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Search: id:A099897
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| A099897 |
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XOR difference triangle, read by rows, of A099898 (in left-most column) such that the main diagonal equals A099898 shift left and divided by 4. |
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+0
3
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| 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
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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Central terms of rows equal powers of 4: T(n,[n/2]) = 4^n for n>=0. The left-most 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.
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FORMULA
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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 analogue of summation under the binary XOR operation and C(k, i)mod 2 = A047999(k, i).
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EXAMPLE
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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.
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PROGRAM
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(PARI) {T(n, k)=if(n<k|k<0, 0, if(k==0, if(n==0, 1, 4*T(n-1, n-1)), bitxor(T(n, k-1), T(n-1, k-1))); )}
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CROSSREFS
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Cf. A099884, A099898, A099899, A099900.
Sequence in context: A032319 A041255 A042835 this_sequence A050251 A125995 A080610
Adjacent sequences: A099894 A099895 A099896 this_sequence A099898 A099899 A099900
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Oct 30 2004
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