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 21-29 of 29 results.

A332143 a(n) = 4*(10^(2*n+1)-1)/9 - 10^n.

Original entry on oeis.org

3, 434, 44344, 4443444, 444434444, 44444344444, 4444443444444, 444444434444444, 44444444344444444, 4444444443444444444, 444444444434444444444, 44444444444344444444444, 4444444444443444444444444, 444444444444434444444444444, 44444444444444344444444444444, 4444444444444443444444444444444
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. A332113 .. A332193 (variants with different repeated digit 1, ..., 9).
Cf. A332140 .. A332149 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332143 := n -> 4*(10^(2*n+1)-1)/9-10^n;
  • Mathematica
    Array[4 (10^(2 # + 1)-1)/9 - 10^# &, 15, 0]
  • PARI
    apply( {A332143(n)=10^(n*2+1)\9*4-10^n}, [0..15])
  • Python
    def A332143(n): return 10**(n*2+1)//9*4-10**n

Formula

a(n) = 4*A138148(n) + 3*10^n = A002278(2n+1) - 10^n.
G.f.: (3 + 101*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.

A332148 a(n) = 4*(10^(2*n+1)-1)/9 + 4*10^n.

Original entry on oeis.org

8, 484, 44844, 4448444, 444484444, 44444844444, 4444448444444, 444444484444444, 44444444844444444, 4444444448444444444, 444444444484444444444, 44444444444844444444444, 4444444444448444444444444, 444444444444484444444444444, 44444444444444844444444444444, 4444444444444448444444444444444
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. A332118 .. A332178, A181965 (variants with different repeated digit 1, ..., 9).
Cf. A332140 .. A332149 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332148 := n -> 4*((10^(2*n+1)-1)/9+10^n);
  • Mathematica
    Array[4 ((10^(2 # + 1)-1)/9 + 10^#) &, 15, 0]
  • PARI
    apply( {A332148(n)=(10^(n*2+1)\9+10^n)*4}, [0..15])
  • Python
    def A332148(n): return (10**(n*2+1)//9+10**n)*4

Formula

a(n) = 4*A138148(n) + 8*10^n = A002278(2n+1) + 4*10^n = 4*A332112(n).
G.f.: (8 - 404*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.

A365644 Array read by ascending antidiagonals: A(n, k) = k*(10^n - 1)/9 with k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 11, 2, 0, 0, 111, 22, 3, 0, 0, 1111, 222, 33, 4, 0, 0, 11111, 2222, 333, 44, 5, 0, 0, 111111, 22222, 3333, 444, 55, 6, 0, 0, 1111111, 222222, 33333, 4444, 555, 66, 7, 0, 0, 11111111, 2222222, 333333, 44444, 5555, 666, 77, 8, 0
Offset: 0

Views

Author

Stefano Spezia, Sep 14 2023

Keywords

Examples

			The array begins:
  0,     0,     0,     0,     0,     0, ...
  0,     1,     2,     3,     4,     5, ...
  0,    11,    22,    33,    44,    55, ...
  0,   111,   222,   333,   444,   555, ...
  0,  1111,  2222,  3333,  4444,  5555, ...
  0, 11111, 22222, 33333, 44444, 55555, ...
  ...

Crossrefs

Cf. A000004 (n=0 or k=0), A001477 (n=1), A002275 (k=1), A002276 (k=2), A002277 (k=3), A002278 (k=4), A002279 (k=5), A002280 (k=6), A002281 (k=7), A002282 (k=8), A002283 (k=9), A008593 (n=2), A053422 (main diagonal), A105279 (k=10), A132583, A177769 (n=3), A365645 (antidiagonal sums), A365646.

Programs

  • Mathematica
    A[n_,k_]:=k(10^n-1)/9; Table[A[n-k,k],{n,0,9},{k,0,n}]//Flatten

Formula

O.g.f.: x*y/((1 - x)*(1 - 10*x)*(1 - y)^2).
E.g.f.: y*exp(x+y)*(exp(9*x) - 1)/9.
A(n, 11) = A132583(n-1) for n > 0.
A(n, 12) = A073551(n+1) for n > 0.

A099658 a(n) is the smallest prime greater than 4(10^n - 1)/9.

Original entry on oeis.org

2, 5, 47, 449, 4447, 44449, 444449, 4444469, 44444453, 444444457, 4444444447, 44444444497, 444444444461, 4444444444493, 44444444444459, 444444444444461, 4444444444444463, 44444444444444461, 444444444444444469, 4444444444444444537, 44444444444444444447
Offset: 0

Views

Author

Labos Elemer, Nov 17 2004

Keywords

Comments

a(n) = smallest prime > A002278(n).

Examples

			n=4: 44 is followed by 47.

Crossrefs

Cf. A003618, A003617, A096497, A096498, A068104, A002275-A002283, A099656-A099668.

