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A099894 XOR BINOMIAL transform of A038712. 3
1, 2, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 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, 64, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

See A099884 for the definitions of the XOR BINOMIAL transform and the XOR difference triangle.

a(n) = A062383(n+1) - A062383(n). - Reinhard Zumkeller, Aug 06 2009

A038712 has offset 1, but we need to use offset 0 for the XOR BINOMIAL. - Michael Somos, Dec 30 2016

LINKS

Table of n, a(n) for n=0..85.

FORMULA

a(2^n-1) = 2^n for n>=0 and a(k)=0 otherwise. a(n) = SumXOR_{i=0..n} (C(n, i)mod 2)*A038712(n-i) and SumXOR is summation under XOR.

a(n) = A048298(n+1). - Michael Somos, Dec 30 2016

EXAMPLE

G.f. = 1 + 2*x + 4*x^3 + 8*x^7 + 16*x^15 + 32*x^31 + 64*x^63 + 128*x^127 + ...

XOR difference triangle of A038712 begins:

[1],

[3,2],

[1,2,0],

[7,6,4,4],

[1,6,0,4,0],

[3,2,4,4,0,0],

[1,2,0,4,0,0,0],

[15,14,12,12,8,8,8,8],...

where A038712 is in the leftmost column and A099894 (this sequence) forms the main diagonal.

a(1) = 1*1 XOR 0*1 = 1, a(2) = 1*1 XOR 0*3 XOR 1*1 = 0, a(3) = 1*1 XOR 1*3 XOR 1*1 XOR 1*7 = 4 where (1, 3, 1, 7) are the first four terms of A038712. - Michael Somos, Dec 30 2016

MATHEMATICA

a[ n_] := With[ {m = n+1}, If[ m >=0 && Total[ IntegerDigits[ m, 2]] == 1, m, 0]]; (* Michael Somos, Dec 30 2016 *)

PROG

(PARI) {a(n)=local(B); B=0; for(i=0, n, B=bitxor(B, binomial(n, i)%2*A038712(n-i) )); B}

(PARI) {a(n) = my(m = n+1); m * ( m>=0 && hammingweight(m) == 1)}; /* Michael Somos, Dec 30 2016 */

CROSSREFS

Cf. A099884, A038712, A099895, A062383, A048298.

Sequence in context: A305212 A104774 A087263 * A048298 A123565 A258701

Adjacent sequences:  A099891 A099892 A099893 * A099895 A099896 A099897

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 29 2004

STATUS

approved

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Last modified May 29 04:27 EDT 2020. Contains 334696 sequences. (Running on oeis4.)