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.

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A338502 Lexicographically earliest sequence of distinct nonnegative integers such that for any n > 0, a(1) XOR ... XOR a(n) is a square (where XOR denotes the bitwise XOR operator).

Original entry on oeis.org

0, 1, 5, 4, 9, 8, 17, 16, 25, 24, 37, 21, 33, 20, 13, 45, 32, 29, 40, 48, 65, 36, 53, 72, 61, 52, 80, 57, 93, 64, 85, 49, 81, 88, 56, 96, 117, 100, 68, 125, 105, 116, 101, 120, 112, 73, 89, 137, 84, 109, 141, 180, 113, 133, 160, 132, 161, 152, 121, 144, 128
Offset: 1

Views

Author

Rémy Sigrist, Oct 31 2020

Keywords

Comments

All terms belong to A042948.

Examples

			The first terms, alongside a(1) XOR ... XOR a(n), are:
  n   a(n)  a(1) AND ... AND a(n)
  --  ----  ---------------------
   1     0                0 = 0^2
   2     1                1 = 1^2
   3     5                4 = 2^2
   4     4                0 = 0^2
   5     9                9 = 3^2
   6     8                1 = 1^2
   7    17               16 = 4^2
   8    16                0 = 0^2
   9    25               25 = 5^2
  10    24                1 = 1^2
  11    37               36 = 6^2
  12    21               49 = 7^2

Links

Crossrefs

Cf. A042948, A042964, A292388 (prime variant), A338503.

Programs

  • PARI
    See Links section.

A377333 Positive integers k such that there does not exist a fully symmetric k-celled polycube, i.e., such that A376971(k) = 0.

Original entry on oeis.org

2, 3, 4, 5, 6, 9, 10, 11, 12, 14, 15, 16, 17, 21, 22, 23, 28, 29, 34, 35, 40, 41, 46, 47, 52, 53, 58, 59, 65, 70, 71, 77
Offset: 1

Views

Author

Pontus von Brömssen, Oct 25 2024

Keywords

Links

Crossrefs

Complement of A377332.
Cf. A042964 (corresponding sequence for polyominoes), A376971, A377337.

A147623 The 3rd Witt transform of A040000.

Original entry on oeis.org

0, 2, 6, 12, 22, 34, 48, 66, 86, 108, 134, 162, 192, 226, 262, 300, 342, 386, 432, 482, 534, 588, 646, 706, 768, 834, 902, 972, 1046, 1122, 1200, 1282, 1366, 1452, 1542, 1634, 1728, 1826, 1926, 2028, 2134, 2242, 2352, 2466, 2582, 2700, 2822, 2946, 3072
Offset: 0

Views

Author

R. J. Mathar, Nov 08 2008

Keywords

Comments

The 2nd Witt transform of A040000 is represented by A042964.

Links

Crossrefs

Cf. A040000, A042964, A049347, A071619.

Programs

  • Magma
    [n le 2 select 1+(-1)^n else 4*(1+(n-2)^2) - Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Oct 24 2022
  • Mathematica
    CoefficientList[Series[2x(1+x)(1 +x^2)/((1-x)^3 (1+x+x^2)), {x,0,40}], x] (* Vincenzo Librandi, Dec 14 2012 *)
LinearRecurrence[{2,-1,1,-2,1},{0,2,6,12,22},50] (* Harvey P. Dale, Jul 04 2021 *)
  • SageMath
    [2*(2*(1+3*n^2) -(2*chebyshev_U(n, -1/2) +chebyshev_U(n-1, -1/2)))/9 for n in range(41)] # G. C. Greubel, Oct 24 2022

Formula

G.f.: 2*x*(1+x)*(1+x^2)/((1-x)^3*(1+x+x^2)).
a(n) = 2*A071619(n).
From G. C. Greubel, Oct 24 2022: (Start)
a(n) = 4*(2 - 2*n + n^2) - a(n-1) - a(n-2).
a(n) = 2*(2*(1 + 3*n^2) - (2*A049347(n) + A049347(n-1)))/9. (End)

A317613 Permutation of the nonnegative integers: lodumo_4 of A047247.