Programs

  • Mathematica
    Table[NextPrime[4*(10^n-1)/9], {n, 0, 35}]

Extensions

Checked by N. J. A. Sloane, Jan 27 2007
Mathematica program edited by Harvey P. Dale, Jul 16 2024

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.

A332145 a(n) = 4*(10^(2*n+1)-1)/9 + 10^n.

Original entry on oeis.org

5, 454, 44544, 4445444, 444454444, 44444544444, 4444445444444, 444444454444444, 44444444544444444, 4444444445444444444, 444444444454444444444, 44444444444544444444444, 4444444444445444444444444, 444444444444454444444444444, 44444444444444544444444444444, 4444444444444445444444444444444
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. A332115 .. A332195 (variants with different repeated digit 1, ..., 9).
Cf. A332140 .. A332149 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332145 := n -> 4*(10^(2*n+1)-1)/9+10^n;
  • Mathematica
    Array[4 (10^(2 # + 1)-1)/9 + 10^# &, 15, 0]
  • PARI
    apply( {A332145(n)=10^(n*2+1)\9*4+10^n}, [0..15])
  • Python
    def A332145(n): return 10**(n*2+1)//9*4+10**n

Formula

a(n) = 4*A138148(n) + 5*10^n = A002278(2n+1) + 10^n.
G.f.: (5 - 101*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.

A332146 a(n) = 4*(10^(2*n+1)-1)/9 + 2*10^n.

Original entry on oeis.org

6, 464, 44644, 4446444, 444464444, 44444644444, 4444446444444, 444444464444444, 44444444644444444, 4444444446444444444, 444444444464444444444, 44444444444644444444444, 4444444444446444444444444, 444444444444464444444444444, 44444444444444644444444444444, 4444444444444446444444444444444
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. A332116 .. A332196 (variants with different repeated digit 2, ..., 9).
Cf. A332140 .. A332149 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332146 := 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( {A332146(n)=10^(n*2+1)\9*4+2*10^n}, [0..15])
  • Python
    def A332146(n): return 10**(n*2+1)//9*4+2*10**n

Formula

a(n) = 4*A138148(n) + 6*10^n = A002278(2n+1) + 2*10^n = 2*A332123(n).
G.f.: (6 - 202*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.

A332147 a(n) = 4*(10^(2*n+1)-1)/9 + 3*10^n.

Original entry on oeis.org

7, 474, 44744, 4447444, 444474444, 44444744444, 4444447444444, 444444474444444, 44444444744444444, 4444444447444444444, 444444444474444444444, 44444444444744444444444, 4444444444447444444444444, 444444444444474444444444444, 44444444444444744444444444444, 4444444444444447444444444444444
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. A332117 .. A332197 (variants with different repeated digit 1, ..., 9).
Cf. A332140 .. A332149 (variants with different middle digit 0, ..., 9).

Programs

  • Maple
    A332147 := n -> 4*(10^(2*n+1)-1)/9+3*10^n;
  • Mathematica
    Array[4 (10^(2 # + 1)-1)/9 + 3*10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{7,474,44744},20] (* Harvey P. Dale, Mar 08 2022 *)
  • PARI
    apply( {A332147(n)=10^(n*2+1)\9*4+3*10^n}, [0..15])
  • Python
    def A332147(n): return 10**(n*2+1)//9*4+3*10**n

Formula

a(n) = 4*A138148(n) + 7*10^n = A002278(2n+1) + 3*10^n.
G.f.: (7 - 303*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.

A205084 a(n) = n 4's sandwiched between two 1's.

Original entry on oeis.org

11, 141, 1441, 14441, 144441, 1444441, 14444441, 144444441, 1444444441, 14444444441, 144444444441, 1444444444441, 14444444444441, 144444444444441, 1444444444444441, 14444444444444441, 144444444444444441, 1444444444444444441, 14444444444444444441
Offset: 0

Views

Author

José María Grau Ribas, Jan 22 2012

Keywords

Links

Crossrefs

Cf. A002278.

Programs

  • Mathematica
    a[0]=11;a[n_]:=a[n-1]*10+31;Table[a[n],{n,0,44}]
  • PARI
    a(n)=(130*10^n-31)/9 \\ Charles R Greathouse IV, Jan 23 2012

Formula

a(0)=11, a(n) = 10*a(n-1) + 31.
a(n) = (130*10^n - 31)/9 (see PARI's code by Charles R Greathouse IV).
From Elmo R. Oliveira, Feb 18 2025: (Start)
G.f.: (11 + 20*x)/((1 - x)*(1 - 10*x)).
E.g.f.: exp(x)*(130*exp(9*x) - 31)/9.
a(n) = 11*a(n-1) - 10*a(n-2). (End)
Previous Showing 21-29 of 29 results.

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

Content is available under The OEIS End-User License Agreement.