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.

Previous Showing 31-35 of 35 results.

A099886 XOR binomial transform of A099885.

Original entry on oeis.org

1, 3, 7, 9, 21, 63, 107, 189, 273, 819, 1911, 2457, 5189, 15567, 28123, 46701, 65793, 197379, 460551, 592137, 1381653, 4144959, 7039851, 12434877, 17829905, 53489715, 124809335, 160469145, 340873541, 1022620623, 1840690907
Offset: 0

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Author

Paul D. Hanna, Oct 28 2004

Keywords

Comments

Many terms of this sequence are equal to 3 times another term! Why? The XOR BINOMIAL transform of this sequence is A099885. A099885 is the central terms of the rows of the XOR difference triangle of the powers of 2 (A099884).

Crossrefs

Cf. A099884, A001317, A099885.

Programs

  • PARI
    {a(n)=my(B);B=0;for(k=0,n, for(i=0,k\2, B=bitxor(B, binomial(n,k)%2*binomial(k\2,i)%2*2^((k+1)\2+k\2-i) )));B}

Formula

a(n) = SumXOR_{k=0..n} SumXOR_{i=0..k\2} (C(n, k)mod 2) * (C(k\2, i) mod 2) * 2^((k+1)\2+k\2-i), where the SumXOR is the analog of summation under XOR and x\y stands for floor(x/y) (see Pari program).

A099887 XOR difference triangle of the powers of 3, read by rows.

Original entry on oeis.org

1, 3, 2, 9, 10, 8, 27, 18, 24, 16, 81, 74, 88, 64, 80, 243, 162, 232, 176, 240, 160, 729, 554, 648, 608, 720, 544, 640, 2187, 2642, 2168, 2800, 2192, 2624, 2144, 2784, 6561, 4394, 7032, 4864, 6640, 4448, 6944, 4928, 6560, 19683, 21826, 17512, 24336, 19472
Offset: 0

Views

Author

Paul D. Hanna, Oct 28 2004

Keywords

Comments

Main diagonal is A099888, the XOR BINOMIAL transform of the powers of 3. See A099884 for the definition of XOR BINOMIAL transform and for the definition of the XOR difference triangle.

Examples

			Rows begin:
[1],
[3,2],
[9,10,8],
[27,18,24,16],
[81,74,88,64,80],
[243,162,232,176,240,160],
[729,554,648,608,720,544,640],
[2187,2642,2168,2800,2192,2624,2144,2784],
[6561,4394,7032,4864,6640,4448,6944,4928,6560],
[19683,21826,17512,24336,19472,21984,17536,24480,19680,21824],...

Crossrefs

Cf. A099884, A047999, A099888.

Programs

  • PARI
    T(n,k)=local(B);B=0;for(i=0,k,B=bitxor(B,binomial(k,i)%2*3^(n-i)));B

Formula

T(n, k) = SumXOR_{i=0..k} (C(k, i)mod 2)*3^i, where SumXOR is the analog of summation under the binary XOR operation and C(k, i)mod 2 = A047999(k, i).

A099891 XOR difference triangle of A003188 (Gray code numbers), read by rows.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 2, 1, 3, 0, 6, 4, 5, 6, 6, 7, 1, 5, 0, 6, 0, 5, 2, 3, 6, 6, 0, 0, 4, 1, 3, 0, 6, 0, 0, 0, 12, 8, 9, 10, 10, 12, 12, 12, 12, 13, 1, 9, 0, 10, 0, 12, 0, 12, 0, 15, 2, 3, 10, 10, 0, 0, 12, 12, 0, 0, 14, 1, 3, 0, 10, 0, 0, 0, 12, 0, 0, 0, 10, 4, 5, 6, 6, 12, 12, 12, 12, 0, 0, 0, 0, 11, 1
Offset: 0

Views

Author

Paul D. Hanna, Oct 29 2004

Keywords

Comments

Main diagonal is A099892, the XOR BINOMIAL transform of A003188. See A099884 for the definitions of the XOR BINOMIAL transform and the XOR difference triangle.

Examples

			Rows begin:
[0],
[1,1],
[3,2,3],
[2,1,3,0],
[6,4,5,6,6],
[7,1,5,0,6,0],
[5,2,3,6,6,0,0],
[4,1,3,0,6,0,0,0],
[12,8,9,10,10,12,12,12,12],
...
where A003188 fills the leftmost column.

Crossrefs

Cf. A047999, A003188 (column k=0), A006519 (column k=1), A099892 (diagonal n=k).
Other triangles: A099884, A099889, A099893.

Programs

  • PARI
    {T(n,k)=local(B);B=0;for(i=0,k,B=bitxor(B,binomial(k,i)%2*(bitxor((n-i),(n-i)\2))));B}

Formula

T(n, k) = SumXOR_{i=0..k} (C(k, i)mod 2)*(A003188(n-i)), where SumXOR is the analog of summation under the binary XOR operation and C(k, i)mod 2 = A047999(k, i). T(2^n, 2^n) = 3*2^(n-1) for n>0, with T(1, 1)=1 and T(k, k)=0 elsewhere.
T(n,1) = A006519(n), the lowest 1-bit of n (see formula by Franklin T. Adams-Watters in A003188). - Kevin Ryde, Jul 02 2020

A099893 XOR BINOMIAL transform of A006068 (inverse Gray code).

Original entry on oeis.org

0, 1, 3, 0, 7, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 63, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 127
Offset: 0

Views

Author

Paul D. Hanna, Oct 29 2004

Keywords

Comments

See A099884 for the definitions of the XOR BINOMIAL transform and the XOR difference triangle.

Crossrefs

Cf. A099884, A006068, A099894.

Programs

  • PARI
    {a(n)=local(B);B=0;for(i=0,n,B=bitxor(B,binomial(n,i)%2*A006068(n-i) ));B}

Formula

a(2^n) = 2^(n+1)-1 for n>0, with a(0)=0 and a(k)=0 otherwise. a(n) = SumXOR_{i=0..n} (C(n, i)mod 2)*A006068(n-i) and SumXOR is summation under XOR.

A099895 XOR BINOMIAL transform of A000069 (Odious numbers).

Original entry on oeis.org

1, 3, 5, 0, 9, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 33, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 65, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 129
Offset: 0

Views

Author

Paul D. Hanna, Oct 29 2004

Keywords

Comments

See A099884 for the definitions of the XOR BINOMIAL transform and the XOR difference triangle.

Examples

			XOR difference triangle of A000069 begins:
[1],
[2,3],
[4,6,5],
[7,3,5,0],
[8,15,12,9,9],
[11,3,12,0,9,0],
[13,6,5,9,9,0,0],
[14,3,5,0,9,0,0,0],
[16,30,29,24,24,17,17,17,17],...
where A000069 is in the leftmost column,
and this sequence forms the main diagonal.

Crossrefs

Cf. A099884, A000069, A099896.

Programs

  • PARI
    {a(n)=local(B);B=0;for(i=0,n,B=bitxor(B,binomial(n,i)%2*A000069(n-i) ));B}

Formula

a(2^n) = 2^(n+1)+1 for n>0, with a(0)=1 and a(k)=0 otherwise. a(n) = SumXOR_{i=0..n} (C(n, i)mod 2)*A000069(n-i) and SumXOR is summation under XOR.
Previous Showing 31-35 of 35 results.

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