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A263132 Positive values of m, arranged in order, such that binomial(4*m - 1, m) is odd. 3
1, 2, 3, 4, 6, 8, 11, 12, 16, 22, 24, 32, 43, 44, 48, 64, 86, 88, 96, 128, 171, 172, 176, 192, 256, 342, 344, 352, 384, 512, 683, 684, 688, 704, 768, 1024, 1366, 1368, 1376, 1408, 1536, 2048, 2731, 2732, 2736, 2752, 2816, 3072, 4096, 5462, 5464, 5472, 5504 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

2*a(n) gives the values of m such that binomial(4*m - 2,m) is odd. 4*a(n) gives the values of m such that binomial(4*m - 3,m) is odd (other than m = 1) and also the values of m such that binomial(4*m - 4,m) is odd.

Compare with A002450, which equals the values of m such that binomial(4*m + 1,m) is odd, A020988 which equals the values of m such that binomial(4*m + 2,m) is odd, A263133, which gives the values of m such that binomial(4*m + 3,m) is odd and A080674, which equals the values of m such that binomial(4*m + 4,m) is odd.

Compare with A118113, which appears to be the values of m such that binomial(3*m - 2,m) is odd. Cf. A003714.

This sequence, when viewed as a set, equals the set of numbers of the form 4^n * ceiling(2^k/3) for n >= 0, k >= 1, i.e., the product subset in Z of A000302 and A005578 regarded as sets. See the example below.

Equivalently, this sequence, when viewed as a set, equals the set of numbers of the form 2^n * (2^(2*k + 1) + 1)/3 for n,k >= 0, i.e., the product subset in Z of A000079 and A007583 regarded as sets. See the example below.

LINKS

Table of n, a(n) for n=1..53.

FORMULA

a(n) = A263133(n) + 1.

m is a term if and only if m AND NOT (4*m-1) = 0 where AND and NOT are bitwise operators. - Chai Wah Wu, Feb 07 2016

EXAMPLE

1) Notice how this sequence can be read from Table 1 below by moving through the table in a sequence of 'knight moves' (1 down and 2 to the left) starting from the first row. For example, Starting at 11 on the top row we move in a series of knights moves 11 -> 12 -> 16, then return to the top row at 22 and move 22 -> 24 -> 32, return to the top row at 43 and move 43 -> 44 -> 48 -> 64, then return to top row at 86 and so on.

........................................................

.   Table 1: 4^n * ceiling(2^k/3) for n >= 0, k >= 1   .

........................................................

n\k|   1    2    3    4     5     6    7    8     9

---+----------------------------------------------------

0  |   1    2    3    6    11    22   43   86   171 ...

1  |   4    8   12   24    44    88  172  ...

2  |  16   32   48   96   176    ...

3  |  64  128  192  ...

4  | 256  ...

...

2) Notice how this sequence can be read from Table 2 below in a sequence of 'knight moves' (2 down and 1 to the left) starting from the first two rows. For example, starting at 43 in the first row we jump 43 -> 44 -> 48 -> 64, then return to the second row at 86 and jump 86 -> 88 -> 96 -> 128, followed by 171 -> 172 -> 176 -> 192 -> 256, and so on.

....................................................

.   Table 2: 2^n * (2^(2*k + 1) + 1)/3, n,k >= 0   .

....................................................

n\k|   0    1     2     3      4      5

---+----------------------------------------------

0  |   1    3    11    43    171    683  ...

1  |   2    6    22    86    342   1366  ...

2  |   4   12    44   172    684   2732  ...

3  |   8   24    88   344   1368   5464  ...

4  |  16   48   176   688   2736  10928  ...

5  |  32   96   352  1376   5472  21856  ...

6  |  64  192   704  2752  10944  43712  ...

7  | 128  384  1408  5504  21888  87424  ...

8  | 256 ...

MAPLE

for n from 1 to 5000 do if mod(binomial(4*n-1, n), 2) = 1 then print(n) end if end do;

MATHEMATICA

Select[Range[6000], OddQ[Binomial[4#-1, #]]&] (* Harvey P. Dale, Dec 26 2015 *)

PROG

(PARI) for(n=1, 1e4, if (binomial(4*n-1, n) % 2 == 1, print1(n", "))) \\ Altug Alkan, Oct 11 2015

(MAGMA) [n: n in [1..6000] | Binomial(4*n-1, n) mod 2 eq 1]; // Vincenzo Librandi, Oct 12 2015

(Python)

A263132_list = [m for m in range(1, 10**6) if not ~(4*m-1) & m] # Chai Wah Wu, Feb 07 2016

CROSSREFS

Cf. A000079, A000302, A002450, A003714, A005578, A007583, A020988, A048716, A118113, A263133.

Sequence in context: A160649 A190203 A034034 * A018502 A023024 A018362

Adjacent sequences:  A263129 A263130 A263131 * A263133 A263134 A263135

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Oct 10 2015

EXTENSIONS

More terms from Vincenzo Librandi, Oct 12 2015

STATUS

approved

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Last modified October 18 23:39 EDT 2019. Contains 328211 sequences. (Running on oeis4.)