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.

A332194 a(n) = 10^(2n+1) - 1 - 5*10^n.

Original entry on oeis.org

4, 949, 99499, 9994999, 999949999, 99999499999, 9999994999999, 999999949999999, 99999999499999999, 9999999994999999999, 999999999949999999999, 99999999999499999999999, 9999999999994999999999999, 999999999999949999999999999, 99999999999999499999999999999, 9999999999999994999999999999999
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Comments

See A183185 = {14, 22, 36, 104, 1136, ...} for the indices of primes.

Links

Crossrefs

Cf. (A077782-1)/2 = A183185: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A002283 (9*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332114 .. A332184 (variants with different repeated digit 1, ..., 8).
Cf. A332190 .. A332197, A181965 (variants with different middle digit 0, ..., 8).

Programs

  • Maple
    A332194 := n -> 10^(n*2+1)-1-5*10^n;
  • Mathematica
    Array[ 10^(2 # + 1) -1 -5*10^# &, 15, 0]
  • PARI
    apply( {A332194(n)=10^(n*2+1)-1-5*10^n}, [0..15])
  • Python
    def A332194(n): return 10**(n*2+1)-1-5*10^n

Formula

a(n) = 9*A138148(n) + 4*10^n = A002283(2n+1) - 5*A011557(n).
G.f.: (4 + 505*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.

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.

A332128 a(n) = 2*(10^(2n+1)-1)/9 + 6*10^n.

Original entry on oeis.org

8, 282, 22822, 2228222, 222282222, 22222822222, 2222228222222, 222222282222222, 22222222822222222, 2222222228222222222, 222222222282222222222, 22222222222822222222222, 2222222222228222222222222, 222222222222282222222222222, 22222222222222822222222222222, 2222222222222228222222222222222
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. A332118 .. A332178, A181965 (variants with different repeated digit 1, ..., 9).
Cf. A332120 .. A332129 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332128 := n -> 2*(10^(2*n+1)-1)/9+6*10^n;
  • Mathematica
    Array[2 (10^(2 # + 1)-1)/9 + 6*10^# &, 15, 0]
  • PARI
    apply( {A332128(n)=10^(n*2+1)\9*2+6*10^n}, [0..15])
  • Python
    def A332128(n): return 10**(n*2+1)//9*2+6*10**n

Formula

a(n) = 2*A138148(n) + 8*10^n = A002276(2n+1) + 6*10^n = 2*A332114(n).
G.f.: (8 - 606*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.

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.

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