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 A263133 #35 Jul 06 2025 22:17:19 %S A263133 0,1,2,3,5,7,10,11,15,21,23,31,42,43,47,63,85,87,95,127,170,171,175, %T A263133 191,255,341,343,351,383,511,682,683,687,703,767,1023,1365,1367,1375, %U A263133 1407,1535,2047,2730,2731,2735,2751,2815,3071,4095,5461,5463,5471,5503 %N A263133 Numbers m such that binomial(4*m + 3, m) is odd. %C A263133 The even terms in the sequence are A020988. 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 A020988 produces all the terms of this sequence. See the example below. %C A263133 2*a(n) gives the values of m such that binomial(4*m + 6, m) is odd. %F A263133 a(n) = A263132(n) - 1. %F A263133 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 %F A263133 a(n) = (2^A000267(n) + 2^A384688(n))/3 - 1, for n >= 1. - _Kevin Ryde_, Jul 06 2025 %e A263133 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. %e A263133 ........................................................... %e A263133 . Table 1. 2^n*ceiling((2^(2*k + 1) - 1)/3) - 1, n,k >= 0 . %e A263133 ........................................................... %e A263133 n\k| 0 1 2 3 4 5 %e A263133 ---+--------------------------------- %e A263133 0 | 0 2 10 42 170 682 ... %e A263133 1 | 1 5 21 85 341 ... %e A263133 2 | 3 11 43 171 683 ... %e A263133 3 | 7 23 87 343 ... %e A263133 4 | 15 47 175 687 ... %e A263133 5 | 31 95 351 ... %e A263133 6 | 63 191 703 ... %e A263133 7 | 127 383 ... %e A263133 8 | 255 767 ... %e A263133 9 | 511 ... %e A263133 ... %e A263133 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. %e A263133 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. %e A263133 ........................................................ %e A263133 . Table 2. (4^n)*ceiling(2^k/3) - 1 for n >= 0, k >= 1 . %e A263133 ........................................................ %e A263133 n\k| 1 2 3 4 5 6 7 8 9 10 %e A263133 ---+--------------------------------------------------------- %e A263133 0| 0 1 2 5 10 21 42 85 170 682... %e A263133 1| 3 7 11 23 43 87 171 343 683 ... %e A263133 2| 15 31 47 95 175 351 687 1375 ... %e A263133 3| 63 127 191 383 703 1407 2751 5503 ... %e A263133 4| 255 511 767 1535 2815 5631 11007 22015 ... %e A263133 5| 1023 2047 3071 6143 11263 22527 44031 88063 ... %e A263133 6| 4095 ... %e A263133 ... %e A263133 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. %p A263133 for n from 1 to 4096 do if mod(binomial(4*n+3, n), 2) = 1 then print(n) end if end do; %t A263133 Select[Range[0,5600],OddQ[Binomial[4#+3,#]]&] (* _Harvey P. Dale_, Apr 15 2019 *) %o A263133 (PARI) for(n=0, 1e4, if (binomial(4*n+3, n) % 2 == 1, print1(n", "))) \\ _Altug Alkan_, Oct 11 2015 %o A263133 (PARI) a(n) = my(r,s=sqrtint(4*n-3,&r)); (1<<s + 1<<(r\/2))/3 - 1; \\ _Kevin Ryde_, Jul 06 2025 %o A263133 (Magma) [n: n in [0..6000] | Binomial(4*n+3, n) mod 2 eq 1]; // _Vincenzo Librandi_, Oct 12 2015 %o A263133 (Python) %o A263133 A263133_list = [m for m in range(10**6) if not ~(4*m+3) & m] # _Chai Wah Wu_, Feb 07 2016 %Y A263133 Cf. A000975. %Y A263133 Other odd binomials: A263132 (4*m-1,m), A002450 (4*m+1,m), A020988 (4*m+2,m), A080674 (4*m+4,m), A118113 (3*m-2,m), A003714 (3*m,m). %K A263133 nonn,easy %O A263133 1,3 %A A263133 _Peter Bala_, Oct 11 2015 %E A263133 More terms from _Vincenzo Librandi_, Oct 12 2015