A099884 -id:A099884 - cp's OEIS frontend

A380112 Lexicographically earliest infinite sequence of positive integers whose XOR difference triangle contains only distinct values.

1, 2, 4, 8, 16, 32, 9, 18, 64, 128, 39, 75, 156, 256, 76, 137, 269, 407, 512, 180, 78, 606, 432, 1024, 63, 771, 1037, 604, 789, 1144, 2048, 31, 564, 1661, 772, 2176, 1286, 2044, 3105, 1638, 377, 2606, 4096, 662, 1857, 4124, 536, 1463, 4188, 2242, 6453, 5302

Offset: 1

Rémy Sigrist, Jan 12 2025

By necessity, all terms are distinct.

This sequence is an infinite variant of A378141.

Rémy Sigrist, PARI program

Cf. A099884, A346298, A338047, A378141, A380148 (XOR difference triangle).

PARI
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A099888 XOR BINOMIAL transform of the powers of 3.

Original entry on oeis.org

1, 2, 8, 16, 80, 160, 640, 2784, 6560, 21824, 65280, 131072, 524800, 1056192, 4201728, 8408000, 43046720, 86135424, 361186816, 1467183104, 3443838080, 11367226112, 32550397696, 70111011712, 282403304960, 569106797952

Offset: 0

Paul D. Hanna, Oct 29 2004

nonn

The diagonal of the XOR difference triangle A099887. The XOR BINOMIAL transform of this sequence returns the powers of 3. See A099884 for the definitions of the XOR BINOMIAL transform and the XOR difference triangle.

Paul D. Hanna, Table of n, a(n) for n = 0..1024

Cf. A099884, A099887.

a(n)=local(B);B=0;for(i=0,n,B=bitxor(B,binomial(n,i)%2*3^(n-i)));B

a(n) = SumXOR_{k=0..n} (C(n, k) mod 2)*3^k, where SumXOR is summation under XOR.

A099890 XOR BINOMIAL transform of the odd numbers; also the main diagonal of the XOR difference triangle A099889.

Original entry on oeis.org

1, 2, 4, 0, 8, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 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, 128

Offset: 0

Paul D. Hanna, Oct 29 2004

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1024

Cf. A099884, A099889.

Join[{1},Table[If[IntegerQ[Log2[n]],2n,0],{n,512}]] (* or *) Join[{1},With[{kk=10},Flatten[ Riffle[ 2^Range[kk],Table[PadRight[{},2^k-1,0],{k,0,kk}]]]]] (* Harvey P. Dale, Aug 11 2024 *)

a(n)=local(B);B=0;for(i=0,n,B=bitxor(B,binomial(n,i)%2*(2*(n-i)+1)));B

a(0) = 1; a(2^n) = 2^(n+1); a(n) = 0 otherwise.

A099892 XOR BINOMIAL transform of A003188 (Gray code numbers); also the main diagonal of the XOR difference triangle A099891.

Original entry on oeis.org

0, 1, 3, 0, 6, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 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, 96, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul D. Hanna, Oct 29 2004

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

Cf. A099884, A003188, A099891.

a[n_] := Module[{e = IntegerExponent[n, 2]}, Switch[n, 0, 0, 1, 1, 2^e, 3*2^(e - 1), , 0]]; Array[a, 100, 0] (* _Amiram Eldar, Aug 31 2023, corrected by Michael Shamos, May 22 2025 *)

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

a(2^n) = 3*2^(n-1) for n>0, with a(0)=0, a(1) = 1 and a(k)=0 otherwise. a(n) = SumXOR_{i=0..n} (C(n, i)mod 2)*A003188(n-i), where A003188(k)=bitxor(k, [k/2]) and SumXOR is summation under XOR.
Multiplicative with a(2^e) = 3*2^(e-1), a(p^e) = 0 otherwise. - David W. Wilson, Jun 12 2005
Dirichlet g.f.: (2^s+1)/(2^s-2). - R. J. Mathar, Apr 14 2011

A099897 XOR difference triangle, read by rows, of A099898 (in leftmost column) such that the main diagonal equals A099898 shift left and divided by 4.

Original entry on oeis.org

1, 4, 5, 20, 16, 21, 84, 64, 80, 69, 276, 320, 256, 336, 277, 1108, 1344, 1024, 1280, 1104, 1349, 5396, 4416, 5120, 4096, 5376, 4432, 5141, 20564, 17728, 21504, 16384, 20480, 17664, 21584, 16453, 65812, 86336, 70656, 81920, 65536, 86016, 70912

Offset: 0

Paul D. Hanna, Oct 30 2004

Central terms of rows equal powers of 4: T(n,[n/2]) = 4^n for n>=0. The leftmost column is A099898. The diagonal forms A099899 and equals the XOR BINOMIAL transform of A099898. See A099884 for the definitions of XOR difference triangle and the XOR BINOMIAL transform.

