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 1-10 of 16 results. Next

A332119 a(n) = (10^(2n+1)-1)/9 + 8*10^n.

Original entry on oeis.org

9, 191, 11911, 1119111, 111191111, 11111911111, 1111119111111, 111111191111111, 11111111911111111, 1111111119111111111, 111111111191111111111, 11111111111911111111111, 1111111111119111111111111, 111111111111191111111111111, 11111111111111911111111111111, 1111111111111119111111111111111
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Comments

See A107649 = {1, 4, 26, 187, 226, 874, ...} for the indices of primes.

Links

Crossrefs

Cf. (A077795-1)/2 = A107649: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332129 .. A332189 (variants with different repeated digit 2, ..., 8).
Cf. A332112 .. A332118 (variants with different middle digit 2, ..., 8).

Programs

  • Maple
    A332119 := n -> (10^(2*n+1)-1)/9+8*10^n;
  • Mathematica
    Array[(10^(2 # + 1)-1)/9 + 8*10^# &, 15, 0]
Table[FromDigits[Join[PadRight[{},n,1],{9},PadRight[{},n,1]]],{n,0,20}] (* or *) LinearRecurrence[ {111,-1110,1000},{9,191,11911},20] (* Harvey P. Dale, Mar 30 2024 *)
  • PARI
    apply( {A332119(n)=10^(n*2+1)\9+8*10^n}, [0..15])
  • Python
    def A332119(n): return 10**(n*2+1)//9+8*10**n

Formula

a(n) = A138148(n) + 9*10^n = A002275(2n+1) + 8*10^n.
G.f.: (9 - 808*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.

A332113 a(n) = (10^(2n+1)-1)/9 + 2*10^n.

Original entry on oeis.org

3, 131, 11311, 1113111, 111131111, 11111311111, 1111113111111, 111111131111111, 11111111311111111, 1111111113111111111, 111111111131111111111, 11111111111311111111111, 1111111111113111111111111, 111111111111131111111111111, 11111111111111311111111111111, 1111111111111113111111111111111
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Comments

See A107123 = {0, 1, 2, 19, 97, 9818, ...} for the indices of primes.

Links

Crossrefs

Cf. (A077779-1)/2 = A107123: indices of primes; A331864 & A331865 (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. A332123 .. A332193 (variants with different repeated digit 2, ..., 9).
Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

Programs

  • Maple
    A332113 := n -> (10^(2*n+1)-1)/9+2*10^n;
  • Mathematica
    Array[(10^(2 # + 1)-1)/9 + 2*10^# &, 15, 0]
  • PARI
    apply( {A332113(n)=10^(n*2+1)\9+2*10^n}, [0..15])
  • Python
    def A332113(n): return 10**(n*2+1)//9+2*10**n

Formula

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

A332117 a(n) = (10^(2n+1)-1)/9 + 6*10^n.

Original entry on oeis.org

7, 171, 11711, 1117111, 111171111, 11111711111, 1111117111111, 111111171111111, 11111111711111111, 1111111117111111111, 111111111171111111111, 11111111111711111111111, 1111111111117111111111111, 111111111111171111111111111, 11111111111111711111111111111, 1111111111111117111111111111111
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Comments

See A107127 = {0, 3, 33, 311, 2933, ...} for the indices of primes.

Links

Crossrefs

Cf. (A077789-1)/2 = A107127: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332127 .. A332197 (variants with different repeated digit 2, ..., 9).
Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

Programs

  • Maple
    A332117 := n -> (10^(2*n+1)-1)/9+6*10^n;
  • Mathematica
    Array[(10^(2 # + 1)-1)/9 + 6*10^# &, 15, 0]
  • PARI
    apply( {A332117(n)=10^(n*2+1)\9+6*10^n}, [0..15])
  • Python
    def A332117(n): return 10**(n*2+1)//9+6*10**n

Formula

a(n) = A138148(n) + 7*10^n = A002275(2n+1) + 6*10^n.
G.f.: (7 - 606*x + 500*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.

A332118 a(n) = (10^(2n+1) - 1)/9 + 7*10^n.

