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.

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)).

A327266 Product of A325907(n) and its 9's complement.

Original entry on oeis.org

18, 2268, 22316868, 2222332266866868, 22222222333322316666886866866868, 2222222222222222333333332222332266666666888866866666886866866868
Offset: 1

Views

Author

Seiichi Manyama, Sep 15 2019

Keywords

Examples

			a(1) =        3 *        6 =         18.
a(2) =       63 *       36 =        2268.
a(3) =     3363 *     6636 =      22316868.
a(4) = 66663363 * 33336636 =  2222332266866868.
-----------------------------------------------
a(1) =        18        =        18        - 2 *        0 +    0 * 10^1.
a(2) =       2268       =       2188       - 2 *       10 +    1 * 10^2.
a(3) =     22316868     =     22218888     - 2 *     1010 +   10 * 10^4.
a(4) = 2222332266866868 = 2222222188888888 - 2 * 11011010 + 1101 * 10^8.

Crossrefs

Cf. A002283, A084021, A178630, A325907, A325910, A325493, A327294.

Programs

  • Ruby
    def A(n)
  a = [3, 6]
  b = ([[3]] + (1..n - 1).map{|i| [a[i % 2]] * (2 ** (i - 1))}).reverse.join.to_i
  b * (10 ** (2 ** (n - 1)) - 1 - b)
end
def A327266(n)
  (1..n).map{|i| A(i)}
end
p A327266(6)

Formula

a(n) = A084021(A325907(n)) = A325907(n) * (A002283(2^(n-1)) - A325907(n)).
a(n) = A327294(n) - 10^(2^(n-1)) = a(n) = (2 * 10^(2^n) - 3 * 10^(2^(n-1)) - 8)/9 - 2 * A325493(n-1) + A325910(n-1) * 10^(2^(n-1)).

A327294 a(n) = (A325907(n) + 1) * (10^(2^(n-1)) - A325907(n)).

Original entry on oeis.org

28, 2368, 22326868, 2222332366866868, 22222222333322326666886866866868, 2222222222222222333333332222332366666666888866866666886866866868
Offset: 1

Views

Author

Seiichi Manyama, Sep 16 2019

Keywords

Comments

a(n) is composed of digits {2,3,6,8}.

Examples

			a(1) =                2 * 10^1  +                8.
a(2) =               23 * 10^2  +               68.
a(3) =             2232 * 10^4  +             6868.
a(4) =         22223323 * 10^8  +         66866868.
a(5) = 2222222233332232 * 10^16 + 6666886866866868.
And
                      2 = 2 * (10^1  - 1)/9 +        0.
                     23 = 2 * (10^2  - 1)/9 +        1.
                   2232 = 2 * (10^4  - 1)/9 +       10.
               22223323 = 2 * (10^8  - 1)/9 +     1101.
       2222222233332232 = 2 * (10^16 - 1)/9 + 11110010.
And
                      8 = 8 * (10^1  - 1)/9 - 2 *                0.
                     68 = 8 * (10^2  - 1)/9 - 2 *               10.
                   6868 = 8 * (10^4  - 1)/9 - 2 *             1010.
               66866868 = 8 * (10^8  - 1)/9 - 2 *         11011010.
       6666886866866868 = 8 * (10^16 - 1)/9 - 2 * 1111001011011010.

Crossrefs

Cf. A325907, A325910, A325493, A327266.

Programs

  • Ruby
    def A(n)
  a = [3, 6]
  b = ([[3]] + (1..n - 1).map{|i| [a[i % 2]] * (2 ** (i - 1))}).reverse.join.to_i
  (b + 1) * (10 ** (2 ** (n - 1)) - b)
end
def A327294(n)
  (1..n).map{|i| A(i)}
end
p A327294(6)

Formula

a(n) = 2 * (10^(2^n) + 3 * 10^(2^(n-1)) - 4)/9 - 2 * A325493(n-1) + A325910(n-1) * 10^(2^(n-1)).
Showing 1-3 of 3 results.

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