This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A263132 #42 Jun 16 2025 00:33:26 %S A263132 1,2,3,4,6,8,11,12,16,22,24,32,43,44,48,64,86,88,96,128,171,172,176, %T A263132 192,256,342,344,352,384,512,683,684,688,704,768,1024,1366,1368,1376, %U A263132 1408,1536,2048,2731,2732,2736,2752,2816,3072,4096,5462,5464,5472,5504 %N A263132 Positive values of m such that binomial(4*m - 1, m) is odd. %C A263132 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. %C A263132 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. %C A263132 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. %F A263132 a(n) = A263133(n) + 1. %F A263132 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 %F A263132 a(n) = (2^A000267(n-1) + 2^A384688(n-1)) / 3. - _Kevin Ryde_, Jun 14 2025 %e A263132 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. %e A263132 ........................................................ %e A263132 . Table 1: 4^n * ceiling(2^k/3) for n >= 0, k >= 1 . %e A263132 ........................................................ %e A263132 n\k| 1 2 3 4 5 6 7 8 9 %e A263132 ---+---------------------------------------------------- %e A263132 0 | 1 2 3 6 11 22 43 86 171 ... %e A263132 1 | 4 8 12 24 44 88 172 ... %e A263132 2 | 16 32 48 96 176 ... %e A263132 3 | 64 128 192 ... %e A263132 4 | 256 ... %e A263132 ... %e A263132 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. %e A263132 .................................................... %e A263132 . Table 2: 2^n * (2^(2*k + 1) + 1)/3, n,k >= 0 . %e A263132 .................................................... %e A263132 n\k| 0 1 2 3 4 5 %e A263132 ---+---------------------------------------------- %e A263132 0 | 1 3 11 43 171 683 ... %e A263132 1 | 2 6 22 86 342 1366 ... %e A263132 2 | 4 12 44 172 684 2732 ... %e A263132 3 | 8 24 88 344 1368 5464 ... %e A263132 4 | 16 48 176 688 2736 10928 ... %e A263132 5 | 32 96 352 1376 5472 21856 ... %e A263132 6 | 64 192 704 2752 10944 43712 ... %e A263132 7 | 128 384 1408 5504 21888 87424 ... %e A263132 8 | 256 ... %p A263132 for n from 1 to 5000 do if mod(binomial(4*n-1, n), 2) = 1 then print(n) end if end do; %t A263132 Select[Range[6000],OddQ[Binomial[4#-1,#]]&] (* _Harvey P. Dale_, Dec 26 2015 *) %o A263132 (PARI) for(n=1, 1e4, if (binomial(4*n-1, n) % 2 == 1, print1(n", "))) \\ _Altug Alkan_, Oct 11 2015 %o A263132 (PARI) a(n) = my(r,s=sqrtint(4*n-3,&r)); (1<<s + 1<<(r\/2))/3; \\ _Kevin Ryde_, Jun 14 2025 %o A263132 (Magma) [n: n in [1..6000] | Binomial(4*n-1, n) mod 2 eq 1]; // _Vincenzo Librandi_, Oct 12 2015 %o A263132 (Python) %o A263132 A263132_list = [m for m in range(1,10**6) if not ~(4*m-1) & m] # _Chai Wah Wu_, Feb 07 2016 %Y A263132 Cf. A000079, A000267, A000302, A005578, A007583, A048716, A384688. %Y A263132 Other odd binomials: A002450 (4*m+1,m), A020988 (4*m+2,m), A263133 (4*m+3,m), A080674 (4*m+4,m), A118113 (3*m-2,m), A003714 (3*m,m). %K A263132 nonn,easy %O A263132 1,2 %A A263132 _Peter Bala_, Oct 10 2015 %E A263132 More terms from _Vincenzo Librandi_, Oct 12 2015