Original entry on oeis.org

2, 3, 0, 1, 4, 5, 6, 7, 10, 11, 8, 9, 12, 13, 14, 15, 18, 19, 16, 17, 20, 21, 22, 23, 26, 27, 24, 25, 28, 29, 30, 31, 34, 35, 32, 33, 36, 37, 38, 39, 42, 43, 40, 41, 44, 45, 46, 47, 50, 51, 48, 49, 52, 53, 54, 55, 58, 59, 56, 57, 60, 61, 62, 63, 66, 67, 64
Offset: 0

Views

Author

Franck Maminirina Ramaharo, Aug 01 2018

Keywords

Comments

Write n in base 8, then apply the following substitution to the rightmost digit: '0'->'2, '1'->'3', and vice versa. Convert back to decimal.
A self-inverse permutation: a(a(n)) = n.
Array whose columns are, in this order, A047463, A047621, A047451 and A047522, read by rows.

Examples

			a(25) = a('3'1') = '3'3' = 27.
a(26) = a('3'2') = '3'0' = 24.
a(27) = a('3'3') = '3'1' = 25.
a(28) = a('3'4') = '3'4' = 28.
a(29) = a('3'5') = '3'5' = 29.
The sequence as array read by rows:
  A047463, A047621, A047451, A047522;
        2,       3,       0,       1;
        4,       5,       6,       7;
       10,      11,       8,       9;
       12,      13,      14,      15;
       18,      19,      16,      17;
       20,      21,      22,      23;
       26,      27,      24,      25;
       28,      29,      30,      31;
  ...

Links

Crossrefs

Cf. A047225, A047243, A047257, A064429, A080412, A159959, A026185.

Programs

  • Magma
    m:=100; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((x^7+x^5+3*x^3-2*x^2-x+2)/((1-x)^2*(x^6+x^4+ x^2+1)))); // G. C. Greubel, Sep 25 2018
  • Mathematica
    Table[(4*(Floor[1/4 Mod[2*n + 4, 8]] - Floor[1/4 Mod[n + 2, 8]]) + 2*n)/2, {n, 0, 100}]
f[n_] := Block[{id = IntegerDigits[n, 8]}, FromDigits[ Join[Most@ id /. {{} -> {0}}, {id[[-1]] /. {0 -> 2, 1 -> 3, 2 -> 0, 3 -> 1}}], 8]]; Array[f, 67, 0] (* or *)
CoefficientList[ Series[(x^7 + x^5 + 3x^3 - 2x^2 - x + 2)/((x - 1)^2 (x^6 + x^4 + x^2 + 1)), {x, 0, 70}], x] (* or *)
LinearRecurrence[{2, -2, 2, -2, 2, -2, 2, -1}, {2, 3, 0, 1, 4, 5, 6, 7}, 70] (* Robert G. Wilson v, Aug 01 2018 *)
  • Maxima
    makelist((4*(floor(mod(2*n + 4, 8)/4) - floor(mod(n + 2, 8)/4)) + 2*n)/2, n, 0, 100);
  • PARI
    my(x='x+O('x^100)); Vec((x^7+x^5+3*x^3-2*x^2-x+2)/((1-x)^2*(x^6+x^4+ x^2+1))) \\ G. C. Greubel, Sep 25 2018

Formula

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - 2*a(n-6) + 2*a(n-7) - a(n-8), n > 7.
a(n) = (4*(floor(((2*n + 4) mod 8)/4) - floor(((n + 2) mod 8)/4)) + 2*n)/2.
a(n) = lod_4(A047247(n+1)).
a(4*n) = A047463(n+1).
a(4*n+1) = A047621(n+1).
a(4*n+2) = A047451(n+1).
a(4*n+3) = A047522(n+1).
a(A042948(n)) = A047596(n+1).
a(A042964(n+1)) = A047551(n+1).
G.f.: (x^7 + x^5 + 3*x^3 - 2*x^2 - x + 2)/((x-1)^2 * (x^2+1) * (x^4+1)).
E.g.f.: x*exp(x) + cos(x) + sin(x) + cos(x/sqrt(2))*cosh(x/sqrt(2)) + (sqrt(2)*cos(x/sqrt(2)) - sin(x/sqrt(2)))*sinh(x/sqrt(2)).
a(n+8) = a(n) + 8 . - Philippe Deléham, Mar 09 2023
Sum_{n>=3} (-1)^(n+1)/a(n) = 1/6 + log(2). - Amiram Eldar, Mar 12 2023
Previous Showing 31-34 of 34 results.

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