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.

Previous Showing 11-15 of 15 results.

A332193 a(n) = 10^(2n+1) - 1 - 6*10^n.

Original entry on oeis.org

3, 939, 99399, 9993999, 999939999, 99999399999, 9999993999999, 999999939999999, 99999999399999999, 9999999993999999999, 999999999939999999999, 99999999999399999999999, 9999999999993999999999999, 999999999999939999999999999, 99999999999999399999999999999, 9999999999999993999999999999999
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

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), A002113 (palindromes).
Cf. A332113 .. A332183 (variants with different repeated digit 1, ..., 8).
Cf. A332190 .. A332197, A181965 (variants with different middle digit 0, ..., 8).

Programs

  • Maple
    A332193 := n -> 10^(n*2+1)-1-6*10^n;
  • Mathematica
    Array[ 10^(2 # + 1) - 1 - 6*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{3,939,99399},20] (* Harvey P. Dale, Jan 19 2024 *)
  • PARI
    apply( {A332193(n)=10^(n*2+1)-1-6*10^n}, [0..15])
  • Python
    def A332193(n): return 10**(n*2+1)-1-6*10^n

Formula

a(n) = 9*A138148(n) + 3*10^n = A002283(2n+1) - 6*10^n.
G.f.: (3 + 606*x - 1500*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.

A332195 a(n) = 10^(2n+1) - 4*10^n - 1.

Original entry on oeis.org

5, 959, 99599, 9995999, 999959999, 99999599999, 9999995999999, 999999959999999, 99999999599999999, 9999999995999999999, 999999999959999999999, 99999999999599999999999, 9999999999995999999999999, 999999999999959999999999999, 99999999999999599999999999999, 9999999999999995999999999999999
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Comments

See A183186 = {88, 112, 198, 622, 4228, ...} for the indices of primes.

Links

Crossrefs

Cf. (A077786-1)/2 = A183186: 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. A332115 .. A332185 (variants with different repeated digit 1, ..., 8).
Cf. A332190 .. A332197, A181965 (variants with different middle digit 0, ..., 8).

Programs

  • Maple
    A332195 := n -> 10^(n*2+1)-4*10^n-1;
  • Mathematica
    Array[ 10^(2 # + 1) - 1 - 4*10^# &, 15, 0]
  • PARI
    apply( {A332195(n)=10^(n*2+1)-1-4*10^n}, [0..15])
  • Python
    def A332195(n): return 10**(n*2+1)-1-4*10^n

Formula

a(n) = 9*A138148(n) + 5*10^n.
G.f.: (5 + 404*x - 1300*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.

A332196 a(n) = 10^(2n+1) - 1 - 3*10^n.

Original entry on oeis.org

6, 969, 99699, 9996999, 999969999, 99999699999, 9999996999999, 999999969999999, 99999999699999999, 9999999996999999999, 999999999969999999999, 99999999999699999999999, 9999999999996999999999999, 999999999999969999999999999, 99999999999999699999999999999
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

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), A002113 (palindromes).
Cf. A332116 .. A332186 (variants with different repeated digit 1, ..., 8).
Cf. A332190 .. A332197, A181965 (variants with different middle digit 0, ..., 8).

Programs

  • Maple
    A332196 := n -> 10^(n*2+1)-1-3*10^n;
  • Mathematica
    Array[ 10^(2 # + 1) - 1 - 3*10^# &, 15, 0]
FromDigits/@Table[Join[PadLeft[{6},n,9],PadRight[{},n-1,9]],{n,30}] (* or *) LinearRecurrence[{111,-1110,1000},{6,969,99699},30] (* Harvey P. Dale, May 03 2021 *)
  • PARI
    apply( {A332196(n)=10^(n*2+1)-1-3*10^n}, [0..15])
  • Python
    def A332196(n): return 10**(n*2+1)-1-3*10^n

Formula

a(n) = 9*A138148(n) + 6*10^n.
G.f.: (6 + 303*x - 1200*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) - 3*exp(9*x) - 1). - Stefano Spezia, Jul 13 2024

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.

A332170 a(n) = 7*(10^(2n+1)-1)/9 - 7*10^n.

Original entry on oeis.org

0, 707, 77077, 7770777, 777707777, 77777077777, 7777770777777, 777777707777777, 77777777077777777, 7777777770777777777, 777777777707777777777, 77777777777077777777777, 7777777777770777777777777, 777777777777707777777777777, 77777777777777077777777777777, 7777777777777770777777777777777
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Links

Crossrefs

Cf. A002275 (repunits R_n = (10^n-1)/9), A002281 (7*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits only), A002113 (palindromes).
Cf. A332120 .. A332190 (variants with different repeated digit 2, ..., 9).
Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).

Programs

  • Maple
    A332170 := n -> 7*(10^(2*n+1)-1)/9-7*10^n;
  • Mathematica
    Array[7 ((10^(2 # + 1)-1)/9 - 10^#) &, 15, 0]
  • PARI
    apply( {A332170(n)=(10^(n*2+1)\9-10^n)*7}, [0..15])
  • Python
    def A332170(n): return (10**(n*2+1)//9-10^n)*7

Formula

a(n) = 7*A138148(n) = A002281(2n+1) - 7*A011557(n).
G.f.: 7*x*(101 - 200*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.
Previous Showing 11-15 of 15 results.

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