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.

A332115 a(n) = (10^(2n+1)-1)/9 + 4*10^n.

Original entry on oeis.org

5, 151, 11511, 1115111, 111151111, 11111511111, 1111115111111, 111111151111111, 11111111511111111, 1111111115111111111, 111111111151111111111, 11111111111511111111111, 1111111111115111111111111, 111111111111151111111111111, 11111111111111511111111111111, 1111111111111115111111111111111
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Comments

See A107125 = {0, 1, 7, 45, 115, 681, 1248, ...} for the indices of primes.

Links

Crossrefs

Cf. (A077783-1)/2 = A107125: indices of primes; A331868 & A331869 (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. A332125 .. A332195 (variants with different repeated digit 2, ..., 9).
Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

Programs

  • Maple
    A332115 := n -> (10^(2*n+1)-1)/9+4*10^n;
  • Mathematica
    Array[(10^(2 # + 1)-1)/9 + 4*10^# &, 15, 0]
  • PARI
    apply( {A332115(n)=10^(n*2+1)\9+4*10^n}, [0..15])
  • Python
    def A332115(n): return 10**(n*2+1)//9+4*10**n

Formula

a(n) = A138148(n) + 5*10^n = A002275(2n+1) + 4*10^n.
G.f.: (5 - 404*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.

A332125 a(n) = 2*(10^(2n+1)-1)/9 + 3*10^n.

Original entry on oeis.org

5, 252, 22522, 2225222, 222252222, 22222522222, 2222225222222, 222222252222222, 22222222522222222, 2222222225222222222, 222222222252222222222, 22222222222522222222222, 2222222222225222222222222, 222222222222252222222222222, 22222222222222522222222222222, 2222222222222225222222222222222
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. A332115 .. A332195 (variants with different repeated digit 1, ..., 9).
Cf. A332120 .. A332129 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332125 := n -> 2*(10^(2*n+1)-1)/9+3*10^n;
  • Mathematica
    Array[2 (10^(2 # + 1)-1)/9 + 3*10^# &, 15, 0]
  • PARI
    apply( {A332125(n)=10^(n*2+1)\9*2+3*10^n}, [0..15])
  • Python
    def A332125(n): return 10**(n*2+1)//9*2+3*10**n

Formula

a(n) = 2*A138148(n) + 5*10^n = A002276(2n+1) + 3*10^n.
G.f.: (5 - 303*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.

A332185 a(n) = 8*(10^(2n+1)-1)/9 - 3*10^n.

Original entry on oeis.org

5, 858, 88588, 8885888, 888858888, 88888588888, 8888885888888, 888888858888888, 88888888588888888, 8888888885888888888, 888888888858888888888, 88888888888588888888888, 8888888888885888888888888, 888888888888858888888888888, 88888888888888588888888888888, 8888888888888885888888888888888
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. A332180 .. A332189 (variants with different middle digit 0, ..., 9).
Cf. A332115 .. A332195 (variants with different "wing" digit 1, ..., 9).

Programs

  • Maple
    A332185 := n -> 8*(10^(2*n+1)-1)/9-3*10^n;
  • Mathematica
    Array[8 (10^(2 # + 1)-1)/9 - 3*10^# &, 15, 0]
  • PARI
    apply( {A332185(n)=10^(n*2+1)\9*8-3*10^n}, [0..15])
  • Python
    def A332185(n): return 10**(n*2+1)//9*8-3*10**n

Formula

a(n) = 8*A138148(n) + 5*10^n = A002282(2n+1) - 3*10^n.
G.f.: (5 + 303*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.

A332135 a(n) = (10^(2n+1)-1)/3 + 2*10^n.

Original entry on oeis.org

5, 353, 33533, 3335333, 333353333, 33333533333, 3333335333333, 333333353333333, 33333333533333333, 3333333335333333333, 333333333353333333333, 33333333333533333333333, 3333333333335333333333333, 333333333333353333333333333, 33333333333333533333333333333, 3333333333333335333333333333333
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Comments

See A183175 = {1, 2, 17, 79, 118, 162, 177, ...} for the indices of primes.

Links

Crossrefs

Cf. (A077784-1)/2 = A183175: indices of primes.
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. A332125 .. A332195 (variants with different repeated digit 2, ..., 9).
Cf. A332130 .. A332139 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332135 := n -> (10^(2*n+1)-1)/3+2*10^n;
  • Mathematica
    Array[ (10^(2 # + 1)-1)/3 + 2*10^# &, 15, 0]
  • PARI
    apply( {A332135(n)=10^(n*2+1)\3+2*10^n}, [0..15])
  • Python
    def A332135(n): return 10**(n*2+1)//3+2*10**n

Formula

a(n) = 3*A138148(n) + 5*10^n = A002277(2n+1) + 2*10^n.
G.f.: (5 - 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.
E.g.f.: exp(x)*(10*exp(99*x) + 6*exp(9*x) - 1)/3. - Stefano Spezia, Sep 24 2024

A332145 a(n) = 4*(10^(2*n+1)-1)/9 + 10^n.

Original entry on oeis.org

5, 454, 44544, 4445444, 444454444, 44444544444, 4444445444444, 444444454444444, 44444444544444444, 4444444445444444444, 444444444454444444444, 44444444444544444444444, 4444444444445444444444444, 444444444444454444444444444, 44444444444444544444444444444, 4444444444444445444444444444444
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. A332115 .. A332195 (variants with different repeated digit 1, ..., 9).
Cf. A332140 .. A332149 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332145 := n -> 4*(10^(2*n+1)-1)/9+10^n;
  • Mathematica
    Array[4 (10^(2 # + 1)-1)/9 + 10^# &, 15, 0]
  • PARI
    apply( {A332145(n)=10^(n*2+1)\9*4+10^n}, [0..15])
  • Python
    def A332145(n): return 10**(n*2+1)//9*4+10**n

Formula

a(n) = 4*A138148(n) + 5*10^n = A002278(2n+1) + 10^n.
G.f.: (5 - 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.

A332165 a(n) = 6*(10^(2*n+1)-1)/9 - 10^n.

Original entry on oeis.org

5, 656, 66566, 6665666, 666656666, 66666566666, 6666665666666, 666666656666666, 66666666566666666, 6666666665666666666, 666666666656666666666, 66666666666566666666666, 6666666666665666666666666, 666666666666656666666666666, 66666666666666566666666666666, 6666666666666665666666666666666
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002280 (6*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332115 .. A332195 (variants with different repeated digit 1, ..., 9).
Cf. A332160 .. A332169 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332165 := n -> 6*(10^(2*n+1)-1)/9-10^n;
  • Mathematica
    Array[6 (10^(2 # + 1)-1)/9 - 10^# &, 15, 0]
  • PARI
    apply( {A332165(n)=10^(n*2+1)\9*6-10^n}, [0..15])
  • Python
    def A332165(n): return 10**(n*2+1)//9*6-10**n

Formula

a(n) = 6*A138148(n) + 5*10^n = A002280(2n+1) - 10^n.
G.f.: (5 + 101*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.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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