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.

A325493 a(n) = Sum_{k=1..n} A325910(k) * 10^(2^(k-1)).

Original entry on oeis.org

0, 10, 1010, 11011010, 1111001011011010, 11111111000011011111001011011010, 1111111111111111000000001111001011111111000011011111001011011010
Offset: 0

Views

Author

Seiichi Manyama, Sep 14 2019

Keywords

Links

Crossrefs

Cf. A309597, A325910.

Programs

  • PARI
    {a(n) = sum(i=1, n, 10^2^(i-1)*((-1)^(i-1)*sum(j=0, i-1, (-1)^j*10^2^j)-(1-(-1)^i)/2)/9)}

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)).

A325906 a(n) = ( (-1)^n * Sum_{k=0..n-2} (-1)^k*10^(2^k) + 10^(2^(n-1)) - ((-1)^n+3)/2 )/9.

Original entry on oeis.org

1, 12, 1121, 11112212, 1111111122221121, 11111111111111112222222211112212, 1111111111111111111111111111111122222222222222221111111122221121
Offset: 1

Views

Author

Seiichi Manyama, Sep 08 2019

Keywords

Examples

			n |       a(n)       |    A325910(n)
--+------------------+-----------------
1 |                1 |                1
2 |               12 |               10
3 |             1121 |             1101
4 |         11112212 |         11110010
5 | 1111111122221121 | 1111111100001101

Links

Crossrefs

Cf. A325907, A325910.

Programs

  • Mathematica
    a[n_] := ((-1)^n * Sum[(-1)^k * 10^(2^k), {k, 0, n - 2}] + 10^(2^(n - 1)) - ((-1)^n + 3)/2)/9; Array[a, 7] (* Amiram Eldar, May 07 2021 *)
  • PARI
    {a(n) = ((-1)^n*sum(k=0, n-2, (-1)^k*10^2^k)+10^2^(n-1)-((-1)^n+3)/2)/9}

A325912 a(n) = (-1)^n * Sum_{k=0..n} (-1)^k*2^(2^k).

Original entry on oeis.org

2, 2, 14, 242, 65294, 4294902002, 18446744069414649614, 340282366920938463444927863362353561842, 115792089237316195423570985008687907852929702298719625576012656144550776078094
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2019

Keywords

Examples

			a(0) = 2^1 = 2.
a(1) = 2^2  - 2^1 = 2.
a(2) = 2^4  - 2^2 + 2^1 = 14.
a(3) = 2^8  - 2^4 + 2^2 - 2^1 = 242.
a(4) = 2^16 - 2^8 + 2^4 - 2^2 + 2^1 = 65294.

Links

Crossrefs

Cf. A001146, A060803, A325910.

Programs

  • Mathematica
    a[n_] := (-1)^n * Sum[(-1)^k * 2^(2^k), {k, 0, n}]; Array[a, 9, 0] (* Amiram Eldar, May 07 2021 *)
  • PARI
    {a(n) = (-1)^n*sum(k=0,n,(-1)^k*2^2^k)}

Formula

a(n) = A001146(n) - a(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 1-6 of 6 results.

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

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