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.

A332112 a(n) = (10^(2n+1)-1)/9 + 10^n.

Original entry on oeis.org

2, 121, 11211, 1112111, 111121111, 11111211111, 1111112111111, 111111121111111, 11111111211111111, 1111111112111111111, 111111111121111111111, 11111111111211111111111, 1111111111112111111111111, 111111111111121111111111111, 11111111111111211111111111111, 1111111111111112111111111111111
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Comments

a(0) = 2 is the only prime in this sequence, since all other terms factor as a(n) = R(n+1)*(10^n+1), where R(n) = (10^n-1)/9.

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332132 .. A332192 (variants with different repeated digit 3, ..., 9).
Cf. A332113 .. A332119 (variants with different middle digit 3, ..., 9).
Cf. A331860 & A331861 (indices of primes in non-palindromic variants).

Programs

  • Maple
    A332112 := n -> (10^(2*n+1)-1)/9+10^n;
  • Mathematica
    Array[ (10^(2 # + 1)-1)/9 + 10^# &, 15, 0]
  • PARI
    apply( {A332112(n)=10^(n*2+1)\9*1+10^n}, [0..15])
  • Python
    def A332112(n): return 10**(n*2+1)//9+10**n

Formula

a(n) = A138148(n) + 2*10^n = A002275(2n+1) + 10^n.
G.f.: (2 - 101*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.

A332132 a(n) = (10^(2n+1)-1)/3 - 10^n.

Original entry on oeis.org

2, 323, 33233, 3332333, 333323333, 33333233333, 3333332333333, 333333323333333, 33333333233333333, 3333333332333333333, 333333333323333333333, 33333333333233333333333, 3333333333332333333333333, 333333333333323333333333333, 33333333333333233333333333333, 3333333333333332333333333333333
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Comments

There are no primes > 2 in this list because a(n) = round(10^n/.6)*(2*10^n-1) = 16...67*19...99.

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. A332112 .. A332192 (variants with different repeated digit 1, ..., 9).
Cf. A332130 .. A332139 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332132 := n -> (10^(2*n+1)-1)/3-10^n;
  • Mathematica
    Array[ (10^(2 # + 1)-1)/3 - 10^# &, 15, 0]
  • PARI
    apply( {A332132(n)=10^(n*2+1)\3-10^n}, [0..15])
  • Python
    def A332132(n): return 10**(n*2+1)//3-10**n

Formula

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

A332182 a(n) = 8*(10^(2n+1)-1)/9 - 6*10^n.

Original entry on oeis.org

2, 828, 88288, 8882888, 888828888, 88888288888, 8888882888888, 888888828888888, 88888888288888888, 8888888882888888888, 888888888828888888888, 88888888888288888888888, 8888888888882888888888888, 888888888888828888888888888, 88888888888888288888888888888, 8888888888888882888888888888888
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).
Cf. A332112 .. A332192 (variants with different repeated digit 1, ..., 9).
Cf. A332180 .. A332189 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332182 := n -> 8*(10^(2*n+1)-1)/9-6*10^n;
  • Mathematica
    Array[8 (10^(2 # + 1)-1)/9 - 6*10^# &, 15, 0]
  • PARI
    apply( {A332182(n)=10^(n*2+1)\9*8-6*10^n}, [0..15])
  • Python
    def A332182(n): return 10**(n*2+1)//9*8-6*10**n

Formula

a(n) = 8*A138148(n) + 2*10^n = A002282(2n+1)- 6*10^n.
G.f.: (2 + 606*x - 1400*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.

A332142 a(n) = 4*(10^(2*n+1)-1)/9 - 2*10^n.

Original entry on oeis.org

2, 424, 44244, 4442444, 444424444, 44444244444, 4444442444444, 444444424444444, 44444444244444444, 4444444442444444444, 444444444424444444444, 44444444444244444444444, 4444444444442444444444444, 444444444444424444444444444, 44444444444444244444444444444, 4444444444444442444444444444444
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. A332112 .. A332192 (variants with different repeated digit 1, ..., 9).
Cf. A332140 .. A332149 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332142 := 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( {A332142(n)=10^(n*2+1)\9*4-2*10^n}, [0..15])
  • Python
    def A332142(n): return 10**(n*2+1)//9*4-2*10**n

Formula

a(n) = 4*A138148(n) + 2*10^n = A002278(2n+1) - 2*10^n = 2*A332121(n).
G.f.: (2 + 202*x - 600*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.

A332152 a(n) = 5*(10^(2*n+1)-1)/9 - 3*10^n.

Original entry on oeis.org

2, 525, 55255, 5552555, 555525555, 55555255555, 5555552555555, 555555525555555, 55555555255555555, 5555555552555555555, 555555555525555555555, 55555555555255555555555, 5555555555552555555555555, 555555555555525555555555555, 55555555555555255555555555555, 5555555555555552555555555555555
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. A332112 .. A332192 (variants with different repeated digit 1, ..., 9).
Cf. A332150 .. A332159 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332152 := n -> 5*(10^(2*n+1)-1)/9-3*10^n;
  • Mathematica
    Array[5 (10^(2 # + 1)-1)/9 - 3*10^# &, 15, 0]
  • PARI
    apply( {A332152(n)=10^(n*2+1)\9*5-3*10^n}, [0..15])
  • Python
    def A332152(n): return 10**(n*2+1)//9*5-3*10**n

Formula

a(n) = 5*A138148(n) + 2*10^n = A002279(2n+1) - 3*10^n.
G.f.: (2 + 303*x - 800*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.

A332162 a(n) = 6*(10^(2*n+1)-1)/9 - 4*10^n.

Original entry on oeis.org

2, 626, 66266, 6662666, 666626666, 66666266666, 6666662666666, 666666626666666, 66666666266666666, 6666666662666666666, 666666666626666666666, 66666666666266666666666, 6666666666662666666666666, 666666666666626666666666666, 66666666666666266666666666666, 6666666666666662666666666666666
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. A332112 .. A332192 (variants with different repeated digit 1, ..., 9).
Cf. A332160 .. A332169 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332162 := n -> 6*(10^(2*n+1)-1)/9-4*10^n;
  • Mathematica
    Array[6 (10^(2 # + 1)-1)/9 - 4*10^# &, 15, 0]
  • PARI
    apply( {A332162(n)=10^(n*2+1)\9*6-4*10^n}, [0..15])
  • Python
    def A332162(n): return 10**(n*2+1)//9*6-4*10**n

Formula

a(n) = 6*A138148(n) + 2*10^n = A002280(2n+1) - 4*10^n = 2*A332131(n).
G.f.: (2 + 404*x - 1000*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.

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