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.

A332146 a(n) = 4*(10^(2*n+1)-1)/9 + 2*10^n.

Original entry on oeis.org

6, 464, 44644, 4446444, 444464444, 44444644444, 4444446444444, 444444464444444, 44444444644444444, 4444444446444444444, 444444444464444444444, 44444444444644444444444, 4444444444446444444444444, 444444444444464444444444444, 44444444444444644444444444444, 4444444444444446444444444444444
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002278 (4*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332116 .. A332196 (variants with different repeated digit 2, ..., 9).
Cf. A332140 .. A332149 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332146 := n -> 4*(10^(2*n+1)-1)/9+2*10^n;
  • Mathematica
    Array[4 (10^(2 # + 1)-1)/9 + 2*10^# &, 15, 0]
  • PARI
    apply( {A332146(n)=10^(n*2+1)\9*4+2*10^n}, [0..15])
  • Python
    def A332146(n): return 10**(n*2+1)//9*4+2*10**n

Formula

a(n) = 4*A138148(n) + 6*10^n = A002278(2n+1) + 2*10^n = 2*A332123(n).
G.f.: (6 - 202*x - 200*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.

A332156 a(n) = 5*(10^(2*n+1)-1)/9 + 10^n.

Original entry on oeis.org

6, 565, 55655, 5556555, 555565555, 55555655555, 5555556555555, 555555565555555, 55555555655555555, 5555555556555555555, 555555555565555555555, 55555555555655555555555, 5555555555556555555555555, 555555555555565555555555555, 55555555555555655555555555555, 5555555555555556555555555555555
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002279 (5*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332116 .. A332196 (variants with different repeated digit 1, ..., 9).
Cf. A332150 .. A332159 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332156 := n -> 5*(10^(2*n+1)-1)/9+10^n;
  • Mathematica
    Array[5 (10^(2 # + 1)-1)/9 + 10^# &, 15, 0]
  • PARI
    apply( {A332156(n)=10^(n*2+1)\9*5+10^n}, [0..15])
  • Python
    def A332156(n): return 10**(n*2+1)//9*5+10**n

Formula

a(n) = 5*A138148(n) + 6*10^n = A002279(2n+1) + 10^n.
G.f.: (6 - 101*x - 400*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.
E.g.f.: exp(x)*(50*exp(99*x) + 9*exp(9*x) - 5)/9. - Stefano Spezia, Jul 13 2024
Showing 1-6 of 6 results.

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