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A340566
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Square array, read by descending antidiagonals; T(n,k) is A001057(n) + A001057(k)*i, converted to complex binary (base -1 + i), where i=sqrt(-1).
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1
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0, 11, 1, 111, 1110, 11101, 1110100, 111010, 10, 1100, 100, 1110101, 110, 1111, 11100, 1110111, 101, 11001, 111011, 11111, 1101, 110011, 1010, 11101001, 1000, 11101011, 111010010, 10001, 1110000, 111110, 1110110, 111000, 11000, 111010110, 11110, 111010000
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OFFSET
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0,2
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COMMENTS
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Complex binary (base -1 + i) has the ability to express all positive or negative, real or complex, integers with only 2 numerical symbols ('0' and '1') as integers, without the need for a sign marking the integers as such.
Converting a real number, n, to complex binary requires one to convert it to base -4 ((n + N) xor N, N = floor(4/5*16^(ceiling(log_4(abs(n))) + 1))), then adding 10 to every digit greater than 1, then treating it as a number in base 16 and converting that to binary. (E.g., -5 => [2,3] => [12,13] => 205 => 11001101.)
Converting a complex number, n + k*i, requires one to convert X = n + k into complex binary and then convert k into the same but shift it one digit to the left. After this, one must add them together. This functions much the same way as binary addition, but the carry is '110' rather than '1' and 11 + 111 = 0.
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REFERENCES
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T. Jamil, Complex Binary Number System, Springer, 2013.
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LINKS
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EXAMPLE
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Square array T(n,k) begins:
\k 0 1 2 3 4 5 6 ...
n\
0| 0 11 111 1110100 100 1110111 110011 ...
1| 1 1110 111010 1110101 101 1010 111110 ...
2| 11101 10 110 11001 11101001 1110110 110010 ...
3| 1100 1111 111011 1000 111000 1011 111111 ...
4| 11100 11111 11101011 11000 11101000 11011 11101111 ...
5| 1101 111010010 111010110 1001 111001 1100110 100010 ...
6| 10001 11110 11101010 1110100101 10101 11010 11101110 ...
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PROG
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(PARI) A340566(n, k)={my(A001057(x)=if(x%2, x\2+1, -x/2), V=vecsum(Vec(matconcat(apply(w->my(Y=if(w, A001057(k), A001057(n)+A001057(k))); if(Y, my(X=floor(4^(2*logint(abs(Y), 4)+5)/5)); Vecrev(binary(shift(fromdigits(apply(z->z+(10*(z>1)), digits(bitxor(Y+X, X), 4)), 16), w)))), [0, 1])~)~))~);
while(vecmax(V)>1, my(Z=Vec(select(x->x>1, V, 1))); for(x=1, #Z, my(z=Z[x]); if(V[z]<=1, , (z+2<=#V)&&(V[z+1]>1)&&V[z+2], for(j=z, z+2, V[j]-=2^(j!=(z+2))), (z+4<=#V)&&vecmin(V[z+2..z+4]), V[z]-=2; for(j=z+2, z+4, V[j]-=1), z+1>#V, V[z]-=2; V=concat(V, [0, 1, 1]), V[z]-=2; for(j=z+2, z+3, if(j<=#V, V[j]+=1, V=concat(V, 1)))))); fromdigits(Vecrev(V))}
(PARI) { T(n, k) = my(z=n\/2*-(-1)^n + k\/2*-(-1)^k*I, ret=List([]));
while(z, my(bit=(real(z)+imag(z))%2);
listput(ret, bit); z=(z-bit)/(I-1));
fromdigits(Vecrev(ret)); } \\ Kevin Ryde, Jan 12 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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