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A263133 Numbers m such that binomial(4*m + 3, m) is odd. 2
0, 1, 2, 3, 5, 7, 10, 11, 15, 21, 23, 31, 42, 43, 47, 63, 85, 87, 95, 127, 170, 171, 175, 191, 255, 341, 343, 351, 383, 511, 682, 683, 687, 703, 767, 1023, 1365, 1367, 1375, 1407, 1535, 2047, 2730, 2731, 2735, 2751, 2815, 3071, 4095, 5461, 5463, 5471, 5503 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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; and A080674, which gives the values of m such that binomial(4*m + 4, m) is odd.

Compare with A263132, which lists the values of m such that binomial(4*m - 1, m) is odd.

The sequence of even values of a(n) is [0, 2, 10, 42, 170, ...] = A020998. If m is a term in the sequence then 2*m + 1 is also a term in the sequence. Repeatedly applying the transformation m -> 2*m + 1 to the terms of A020998 produces all the terms of this sequence. See the example below.

2*a(n) gives the values of m such that binomial(4*m + 6, m) is odd.

LINKS

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

FORMULA

a(n) = A263132(n) - 1.

m is a term if and only if m AND NOT (4*m+3) = 0 where AND and NOT are bitwise operators. - Chai Wah Wu, Feb 07 2016

EXAMPLE

1) This sequence can be read from Table 1 below in a sequence of 'knight moves' (2 down and 1 to the left) starting from the first two rows. For example, starting at 42 in the first row we jump 42 -> 43 -> 47 -> 63, then return to the second row at 85 and jump 85 -> 87 -> 95 -> 127, followed by 170 -> 171 -> 175 -> 191 -> 255, and so on.

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

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

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

  n\k|   0    1    2    3    4    5

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

   0 |   0    2   10   42  170  682 ...

   1 |   1    5   21   85  341  ...

   2 |   3   11   43  171  683  ...

   3 |   7   23   87  343  ...

   4 |  15   47  175  687  ...

   5 |  31   95  351  ...

   6 |  63  191  703  ...

   7 | 127  383  ...

   8 | 255  767  ...

   9 | 511  ...

   ...

The first row of the table is A020988. The columns of the table are obtained by repeatedly applying the transformation m -> 2*m + 1 to the entries in the first row.

2) Alternatively, this sequence can be read from Table 2 below by starting with a number on the top row and moving in a series of 'knight moves' (1 down and 2 to the left) through the table as far as you can, before returning to the next number in the top row and repeating the process. For example, starting at 10 in the first row we move 10 -> 11 -> 15, then return to the top row at 21 and move 21 -> 23 -> 31, before returning to the top row at 42 and so on.

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

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

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

n\k|    1    2    3    4     5     6     7     8    9   10

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

  0|    0    1    2    5    10    21    42    85  170  682...

  1|    3    7   11   23    43    87   171   343  683  ...

  2|   15   31   47   95   175   351   687  1375  ...

  3|   63  127  191  383   703  1407  2751  5503  ...

  4|  255  511  767 1535  2815  5631 11007 22015  ...

  5| 1023 2047 3071 6143 11263 22527 44031 88063  ...

  6| 4095 ...

  ...

The first row of the table is A000975. The columns of the table are obtained by repeatedly applying the transformation m -> 4*m + 3 to the entries in the first row.

MAPLE

for n from 1 to 4096 do if mod(binomial(4*n+3, n), 2) = 1 then print(n) end if end do;

MATHEMATICA

Select[Range[0, 5600], OddQ[Binomial[4#+3, #]]&] (* Harvey P. Dale, Apr 15 2019 *)

PROG

(PARI) for(n=0, 1e4, if (binomial(4*n+3, n) % 2 == 1, print1(n", "))) \\ Altug Alkan, Oct 11 2015

(MAGMA) [n: n in [0..6000] | Binomial(4*n+3, n) mod 2 eq 1]; // Vincenzo Librandi, Oct 12 2015

(Python)

A263133_list = [m for m in range(10**6) if not ~(4*m+3) & m] # Chai Wah Wu, Feb 07 2016

CROSSREFS

Cf. A000975, A002450, A020988, A080674, A263133.

Sequence in context: A285257 A317407 A191211 * A254860 A144726 A123885

Adjacent sequences:  A263130 A263131 A263132 * A263134 A263135 A263136

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Oct 11 2015

EXTENSIONS

More terms from Vincenzo Librandi, Oct 12 2015

STATUS

approved

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Last modified January 20 23:47 EST 2021. Contains 340332 sequences. (Running on oeis4.)