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 1-3 of 3 results.

A093143 Expansion of (1-5*x)/(1-10*x).

Original entry on oeis.org

1, 5, 50, 500, 5000, 50000, 500000, 5000000, 50000000, 500000000, 5000000000, 50000000000, 500000000000, 5000000000000, 50000000000000, 500000000000000, 5000000000000000, 50000000000000000, 500000000000000000, 5000000000000000000, 50000000000000000000, 500000000000000000000
Offset: 0

Views

Author

Paul Barry, Mar 24 2004

Keywords

Comments

Partial sums are A093142. A convex combination of 10^n and 0^n.
a(n) is the number of compositions of even natural numbers in n parts <= 9 (0 is counted as a part); also the number of ways of placing of an even number of indistinguishable objects into n distinguishable boxes with the condition that at most 9 objects can be placed in each box. - Adi Dani, May 17 2011
See an A246057 comment with a reference for the k-family satisfying a so-called curious cubic identity involving A246057(k-1), a(k) and A002277(k). - Wolfdieter Lang, Feb 07 2017

Examples

			From _Adi Dani_, May 17 2011: (Start)
a(2)=50: there are 50 compositions of even numbers into 2 parts <= 9:
(0,0);
(0,2),(2,0),(1,1);
(0,4),(4,0),(1,3),(3,1),(2,2);
(0,6),(6,0),(1,5),(5,1),(2,4),(4,2),(3,3);
(0,8),(8,0),(1,7),(7,1),(2,6),(6,2),(3,5),(5,3),(4,4);
(1,9),(9,1),(2,8),(8,2),(3,7),(7,3),(4,6),(6,4),(5,5);
(3,9),(9,3),(4,8),(8,4),(5,7),(7,5),(6,6);
(5,9),(9,5),(6,8),(8,6),(7,7);
(7,9),(9,7),(8,8);
(9,9).
(End)
Curious cubic identities (see a comment above): 1^3 + 5^3 + 3^3 = 153, 16^3 + 50^3 + 33^3 = 165033, 166^3 + 500^3 + 333^3 = 166500333, ... - _Wolfdieter Lang_, Feb 07 2017

Links

Crossrefs

Cf. A002277, A055372, A093142, A134309, A246057.

Programs

  • Mathematica
    Table[Ceiling[1/2*10^n],{n,0,30}] (* Adi Dani, Jun 20 2011 *)
Join[{1},NestList[10#&,5,20]] (* Harvey P. Dale, Apr 10 2021 *)
  • PARI
    Vec((1-5*x)/(1-10*x) + O(x^100)) \\ Altug Alkan, Nov 01 2015

Formula

a(n) = 5*10^n/10 for n > 0.
a(n) = Sum_{k=0..n} A134309(n,k)*5^k = Sum_{k=0..n} A055372(n,k)*4^k. - Philippe Deléham, Feb 04 2012
From Elmo R. Oliveira, Aug 21 2024: (Start)
E.g.f.: (exp(10*x) + 1)/2.
a(n) = 10*a(n-1) for n > 1. (End)

Extensions

a(19)-a(21) from Elmo R. Oliveira, Aug 21 2024

A309597 a(n) is the A325907(n)-th triangular number.

Original entry on oeis.org

6, 666, 5656566, 555665666566566, 5555555666655656666556566566566, 555555555555555666666665555665666666666555566566666556566566566
Offset: 1

Views

Author

Seiichi Manyama, Sep 14 2019

Keywords

Comments

a(n) decimal expansion includes A141023(n-1) 5's and A052950(n) 6's in digits.
All terms are elements of A213516.

Examples

			a(1) =               6 =               6 +        0 +    0 * 10^1.
a(2) =             666 =             556 +       10 +    1 * 10^2.
a(3) =         5656566 =         5555556 +     1010 +   10 * 10^4.
a(4) = 555665666566566 = 555555555555556 + 11011010 + 1101 * 10^8.
------------------------------------------------------------------
a(2) =                                 6 6 6. (            3 6's)
                                       - -
a(3) =                           5 65 65  66. ( 3 5's and  4 6's)
                                 - -- --
a(4) =                  555 6656 6656   6566. ( 6 5's and  9 6's)
                        --- ---- ----
a(5) = 5555555 66665565 66665565    66566566. (15 5's and 16 6's)
       ------- -------- --------

Links

Crossrefs

Cf. A000217, A052950, A093142, A141023, A213516, A325907, A325910, A325493.

Programs

  • Ruby
    def A325907(n)
  a = [3]
  (2..n).each{|i|
    j = 10 ** (2 ** (i - 2))
    a << (j + 3) * (j - 1) / 3 - a[-1]
  }
  a
end
def A309597(n)
  A325907(n).map{|i| i * (i + 1) / 2}
end
p A309597(10)

Formula

a(n) = A000217(A325907(n)).
a(n) = A093142(2^n - 1) + A325493(n-1) + A325910(n-1) * 10^(2^(n-1)).

A003555 Sum_{i=1..(10^n - 1)/9} i, or ((10^n -1)/9)*((10^n -1)/9 +1)/2 (n-th term is the middle 2(n-1) digits of the (n+9)-th term for n > 1).

Original entry on oeis.org

1, 66, 6216, 617716, 61732716, 6172882716, 617284382716, 61728399382716, 6172839549382716, 617283951049382716, 61728395066049382716, 6172839506216049382716, 617283950617716049382716, 61728395061732716049382716, 6172839506172882716049382716
Offset: 1

Views

Author

Daniel Lawson (dlawson(AT)cats.ucsc.edu)

Keywords

Comments

Patterned, or almost palindromic, numbers obtained by modification of "half-repdigit" numbers. a[n]=A*(((100^n-1)/9)+B*(10^n - 1)/9)/18, where A=1, B=8-A=7. For example, n = 20: a[20] = 1111111111111111111188888888888888888888/18 = 61728395061728395066049382716049382716, an "almost-palindromic-number". - Labos Elemer, Oct 28 2004
n is "almost palindromic" if digitlist-Rev[digitlist], besides zeros, contains +1 or -1 in "regular" positions. - Labos Elemer, Oct 28 2004

Links

Crossrefs

Cf. A002275, A093142, A098210, A099638.

Programs

  • Mathematica
    f[x_] := 1*((100^x-1)/9) + 7*(10^x-1)/9 Table[f[w], {w, 1, 20}]/18 (* Labos Elemer, Oct 28 2004 *)
  • PARI
    Vec(x*(-1+45*x)/((x-1)*(100*x-1)*(10*x-1))+O(x^99)) \\ Charles R Greathouse IV, Jun 23 2020

Formula

a(n) = A002275(n) * A093142(n).
G.f.: x*(-1+45*x) / ( (x-1)*(100*x-1)*(10*x-1) ). - R. J. Mathar, Mar 10 2011

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 31 2007
Showing 1-3 of 3 results.

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