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A263132
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Positive values of m, arranged in order, such that binomial(4*m - 1, m) is odd.
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8
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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. Table 1: 4^n * ceiling(2^k/3) for n >= 0, k >= 1 .
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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.
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. Table 2: 2^n * (2^(2*k + 1) + 1)/3, n,k >= 0 .
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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 ...
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MAPLE
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for n from 1 to 5000 do if mod(binomial(4*n-1, n), 2) = 1 then print(n) end if end do;
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MATHEMATICA
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Select[Range[6000], OddQ[Binomial[4#-1, #]]&] (* Harvey P. Dale, Dec 26 2015 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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