A099899 -id:A099899 - cp's OEIS frontend

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

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).

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
```
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).

cp's OEIS Frontend

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

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