A099898 -id:A099898 - cp's OEIS frontend

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.

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
```
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).

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).

cp's OEIS Frontend

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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