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 1-8 of 8 results.

A086574 a(n) = 3*(10^n - 1).

Original entry on oeis.org

0, 27, 297, 2997, 29997, 299997, 2999997, 29999997, 299999997, 2999999997, 29999999997, 299999999997, 2999999999997, 29999999999997, 299999999999997, 2999999999999997, 29999999999999997, 299999999999999997, 2999999999999999997, 29999999999999999997, 299999999999999999997
Offset: 0

Views

Author

Ray Chandler, Jul 22 2003

Keywords

Comments

Original definition: a(n) = k where R(k+3) = 3. [Here R = reverse = A004086, not R = repunit = A002275 as in some other places.]
The restriction to positive indices yields A083813. - M. F. Hasler, Jul 29 2016

Links

Crossrefs

Cf. A002275, A004086 (R(n)), A083813.
Cf. One of family of sequences of form a(n) = k, where R(k+m) = m, m=1 to 9; m=1: A002283, m=2: A086573, m=3: A086574, m=4: A086575, m=5: A086576, m=6: A086577, m=7: A086578, m=8: A086579, m=9: A086580.

Programs

  • Mathematica
    Join[{0},Table[FromDigits[Join[PadRight[{2},n,9],{7}]],{n,20}]] (* or *) LinearRecurrence[{11,-10},{0,27},30] (* Harvey P. Dale, Nov 28 2015 *)
  • PARI
    a(n)=10^n*3-3 \\ M. F. Hasler, Jul 29 2016

Formula

a(n) = 3*9*A002275(n) = 3*A002283(n).
R(a(n)) = A086579(n).
a(n) = 11*a(n-1) - 10*a(n-2) with a(0)=0 and a(1)=27. - Harvey P. Dale, Nov 28 2015
G.f.: 27*x/((1 - x)*(1 - 10*x)). - Ilya Gutkovskiy, Jul 29 2016
E.g.f.: 3*exp(x)*(exp(9*x) - 1). - Elmo R. Oliveira, Sep 12 2024

Extensions

Edited by M. F. Hasler, Jul 29 2016

A086573 a(n) = 2*(10^n - 1).

Original entry on oeis.org

0, 18, 198, 1998, 19998, 199998, 1999998, 19999998, 199999998, 1999999998, 19999999998, 199999999998, 1999999999998, 19999999999998, 199999999999998, 1999999999999998, 19999999999999998, 199999999999999998, 1999999999999999998, 19999999999999999998, 199999999999999999998
Offset: 0

Views

Author

Ray Chandler, Jul 22 2003

Keywords

Comments

Original definition: a(n) = k where R(k+2) = 2.

Links

Crossrefs

Cf. A002275, A004086 (R(n)), A083812.
One of family of sequences of form a(n) = k, where R(k+m) = m, m=1 to 9; m=1: A002283, m=2: A086573, m=3: A086574, m=4: A086575, m=5: A086576, m=6: A086577, m=7: A086578, m=8: A086579, m=9: A086580.

Formula

a(n) = 2*9*A002275(n) = 2*A002283(n).
R(a(n)) = A086580(n).
From Chai Wah Wu, Jul 08 2016: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
G.f.: 18*x/((1 - x)*(1 - 10*x)). (End)
E.g.f.: 2*exp(x)*(exp(9*x) - 1). - Elmo R. Oliveira, Sep 12 2024

Extensions

Edited by Jinyuan Wang, Aug 04 2021

A086578 a(n) = 7*(10^n - 1).

Original entry on oeis.org

0, 63, 693, 6993, 69993, 699993, 6999993, 69999993, 699999993, 6999999993, 69999999993, 699999999993, 6999999999993, 69999999999993, 699999999999993, 6999999999999993, 69999999999999993, 699999999999999993, 6999999999999999993, 69999999999999999993, 699999999999999999993
Offset: 0

Views

Author

Ray Chandler, Jul 22 2003

Keywords

Comments

Original definition: a(n) = k where R(k+7) = 7.

Links

Crossrefs

Cf. A002275, A004086 (R(n)).
One of family of sequences of form a(n) = k, where R(k+m) = m, m=1 to 9; m=1: A002283, m=2: A086573, m=3: A086574, m=4: A086575, m=5: A086576, m=6: A086577, m=7: A086578, m=8: A086579, m=9: A086580.
Sequences of the form m*10^n - 7: 3*A033175 (m=1, 10), A086943 (m=3), 3*A185127 (m=4), this sequence (m=7), A100412 (m=8).

