A101120 -id:A101120 - cp's OEIS frontend

A101121 Bitwise XOR of adjacent terms of A101120; also the nonzero terms of A101122.

7, 17, 34, 68, 159, 257, 514, 1028, 2063, 4097, 8194, 16388, 32831, 65537, 131074, 262148, 524303, 1048577, 2097154, 4194308, 8388639, 16777217, 33554434, 67108868, 134217743, 268435457, 536870914, 1073741828, 2147483775

Offset: 1

Simon Plouffe and Paul D. Hanna, Dec 02 2004

nonn

A101120 gives the records in A101119, which equals the nonzero differences of A006519 and A003484. A101122 is the XOR BINOMIAL transform of A101119 and has zeros everywhere except at positions equal to powers of 2.

			a(5) = 159 since A101120(4)=112, A101120(5)=239 and 159 = 112 XOR 239.

Cf. A003484, A006519, A101119, A101120, A101122.

{a(n)=bitxor(2^(n+2)-2^((n-2)%4)-8*((n+2)\4),2^(n+3)-2^((n-1)%4)-8*((n+3)\4))}

def A101121(n): return ((1<<(n+2))-(1<<((n-2)&3))-(((n+2)&-4)<<1))^((1<<(n+3))-(1<<((n-1)&3))-(((n+3)&-4)<<1)) # Chai Wah Wu, Jul 10 2022

a(n) = A101120(n-1) XOR A101120(n) for n>1, with a(1) = A101120(1), where A101120(n) = 2^(n+3) - 2^((n-1)(Mod 4)) - 8*floor((n+3)/4).

A101119 Nonzero differences of A006519 (highest power of 2 dividing n) and A003484 (Radon function).

Original entry on oeis.org

7, 22, 7, 52, 7, 22, 7, 112, 7, 22, 7, 52, 7, 22, 7, 239, 7, 22, 7, 52, 7, 22, 7, 112, 7, 22, 7, 52, 7, 22, 7, 494, 7, 22, 7, 52, 7, 22, 7, 112, 7, 22, 7, 52, 7, 22, 7, 239, 7, 22, 7, 52, 7, 22, 7, 112, 7, 22, 7, 52, 7, 22, 7, 1004, 7, 22, 7, 52, 7, 22, 7, 112, 7, 22, 7, 52, 7, 22, 7, 239

Offset: 1

Simon Plouffe and Paul D. Hanna, Dec 02 2004

nonn

A006519 and A003484 differ only at every 16th term; this sequence forms the nonzero differences. Records form A101120. Equals the XOR BINOMIAL transform of A101122.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003484, A006519, A101120, A101122.

[2^Valuation(16*n,2) - 8*Floor(Valuation(16*n,2)/4) - 2^(Valuation(16*n,2) mod 4): n in [1..50]]; // G. C. Greubel, Nov 01 2018

Table[2^(IntegerExponent[16*n, 2]) - 8*Floor[IntegerExponent[16*n, 2]/4] - 2^(Mod[IntegerExponent[16*n, 2], 4]), {n, 1, 50}] (* G. C. Greubel, Nov 01 2018 *)

{a(n)=2^valuation(16*n,2)-(8*(valuation(16*n,2)\4)+2^(valuation(16*n,2)%4))}

def A101119(n): return (1<<(m:=(~n&n-1).bit_length()+4))-((m&-4)<<1)-(1<<(m&3)) # Chai Wah Wu, Jul 10 2022

a(n) = A006519(16*n) - A003484(16*n) for n>=1. a(2*n-1) = 7 for n>=1.

A101122 XOR BINOMIAL transform of A101119.

Original entry on oeis.org

7, 17, 0, 34, 0, 0, 0, 68, 0, 0, 0, 0, 0, 0, 0, 159, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 257, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 514, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Simon Plouffe and Paul D. Hanna, Dec 02 2004

nonn

Nonzero terms form A101121 and occur at positions 2^k for k >= 0. A101119 equals the nonzero differences of A006519 and A003484. See A099884 for the definition of the XOR BINOMIAL transform.

Cf. A003484, A006519, A101119, A101120, A101121.

{a(n)=local(B);B=0;for(i=0,n-1,B=bitxor(B,binomial(n-1,i)%2* (16*2^valuation(n-i,2)-2^(valuation(n-i,2)%4)-8*(valuation(n-i,2)\4)-8)));B}

from operator import xor
from functools import reduce
def A101122(n): return reduce(xor,(((1<<(m:=(~(k+1)&k).bit_length()+4))-((m&-4)<<1)-(1<<(m&3)))&-int(not k&~(n-1)) for k in range(n))) # Chai Wah Wu, Jul 10 2022

a(n) = SumXOR_{k=0..n} (C(n, k) mod 2)*A101119(k), where SumXOR is summation under XOR. A101119(n) = SumXOR_{k=0..n} (C(n, k) mod 2)*a(k). a(2^(n-1)) = A101121(n) for n >= 1 and a(k)=0 when k is not a power of 2.

cp's OEIS Frontend

A101121 Bitwise XOR of adjacent terms of A101120; also the nonzero terms of A101122.

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A101119 Nonzero differences of A006519 (highest power of 2 dividing n) and A003484 (Radon function).

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A101122 XOR BINOMIAL transform of A101119.

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