			Rows begin:
[_1],
[_4,5],
[20,_16,21],
[84,_64,80,69],
[276,320,_256,336,277],
[1108,1344,_1024,1280,1104,1349],
[5396,4416,5120,_4096,5376,4432,5141],
[20564,17728,21504,_16384,20480,17664,21584,16453],
[65812,86336,70656,81920,_65536,86016,70912,82256,65813],...
notice that the column terms equal 4 times the diagonal (with offset), and that the central terms in the rows form the powers of 4.

Cf. A099884, A099898, A099899, A099900.

PARI
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T(n,k)=if(n
				
```

T(n, [n/2]) = 4^n. T(n+1, 0) = 4*T(n, n) (n>=0); T(0, 0)=1; T(n, k) = T(n, k-1) XOR T(n-1, k-1) for n>k>0. T(n, k) = SumXOR_{i=0..k} (C(k, i)mod 2)*T(n-i, 0), where SumXOR is the analog of summation under the binary XOR operation and C(k, i)mod 2 = A047999(k, i).

A099898 Shifts left and divides by 4 under the XOR BINOMIAL transform (A099899).

Original entry on oeis.org

1, 4, 20, 84, 276, 1108, 5396, 20564, 65812, 263252, 1316116, 5525588, 18153748, 72352852, 352326932, 1342197844, 4295033108, 17180132436, 85900662036, 360782778452, 1185429127444, 4758896116820, 23175995856148, 88323049672788

Offset: 0

Paul D. Hanna, Oct 30 2004

nonn

Equals the XOR BINOMIAL transform of A099899. Also, equals the leftmost column of the XOR difference triangle A099897, in which the central terms of the rows forms the powers of 4. See A099884 for the definitions of XOR difference triangle and the XOR BINOMIAL transform.

Cf. A099884, A099897, A099899.

{a(n)=local(B);B=0;if(n==0,B=1, for(k=0,n-1, B=bitxor(B,binomial(n-1-k+k\2,k\2)%2*4^(k+1))));B}

a(0)=1; a(n) = SumXOR_{k=0..n-1} (C(n-1-k+[k/2], [k/2])mod 2)*4^(k+1) for n>0. a(n) = SumXOR_{i=0..n} (C(n, i)mod 2)*A099899(n-i), where SumXOR is the analog of summation under the binary XOR operation and C(k, i)mod 2 = A047999(k, i).

A099899 Multiplies by 4 and shifts right under the XOR BINOMIAL transform (A099898).

Original entry on oeis.org

1, 5, 21, 69, 277, 1349, 5141, 16453, 65813, 329029, 1381397, 4538437, 18088213, 88081733, 335549461, 1073758277, 4295033109, 21475165509, 90195694613, 296357281861, 1189724029205, 5793998964037, 22080762418197, 70666170679365

Offset: 0

Paul D. Hanna, Oct 30 2004

nonn

Equals the XOR BINOMIAL transform of A099898. Also, equals the main diagonal of the XOR difference triangle A099897, in which the central terms of the rows form the powers of 4. See A099884 for the definitions of XOR difference triangle and the XOR BINOMIAL transform.

Cf. A099884, A099897, A099898.

{a(n)=local(B);B=0;for(k=0,n,B=bitxor(B,binomial(n-k+k\2,k\2)%2*4^k));B}

a(n) = SumXOR_{k=0..n} (C(n-k+[k/2], [k/2])mod 2)*2^k for n>=0. a(n) = SumXOR_{i=0..n} (C(n, i)mod 2)*A099898(n-i), where SumXOR is the analog of summation under the binary XOR operation and C(i, j)mod 2 = A047999(i, j).

A099901 Shifts left and divides by 2 under the XOR BINOMIAL transform (A099902).

Original entry on oeis.org

1, 2, 6, 14, 22, 46, 118, 206, 278, 558, 1654, 3790, 5910, 11310, 28790, 49358, 65814, 131630, 394870, 921294, 1447702, 3025966, 7762038, 13549774, 18284822, 36438574, 108004982, 247467726, 385881878, 738208814, 1879076982, 3221274830

Offset: 0

Paul D. Hanna, Oct 30 2004

Equals the XOR BINOMIAL transform of A099902. Also, equals the leftmost column of the XOR difference triangle A099900, in which the central terms of the rows forms the powers of 2.

Cf. A099884, A099900, A099902.

{a(n)=local(B);B=0;if(n==0,B=1, for(k=0,n-1, B=bitxor(B,binomial(n-1-k+k\2,k\2)%2*2^(k+1))));B}

a(0)=1; a(n) = SumXOR_{k=0..n-1} (C(n-1-k+[k/2], [k/2])mod 2)*2^(k+1) for n>0. a(n) = SumXOR_{i=0..n} (C(n, i)mod 2)*A099902(n-i), where SumXOR is the analog of summation under the binary XOR operation and C(k, i)mod 2 = A047999(k, i).

A378141 For any n > 0, a(n) is the least positive integer such that the XOR difference triangle with bottom row (a(1), ..., a(n)) has distinct values.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 9, 18, 64, 128, 39, 75, 156, 256, 76, 137, 259

Offset: 1

Rémy Sigrist, Jan 09 2025

By necessity, all terms are distinct.