Programs

  • Magma
    [7*(10^n -1): n in [0..20]]; // G. C. Greubel, Apr 14 2023
  • Mathematica
    LinearRecurrence[{11,-10}, {0,63}, 31] (* G. C. Greubel, Apr 14 2023 *)
  • SageMath
    [7*(10^n -1) for n in range(21)] # G. C. Greubel, Apr 14 2023

Formula

a(n) = 7*9*A002275(n) = 7*A002283(n).
R(a(n)) = A086575(n).
From Chai Wah Wu, Jul 08 2016: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
G.f.: 63*x/((1 - x)*(1 - 10*x)). (End)
E.g.f.: 7*(exp(10*x) - exp(x)). - G. C. Greubel, Apr 14 2023

Extensions

Edited by Jinyuan Wang, Aug 04 2021

A086575 a(n) = 4*(10^n - 1).

Original entry on oeis.org

0, 36, 396, 3996, 39996, 399996, 3999996, 39999996, 399999996, 3999999996, 39999999996, 399999999996, 3999999999996, 39999999999996, 399999999999996, 3999999999999996, 39999999999999996, 399999999999999996, 3999999999999999996, 39999999999999999996, 399999999999999999996
Offset: 0

Views

Author

Ray Chandler, Jul 22 2003

Keywords

Comments

Original definition: a(n) = k where R(k+4) = 4.

Links

Crossrefs

Cf. A002275, A004086 (R(n)), A083811.
One of family of sequences of form a(n) = k, where R(k+m) = m, m=1 to 9; m=1: A002283, m=2: A086573, m=3: A086574, m=4: A086575, m=5: A086576, m=6: A086577, m=7: A086578, m=8: A086579, m=9: A086580.

Programs

  • Mathematica
    LinearRecurrence[{11,-10},{0,36},20] (* Harvey P. Dale, May 14 2017 *)

Formula

a(n) = 4*9*A002275(n) = 4*A002283(n).
R(a(n)) = A086578(n).
From Chai Wah Wu, Jul 08 2016: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
G.f.: 36*x/((1 - x)*(1 - 10*x)). (End)
E.g.f.: 4*exp(x)*(exp(9*x) - 1). - Elmo R. Oliveira, Sep 12 2024

Extensions

Edited by Jinyuan Wang, Aug 04 2021

A086576 a(n) = 5*(10^n - 1).

Original entry on oeis.org

0, 45, 495, 4995, 49995, 499995, 4999995, 49999995, 499999995, 4999999995, 49999999995, 499999999995, 4999999999995, 49999999999995, 499999999999995, 4999999999999995, 49999999999999995, 499999999999999995, 4999999999999999995, 49999999999999999995, 499999999999999999995
Offset: 0

Views

Author

Ray Chandler, Jul 22 2003

Keywords

Comments

Original definition: a(n) = k where R(k+5) = 5.

Examples

			From _John Elias_, Jun 23 2021: (Start)
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45;
11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99 = 495;
111 + 222 + 333 + 444 + 555 + 666 + 777 + 888 + 999 = 4995;
1111 + 2222 + 3333 + 4444 + 5555 + 6666 + 7777 + 8888 + 9999 = 49995;
11111 + 22222 + 33333 + 44444 + 55555 + 66666 + 77777 + 88888 + 99999 = 499995; etc. (End)

Links

Crossrefs

Cf. A002275, A004086 (R(n)).
Cf. One of family of sequences of form a(n)=k, where R(k+m)=m, m = 1 to 9; m=1: A002283, m=2: A086573, m=3: A086574, m=4: A086575, m=5: A086576, m=6: A086577, m=7: A086578, m=8: A086579, m=9: A086580.

Programs

  • Mathematica
    Join[{0}, LinearRecurrence[{11,-10},{45,495},50]] (* G. C. Greubel, Jul 08 2016 *)

Formula

a(n) = 5*9*A002275(n) = 5*A002283(n).
R(a(n)) = A086577(n).
From Chai Wah Wu, Jul 08 2016: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
G.f.: 45*x/((1 - x)*(1 - 10*x)). (End)
E.g.f.: 5*exp(x)*(exp(9*x) - 1). - Elmo R. Oliveira, Sep 12 2024

Extensions

Name edited by Jinyuan Wang, Aug 04 2021

A086577 a(n) = 6*(10^n - 1).

Original entry on oeis.org

0, 54, 594, 5994, 59994, 599994, 5999994, 59999994, 599999994, 5999999994, 59999999994, 599999999994, 5999999999994, 59999999999994, 599999999999994, 5999999999999994, 59999999999999994, 599999999999999994, 5999999999999999994, 59999999999999999994, 599999999999999999994
Offset: 0

Views

Author

Ray Chandler, Jul 22 2003

Keywords

Comments

Original definition: a(n) = k where R(k+6) = 6.

