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 1-5 of 5 results.

A332114 a(n) = (10^(2n+1)-1)/9 + 3*10^n.

Original entry on oeis.org

4, 141, 11411, 1114111, 111141111, 11111411111, 1111114111111, 111111141111111, 11111111411111111, 1111111114111111111, 111111111141111111111, 11111111111411111111111, 1111111111114111111111111, 111111111111141111111111111, 11111111111111411111111111111, 1111111111111114111111111111111
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Comments

See A107124 = {2, 3, 32, 45, 1544, ...} for the indices of primes.

Links

Crossrefs

Cf. (A077780-1)/2 = A107124: indices of primes; A331866 & A331867 (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. A332124 .. A332194 (variants with different repeated digit 2, ..., 9).
Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

Programs

  • Maple
    A332114 := n -> (10^(2*n+1)-1)/9+3*10^n;
  • Mathematica
    Array[(10^(2 # + 1)-1)/9 + 3*10^# &, 15, 0]
  • PARI
    apply( {A332114(n)=10^(n*2+1)\9+3*10^n}, [0..15])
  • Python
    def A332114(n): return 10**(n*2+1)//9+3*10**n

Formula

a(n) = A138148(n) + 4*10^n = A002275(2n+1) + 3*10^n.
G.f.: (4 - 303*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.

A332124 a(n) = 2*(10^(2n+1)-1)/9 + 2*10^n.

Original entry on oeis.org

4, 242, 22422, 2224222, 222242222, 22222422222, 2222224222222, 222222242222222, 22222222422222222, 2222222224222222222, 222222222242222222222, 22222222222422222222222, 2222222222224222222222222, 222222222222242222222222222, 22222222222222422222222222222, 2222222222222224222222222222222
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. A332114 .. A332194 (variants with different repeated digit 1, ..., 9).
Cf. A332120 .. A332129 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332124 := n -> 2*((10^(2*n+1)-1)/9+10^n);
  • Mathematica
    Array[2 ((10^(2 # + 1)-1)/9 + 10^#) &, 15, 0]
Table[FromDigits[Join[PadRight[{},n,2],{4},PadRight[{},n,2]]],{n,0,20}] (* or *) LinearRecurrence[{111,-1110,1000},{4,242,22422},20](* Harvey P. Dale, Mar 06 2023 *)
  • PARI
    apply( {A332124(n)=(10^(n*2+1)\9+10^n)*2}, [0..15])
  • Python
    def A332124(n): return (10**(n*2+1)//9+10**n)*2

Formula

a(n) = 2*A138148(n) + 4*10^n = A002276(2n+1) + 2*10^n = 2*A332112(n).
G.f.: (4 - 202*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.

A332134 a(n) = (10^(2n+1)-1)/3 + 10^n.

Original entry on oeis.org

4, 343, 33433, 3334333, 333343333, 33333433333, 3333334333333, 333333343333333, 33333333433333333, 3333333334333333333, 333333333343333333333, 33333333333433333333333, 3333333333334333333333333, 333333333333343333333333333, 33333333333333433333333333333, 3333333333333334333333333333333
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Comments

There are no primes in this sequence because a(n) = round(n*2/3)*(5*10^n-1).

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. A332124 .. A332194 (variants with different repeated digit 2, ..., 9).
Cf. A332130 .. A332139 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332134 := n -> (10^(2*n+1)-1)/3+10^n;
  • Mathematica
    Array[ (10^(2 # + 1)-1)/3 + 10^# &, 15, 0]
  • PARI
    apply( {A332134(n)=10^(n*2+1)\3+10^n}, [0..15])
  • Python
    def A332134(n): return 10**(n*2+1)//3+10**n

Formula

a(n) = 3*A138148(n) + 4*10^n = A002277(2n+1) + 10^n.
G.f.: (4 - 101*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.

A332154 a(n) = 5*(10^(2*n+1)-1)/9 - 10^n.

Original entry on oeis.org

4, 545, 55455, 5554555, 555545555, 55555455555, 5555554555555, 555555545555555, 55555555455555555, 5555555554555555555, 555555555545555555555, 55555555555455555555555, 5555555555554555555555555, 555555555555545555555555555, 55555555555555455555555555555, 5555555555555554555555555555555
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. A332114 .. A332194 (variants with different repeated digit 1, ..., 9).
Cf. A332150 .. A332159 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332154 := n -> 5*(10^(2*n+1)-1)/9-10^n;
  • Mathematica
    Array[5 (10^(2 # + 1)-1)/9 - 10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{4,545,55455},20] (* or *) Table[FromDigits[Join[PadRight[{},n,5],{4},PadRight[{},n,5]]],{n,0,20}] (* Harvey P. Dale, Mar 09 2025 *)
  • PARI
    apply( {A332154(n)=10^(n*2+1)\9*5-10^n}, [0..15])
  • Python
    def A332154(n): return 10**(n*2+1)//9*5-10**n

Formula

a(n) = 5*A138148(n) + 4*10^n = A002279(2n+1) - 10^n.
G.f.: (4 + 101*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.

A332164 a(n) = 6*(10^(2*n+1)-1)/9 - 2*10^n.

Original entry on oeis.org

4, 646, 66466, 6664666, 666646666, 66666466666, 6666664666666, 666666646666666, 66666666466666666, 6666666664666666666, 666666666646666666666, 66666666666466666666666, 6666666666664666666666666, 666666666666646666666666666, 66666666666666466666666666666, 6666666666666664666666666666666
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. A332114 .. A332194 (variants with different repeated digit 1, ..., 9).
Cf. A332160 .. A332169 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332164 := n -> 6*(10^(2*n+1)-1)/9-2*10^n;
  • Mathematica
    Array[6 (10^(2 # + 1)-1)/9 - 2*10^# &, 15, 0]
  • PARI
    apply( {A332164(n)=10^(n*2+1)\9*6-2*10^n}, [0..15])
  • Python
    def A332164(n): return 10**(n*2+1)//9*6-2*10**n

Formula

a(n) = 6*A138148(n) + 4*10^n = A002280(2n+1) - 2*10^n = 2*A332132(n).
G.f.: (4 + 202*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.
Showing 1-5 of 5 results.

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

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