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A334593 Number of ones in XOR-triangle with first row generated from the binary expansion of n. 6
1, 2, 2, 3, 4, 4, 3, 4, 6, 5, 7, 5, 7, 6, 4, 5, 8, 9, 8, 8, 7, 10, 9, 7, 8, 9, 10, 8, 9, 8, 5, 6, 10, 10, 12, 12, 12, 10, 12, 10, 12, 8, 12, 12, 14, 12, 12, 8, 12, 12, 10, 12, 12, 14, 12, 10, 12, 12, 12, 10, 12, 10, 6, 7, 12, 14, 13, 12, 15, 15, 16, 14, 17, 15 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
An XOR-triangle is an inverted 0-1 triangle formed by choosing a top row and having each entry in the subsequent rows be the XOR of the two values above it.
Records occur at 1, 2, 4, 5, 9, 11, 17, 18, 22, 35, 45, 69, 71, 73, 91, 139, 142, 146, 182, ...
LINKS
MathOverflow user DSM, Number triangle
FORMULA
a(n) = A000217(A070939(n)) - A334592(n).
EXAMPLE
For n = 53, a(53) = 12 because 53 = 110101_2 in binary, and the corresponding XOR-triangle has 12 ones:
1 1 0 1 0 1
0 1 1 1 1
1 0 0 0
1 0 0
1 0
1
MATHEMATICA
Array[Total@ Flatten@ NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[#, 2], Length@ # > 1 &] &, 74] (* Michael De Vlieger, May 08 2020 *)
PROG
(PARI) a(n) = {my(b=binary(n), nb=hammingweight(n)); for (n=1, #b-1, b = vector(#b-1, k, bitxor(b[k], b[k+1])); nb += vecsum(b); ); nb; } \\ Michel Marcus, May 08 2020
CROSSREFS
Sequence in context: A339695 A326846 A243220 * A293596 A301977 A085430
KEYWORD
nonn,base
AUTHOR
Peter Kagey, May 07 2020
STATUS
approved

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Last modified February 27 12:56 EST 2024. Contains 370375 sequences. (Running on oeis4.)