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 1-6 of 6 results.

A099675 Partial sums of repdigits of A002282.

Original entry on oeis.org

8, 96, 984, 9872, 98760, 987648, 9876536, 98765424, 987654312, 9876543200, 98765432088, 987654320976, 9876543209864, 98765432098752, 987654320987640, 9876543209876528, 98765432098765416, 987654320987654304, 9876543209876543192, 98765432098765432080, 987654320987654320968
Offset: 1

Views

Author

Labos Elemer, Nov 17 2004

Keywords

Examples

			8 + 88 + 888 + 8888 + 88888 = a(5) = 98760.

Links

Crossrefs

Cf. A002275-A002283, A014824, A057932, A099669-A099675.

Programs

  • Mathematica
    <Robert G. Wilson v, Nov 20 2004 *)

Formula

a(n) = (8/81)*(10^(n+1) - 9*n - 10). - R. Piyo (nagoya314(AT)yahoo.com), Dec 10 2004
a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3). - Wesley Ivan Hurt, Jan 20 2024
From Elmo R. Oliveira, Apr 02 2025: (Start)
G.f.: 8*x/((1 - x)^2*(1 - 10*x)).
E.g.f.: 8*exp(x)*(10*exp(9*x) - 9*x - 10)/81.
a(n) = 8*A014824(n). (End)

Extensions

More terms from Elmo R. Oliveira, Apr 02 2025

A099674 Partial sums of repdigits of A002281.

Original entry on oeis.org

0, 7, 84, 861, 8638, 86415, 864192, 8641969, 86419746, 864197523, 8641975300, 86419753077, 864197530854, 8641975308631, 86419753086408, 864197530864185, 8641975308641962, 86419753086419739, 864197530864197516, 8641975308641975293, 86419753086419753070, 864197530864197530847
Offset: 0

Views

Author

Labos Elemer, Nov 17 2004

Keywords

Examples

			7 + 77 + 777 + 7777 + 77777 = a(5) = 86415.

Links

Crossrefs

Cf. A002275-A002283, A014824, A057932, A099669-A099675.

Programs

  • Mathematica
    <Robert G. Wilson v, Nov 20 2004 *)
Accumulate[LinearRecurrence[{11,-10},{0,7},25]] (* Harvey P. Dale, Jul 22 2025 *)

Formula

a(n) = (7/81)*(10^(n+1) - 9*n - 10). - R. Piyo (nagoya314(AT)yahoo.com), Dec 10 2004
From Elmo R. Oliveira, Apr 02 2025: (Start)
G.f.: 7*x/((1 - x)^2*(1 - 10*x)).
E.g.f.: 7*exp(x)*(10*exp(9*x) - 9*x - 10)/81.
a(n) = 7*A014824(n).
a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3) for n > 3. (End)

Extensions

More terms from Elmo R. Oliveira, Apr 02 2025
a(0)=0 prepended by Harvey P. Dale, Jul 22 2025

A099676 Partial sums of repdigits of A002283.

Original entry on oeis.org

9, 108, 1107, 11106, 111105, 1111104, 11111103, 111111102, 1111111101, 11111111100, 111111111099, 1111111111098, 11111111111097, 111111111111096, 1111111111111095, 11111111111111094, 111111111111111093, 1111111111111111092, 11111111111111111091
Offset: 1

Views

Author

Labos Elemer, Nov 17 2004

Keywords

Comments

a(n) is the maximal positive integer k such that the sequence 1, 2, 3, 4, ..., k-1, k has a total of n*k digits. - Bui Quang Tuan, Mar 12 2015

Examples

			9 + 99 + 999 + 9999 + 99999 = a(5) = 111105.

Links

Crossrefs

Cf. A057932, A002275-A002283, A099669-A099674.

Programs

  • Magma
    [(10/9)*(10^n-1)-n: n in [1..20]]; // Vincenzo Librandi, Mar 14 2014
  • Maple
    a:=n->sum((10^(n-j)-1^(n-j)), j=0..n): seq(a(n), n=1..17); # Zerinvary Lajos, Jan 15 2007
  • Mathematica
    <Vincenzo Librandi, Mar 14 2014 *)
LinearRecurrence[{12,-21,10},{9,108,1107},20] (* Harvey P. Dale, Apr 18 2015 *)
  • PARI
    Vec(-9*x/((x-1)^2*(10*x-1)) + O(x^100)) \\ Colin Barker, Mar 12 2014
  • Sage
    [gaussian_binomial(n,1,10)-n for n in range(2,19)] # Zerinvary Lajos, May 29 2009

Formula

a(n) = (10/9)*(10^n-1) - n. - R. Piyo (nagoya314(AT)yahoo.com), Dec 10 2004
From Colin Barker, Mar 12 2014: (Start)
a(n) = 12*a(n-1)-21*a(n-2)+10*a(n-3).
G.f.: -9*x / ((x-1)^2*(10*x-1)). (End)
E.g.f.: exp(x)*(10*(exp(9*x) - 1) - 9*x)/9. - Stefano Spezia, Sep 13 2023

A099672 Partial sums of repdigits of A002279.