This sequence is finite; see A380112 for the infinite variant.

			The XOR difference triangle for the whole sequence is:
                                    258
                                  465 211
                                171 378 425
                              367 452 190 279
                            205 418 102 216 463
                          163 110 460 426 370 189
                         98 193 175 355 201 443 262
                      195 161  96 207 428 357 222 472
                     65 130  35  67 140 288  69 155 323
                   36 101 231 196 135  11 299 366 501 182
                 28  56  93 186 126 249 242 473 183 322 500
               51  47  23  74 240 142 119 133 348 491 169 349
             17  34  13  26  80 160  46  89 220 384 107 194 415
           15  30  60  49  43 123 219 245 172 112 496 411 345 198
         5   10  20  40  25  50  73 146 103 203 187 331 208 393 335
       3   6   12  24  48  41  27  82 192 167 108 215 412 332 197 394
     1   2   4   8   16  32  9   18  64 128  39  75 156 256  76 137 259

Rémy Sigrist, PARI program

Cf. A099884, A380112.

PARI
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\\ See Links section.
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A380148 Triangle T(n, k), n > 0, k = 1..n, read by rows; T(n, 1) = A380112(n), and for any k in 2..n, T(n, k) = T(n, k-1) XOR T(n-1, k-1) (where XOR denotes the bitwise XOR operator).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 12, 10, 15, 16, 24, 20, 30, 17, 32, 48, 40, 60, 34, 51, 9, 41, 25, 49, 13, 47, 28, 18, 27, 50, 43, 26, 23, 56, 36, 64, 82, 73, 123, 80, 74, 93, 101, 65, 128, 192, 146, 219, 160, 240, 186, 231, 130, 195, 39, 167, 103, 245, 46, 142, 126, 196, 35, 161, 98

Offset: 1

Rémy Sigrist, Jan 13 2025

This sequence corresponds to the XOR difference triangle associated with A380112. All terms are distinct.

			Triangle T(n, k) begins:
                        1
                      2   3
                    4   6   5
                  8   12  10  15
                16  24  20  30  17
              32  48  40  60  34  51
            9   41  25  49  13  47  28
          18  27  50  43  26  23  56  36
        64  82  73 123  80  74  93 101  65
     128 192 146 219 160 240 186 231 130 195

Rémy Sigrist, PARI program

Cf. A099884, A380112.

PARI
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\\ See Links section.
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cp's OEIS Frontend

A380112 Lexicographically earliest infinite sequence of positive integers whose XOR difference triangle contains only distinct values.

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A099888 XOR BINOMIAL transform of the powers of 3.

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Formula

A099890 XOR BINOMIAL transform of the odd numbers; also the main diagonal of the XOR difference triangle A099889.

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Mathematica

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A099892 XOR BINOMIAL transform of A003188 (Gray code numbers); also the main diagonal of the XOR difference triangle A099891.

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Mathematica

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A099897 XOR difference triangle, read by rows, of A099898 (in leftmost column) such that the main diagonal equals A099898 shift left and divided by 4.

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A099898 Shifts left and divides by 4 under the XOR BINOMIAL transform (A099899).

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A099899 Multiplies by 4 and shifts right under the XOR BINOMIAL transform (A099898).

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A099901 Shifts left and divides by 2 under the XOR BINOMIAL transform (A099902).

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A378141 For any n > 0, a(n) is the least positive integer such that the XOR difference triangle with bottom row (a(1), ..., a(n)) has distinct values.

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