Original entry on oeis.org

8, 181, 11811, 1118111, 111181111, 11111811111, 1111118111111, 111111181111111, 11111111811111111, 1111111118111111111, 111111111181111111111, 11111111111811111111111, 1111111111118111111111111, 111111111111181111111111111, 11111111111111811111111111111, 1111111111111118111111111111111
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Comments

See A107648 = {1, 4, 6, 7, 384, 666, ...} for the indices of primes.

Links

Crossrefs

Cf. (A077791-1)/2 = A107648: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes), A077798 (palindromic wing primes), A088281 (primes 1..1x1..1), A068160 (smallest of given length), A053701 (vertically symmetric numbers).
Cf. A332128 .. A332178, A181965 (variants with different repeated digit 2, ..., 9).
Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

Programs

  • Maple
    A332118 := n -> (10^(2*n+1)-1)/9+7*10^n;
  • Mathematica
    Array[(10^(2 # + 1)-1)/9 + 7*10^# &, 15, 0]
  • PARI
    apply( {A332118(n)=10^(n*2+1)\9+7*10^n}, [0..15])
  • Python
    def A332118(n): return 10**(n*2+1)//9+7*10**n

Formula

a(n) = A138148(n) + 8*10^n = A002275(2n+1) + 7*10^n.
G.f.: (8 - 707*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.

A332192 a(n) = 10^(2n+1) - 1 - 7*10^n.

Original entry on oeis.org

2, 929, 99299, 9992999, 999929999, 99999299999, 9999992999999, 999999929999999, 99999999299999999, 9999999992999999999, 999999999929999999999, 99999999999299999999999, 9999999999992999999999999, 999999999999929999999999999, 99999999999999299999999999999, 9999999999999992999999999999999
Offset: 0

Views

Author

M. F. Hasler, Feb 08 2020

Keywords

Comments

See A115073 = {1, 8, 9, 352, 530, 697, ...} for the indices of primes.

Links

Crossrefs

Cf. (A077778-1)/2 = A115073: 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. A332190 .. A332197, A181965 (variants with different middle digit 0, ..., 8).
Cf. A332112 .. A332182 (variants with different repeated digit 1, ..., 8).

Programs

  • Maple
    A332192 := n -> 10^(n*2+1)-1-7*10^n;
  • Mathematica
    Array[ 10^(2 # +1) -1 -7*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{2,929,99299},20] (* Harvey P. Dale, Nov 07 2022 *)
  • PARI
    apply( {A332192(n)=10^(n*2+1)-1-7*10^n}, [0..15])
  • Python
    def A332192(n): return 10**(n*2+1)-1-7*10^n

Formula

a(n) = 9*A138148(n) + 2*10^n = A002283(2n+1) - 7*10^n.
G.f.: (2 + 707*x - 1600*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.

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.

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.

A332116 a(n) = (10^(2n+1)-1)/9 + 5*10^n.

Original entry on oeis.org

6, 161, 11611, 1116111, 111161111, 11111611111, 1111116111111, 111111161111111, 11111111611111111, 1111111116111111111, 111111111161111111111, 11111111111611111111111, 1111111111116111111111111, 111111111111161111111111111, 11111111111111611111111111111, 1111111111111116111111111111111
Offset: 0

Views

Author

M. F. Hasler, Feb 09 2020

Keywords

Comments

See A107126 = {10, 14, 40, 59, 160, 412, ...} for the indices of primes.

Links

Crossrefs

Cf. (A077706-1)/2 = A107126: indices of primes.
Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332126 .. A332196 (variants with different repeated digit 2, ..., 9).
Cf. A332112 .. A332119 (variants with different middle digit 2, ..., 9).

Programs

  • Maple
    A332116 := n -> (10^(2*n+1)-1)/9+5*10^n;
  • Mathematica
    Array[(10^(2 # + 1)-1)/9 + 5*10^# &, 15, 0]
  • PARI
    apply( {A332116(n)=10^(n*2+1)\9+5*10^n}, [0..15])
  • Python
    def A332116(n): return 10**(n*2+1)//9+5*10**n

Formula

a(n) = A138148(n) + 6*10^n = A002275(2n+1) + 5*10^n.
G.f.: (6 - 505*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.
E.g.f.: exp(x)*(10*exp(99*x) + 45*exp(9*x) - 1)/9. - Stefano Spezia, Jul 13 2024

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.

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.
Showing 1-10 of 16 results. Next

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