Links

Crossrefs

Cf. A002275, A004086 (R(n)).
One of a family of sequences of the form "Numbers k such that reverse(k+m) = m", m = 1 to 9; m=1: A002283, m=2: A086573, m=3: A086574, m=4: A086575, m=5: A086576, m=6: A086577, m=7: A086578, m=8: A086579, m=9: A086580.

Programs

  • Mathematica
    LinearRecurrence[{11,-10},{0,54},30] (* Harvey P. Dale, Nov 27 2022 *)

Formula

a(n) = 6*9*A002275(n) = 6*A002283(n).
R(a(n)) = A086576(n).
From Chai Wah Wu, Jul 08 2016: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
G.f.: 54*x/((1 - x)*(1 - 10*x)). (End)
E.g.f.: 6*exp(x)*(exp(9*x) - 1). - Elmo R. Oliveira, Sep 12 2024

Extensions

Name edited by Jinyuan Wang, Aug 04 2021

A086579 a(n) = 8*(10^n - 1).

Original entry on oeis.org

0, 72, 792, 7992, 79992, 799992, 7999992, 79999992, 799999992, 7999999992, 79999999992, 799999999992, 7999999999992, 79999999999992, 799999999999992, 7999999999999992, 79999999999999992, 799999999999999992, 7999999999999999992, 79999999999999999992, 799999999999999999992
Offset: 0

Views

Author

Ray Chandler, Jul 22 2003

Keywords

Comments

Original definition: a(n) = k where R(k+8) = 8.

Links

Crossrefs

Cf. A002275, A004086 (R(n)).
One of family of sequences of form a(n) = k, where R(k+m) = m, m=1 to 9; m=1: A002283, m=2: A086573, m=3: A086574, m=4: A086575, m=5: A086576, m=6: A086577, m=7: A086578, m=8: A086579, m=9: A086580.

Programs

  • Mathematica
    LinearRecurrence[{11,-10},{0,72},30] (* Harvey P. Dale, Mar 22 2025 *)

Formula

a(n) = 8*9*A002275(n) = 8*A002283(n).
R(a(n)) = A086574(n).
From Chai Wah Wu, Jul 08 2016: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
G.f.: 72*x/((1 - x)*(1 - 10*x)). (End)
E.g.f.: 8*exp(x)*(exp(9*x) - 1). - Elmo R. Oliveira, Sep 12 2024

Extensions

Edited by Jinyuan Wang, Aug 04 2021

A328683 Positive integers that are equal to 99...99 (repdigit with n digits 9) times the sum of their digits.

Original entry on oeis.org

81, 1782, 26973, 359964, 4499955, 53999946, 629999937, 7199999928, 80999999919, 899999999910, 9899999999901, 107999999999892, 1169999999999883, 12599999999999874, 134999999999999865, 1439999999999999856, 15299999999999999847, 161999999999999999838
Offset: 1

Views

Author

Bernard Schott, Feb 25 2020

Keywords

Comments

The idea of this sequence comes from a problem during the annual Moscow Mathematical Olympiad (MMO) in 2001 (see reference).

Examples

			359964 = 36 * 9999 and the digital sum of 359964 = 36 , so 359964 = a(4).

References

  • Roman Fedorov, Alexei Belov, Alexander Kovaldzhi, Ivan Yashchenko, Moscow Mathematical Olympiads, 2000-2005, Level B, Problem 5, 2001, MSRI, 2011, p. 8 and 70/71.

Links

Crossrefs

Cf. A057147, A086580, A110807.

Programs

  • Maple
    C:=seq(9*n*(10^n-1),n=1..20);
  • Mathematica
    Table[9*n*(10^n - 1), {n, 1, 18}] (* Amiram Eldar, Feb 25 2020 *)
LinearRecurrence[{22,-141,220,-100},{81,1782,26973,359964},20] (* Harvey P. Dale, Feb 02 2025 *)
  • PARI
    Vec(81*x*(1 - 10*x^2) / ((1 - x)^2*(1 - 10*x)^2) + O(x^20)) \\ Colin Barker, Feb 25 2020

Formula

a(n) = 9 * n * (10^n - 1).
From Colin Barker, Feb 25 2020: (Start)
G.f.: 81*x*(1 - 10*x^2) / ((1 - x)^2*(1 - 10*x)^2).
a(n) = 22*a(n-1) - 141*a(n-2) + 220*a(n-3) - 100*a(n-4) for n>4.
(End)
From Michel Marcus, Feb 25 2020: (Start)
a(n) = 9*A110807(n).
a(n) = n*A086580(n). (End)
Showing 1-8 of 8 results.

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