Original entry on oeis.org

5, 60, 615, 6170, 61725, 617280, 6172835, 61728390, 617283945, 6172839500, 61728395055, 617283950610, 6172839506165, 61728395061720, 617283950617275, 6172839506172830, 61728395061728385, 617283950617283940, 6172839506172839495, 61728395061728395050, 617283950617283950605
Offset: 1

Views

Author

Labos Elemer, Nov 17 2004

Keywords

Examples

			5 + 55 + 555 + 5555 + 55555 = a(5) = 61725.

Links

Crossrefs

Cf. A002275-A002283, A014824, A057932, A099669-A099675.

Programs

  • Mathematica
    <Robert G. Wilson v, Nov 20 2004 *)
Accumulate[Table[FromDigits[PadRight[{},n,5]],{n,0,20}]] (* Harvey P. Dale, Oct 05 2013 *)
  • PARI
    Vec(5*x/((1 - x)^2*(1 - 10*x)) + O(x^40)) \\ Colin Barker, Nov 30 2017

Formula

a(n) = (5/81)*(10^(n+1) - 9*n - 10). - R. Piyo (nagoya314(AT)yahoo.com), Dec 10 2004.
From Colin Barker, Nov 30 2017: (Start)
G.f.: 5*x/((1 - x)^2*(1 - 10*x)).
a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3) for n > 3. (End)
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: 5*exp(x)*(10*exp(9*x) - 9*x - 10)/81.
a(n) = 5*A014824(n). (End)

A099663 a(n) is the largest prime before A002276(n).

Original entry on oeis.org

19, 211, 2221, 22193, 222199, 2222219, 22222199, 222222193, 2222222137, 22222222189, 222222222169, 2222222222197, 22222222222201, 222222222222151, 2222222222222203, 22222222222222153, 222222222222222221, 2222222222222222177, 22222222222222222169, 222222222222222222149, 2222222222222222222161
Offset: 2

Views

Author

Labos Elemer, Nov 17 2004

Keywords

Examples

			n=2: 19 is before 22.

Crossrefs

Cf. A007917, A002276.
Cf. A003618, A003617, A096497, A096498, A068104, A002275-A002283, A099656-A099669.

Programs

  • Mathematica
    Table[NextPrime[2(10^n-1)/9, -1], {n, 2, 35}]
Drop[NextPrime[#,-1]&/@LinearRecurrence[{11,-10},{0,2},20],2] (* Harvey P. Dale, Dec 19 2020 *)

Formula

a(n) = A007917(A002276(n)). - Michel Marcus, Jun 29 2025

Extensions

More terms from Michel Marcus, Jun 29 2025

A099673 Partial sums of repdigits of A002280.

Original entry on oeis.org

6, 72, 738, 7404, 74070, 740736, 7407402, 74074068, 740740734, 7407407400, 74074074066, 740740740732, 7407407407398, 74074074074064, 740740740740730, 7407407407407396, 74074074074074062, 740740740740740728, 7407407407407407394, 74074074074074074060, 740740740740740740726
Offset: 1

Views

Author

Labos Elemer, Nov 17 2004

Keywords

Examples

			6 + 66 + 666 + 6666 + 66666 = a(5) = 74070.

Links

Crossrefs

Cf. A002275-A002283, A014824, A057932, A099669-A099675.

Programs

  • Mathematica
    <Robert G. Wilson v, Nov 20 2004 *)

Formula

a(n) = (2/27)*(10^(n+1) - 9*n - 10). - R. Piyo (nagoya314(AT)yahoo.com), Dec 10 2004
From Elmo R. Oliveira, Apr 02 2025: (Start)
G.f.: 6*x/((1 - x)^2*(1 - 10*x)).
a(n) = 6*A014824(n).
E.g.f.: 2*exp(x)*(10*exp(9*x) - 9*x - 10)/27.
a(n) = 12*a(n-1) - 21*a(n-2) + 10*a(n-3) for n > 3. (End)

Extensions

More terms from Elmo R. Oliveira, Apr 02 2025
Showing 1-6 of 6 results.

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