Marc A. A. van Leeuwen - cp's OEIS frontend

User: Marc A. A. van Leeuwen

Marc A. A. van Leeuwen has authored 2 sequences.

A341242 Numbers whose binary representation encodes a subset S of the natural numbers such that the XOR of the binary representations of all s in S gives 0.

0, 1, 14, 15, 50, 51, 60, 61, 84, 85, 90, 91, 102, 103, 104, 105, 150, 151, 152, 153, 164, 165, 170, 171, 194, 195, 204, 205, 240, 241, 254, 255, 770, 771, 780, 781, 816, 817, 830, 831, 854, 855, 856, 857, 868, 869, 874, 875, 916, 917, 922, 923, 934, 935, 936

Offset: 1

Marc A. A. van Leeuwen, Feb 07 2021

The numbers for which the set S of positions of bits 1 in the binary representation, interpreted as a set of distinct-sized Nim heaps (including a possible uninteresting size 0 heap for the least significant bit) is losing for the player to move.
Viewing the list as a set of valid code words, every natural number N can be "corrected" to a valid code word by changing exactly one bit, in exactly one way. The position of that bit is found by computing for N the XOR of its raised-bit positions of the title (if the result is 0, then N is already valid but flipping the irrelevant bit 0 makes it valid again).
The "error correcting" interpretation, applied to 64-bit numbers interpreted as orientation of 64 coins, corresponds to a solution of the "coins on a chessboard" puzzle described in the Nick Berry's blog, and also mentioned at A253315.
Numbers 2*n and 2*n+1 for n = A075926(m).
Numbers m such that A253315(m) = 0. - Rémy Sigrist, Feb 09 2021

Nick Berry, Impossible Escape?, DataGenetics blog, December 2014.

Cf. A075926, A253315.

def ok(n):
  xor, b = 0, (bin(n)[2:])[::-1]
  for i, c in enumerate(b):
    if c == '1': xor ^= i
  return xor == 0
print([m for m in range(937) if ok(m)]) # Michael S. Branicky, Feb 07 2021

a(2*n+1) = 2*A075926(n), a(2*n+2) = 2*A075926(n) + 1 for any n >= 0. - Rémy Sigrist, Feb 09 2021

A139594 Number of different n X n symmetric matrices with nonnegative entries summing to 4. Also number of symmetric oriented graphs with 4 arcs on n points.

Original entry on oeis.org

0, 1, 9, 39, 116, 275, 561, 1029, 1744, 2781, 4225, 6171, 8724, 11999, 16121, 21225, 27456, 34969, 43929, 54511, 66900, 81291, 97889, 116909, 138576, 163125, 190801, 221859, 256564, 295191, 338025, 385361, 437504, 494769, 557481, 625975, 700596

Offset: 0

Marc A. A. van Leeuwen, Jun 12 2008

a(n) is also the number of semistandard Young tableaux over all partitions of 4 with maximal element <= n. - Alois P. Heinz, Mar 22 2012

Starting from 1 the partial sums give A244864. - J. M. Bergot, Sep 17 2016

			From _Michael B. Porter_, Sep 18 2016: (Start)
The nine 2 X 2 matrices summing to 4 are:
4 0  3 0  2 0  1 0  0 0  2 1  1 1  0 1  0 2
0 0  0 1  0 2  0 3  0 4  1 0  1 1  1 2  2 0
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

For 3 in place of 4 this gives A005900.
Row n=4 of A210391. - Alois P. Heinz, Mar 22 2012
Partial sums of A063489.

dd := proc(n,m) coeftayl(1/((1-X)^m*(1-X^2)^binomial(m,2)),X=0,n); seq(dd(4,m),m=0..N);

gf[k_] := 1/((1-x)^k (1-x^2)^(k(k-1)/2));
T[n_, k_] := SeriesCoefficient[gf[k], {x, 0, n}];
a[k_] := T[4, k];
a /@ Range[0, 40] (* Jean-François Alcover, Nov 07 2020 *)

a(n) = coefficient of x^4 in 1/((1-x)^n * (1-x^2)^binomial(n,2)).

a(n) = (n^2*(7+5*n^2))/12. G.f.: x*(1+x)*(1+3*x+x^2)/(1-x)^5. [Colin Barker, Mar 18 2012]

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User: Marc A. A. van Leeuwen

A341242 Numbers whose binary representation encodes a subset S of the natural numbers such that the XOR of the binary representations of all s in S gives 0.

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