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A099889 XOR difference triangle of the odd numbers, read by rows. 3
1, 3, 2, 5, 6, 4, 7, 2, 4, 0, 9, 14, 12, 8, 8, 11, 2, 12, 0, 8, 0, 13, 6, 4, 8, 8, 0, 0, 15, 2, 4, 0, 8, 0, 0, 0, 17, 30, 28, 24, 24, 16, 16, 16, 16, 19, 2, 28, 0, 24, 0, 16, 0, 16, 0, 21, 6, 4, 24, 24, 0, 0, 16, 16, 0, 0, 23, 2, 4, 0, 24, 0, 0, 0, 16, 0, 0, 0, 25, 14, 12, 8, 8, 16, 16, 16, 16, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Main diagonal is A099890, the XOR BINOMIAL transform of the odd numbers. See A099884 for the definitions of the XOR BINOMIAL transform and the XOR difference triangle.
LINKS
FORMULA
T(n, k) = SumXOR_{i=0..k} (C(k, i)mod 2)*(2*(n-i)+1), 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) = 2^(n+1) for n>=0, with T(0, 0)=1.
EXAMPLE
Rows begin:
[1],
[3,2],
[5,6,4],
[7,2,4,0],
[9,14,12,8,8],
[11,2,12,0,8,0],
[13,6,4,8,8,0,0],
[15,2,4,0,8,0,0,0],
[17,30,28,24,24,16,16,16,16],...
MATHEMATICA
mx = 14; Flatten@Table[NestList[BitXor @@@ Transpose[{Most@#, Rest@#}] &, Range[1, 2 mx, 2], mx][[k, n - k]], {n, 2, mx}, {k, n - 1}] (* Ivan Neretin, Sep 01 2016 *)
PROG
(PARI) T(n, k)=local(B); B=0; for(i=0, k, B=bitxor(B, binomial(k, i)%2*(2*(n-i)+1))); B
CROSSREFS
Sequence in context: A341521 A227192 A360260 * A115511 A303768 A182801
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Oct 29 2004
STATUS
approved

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Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)