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 A265705 #50 Feb 16 2025 08:33:27
%S A265705 0,1,1,3,2,3,3,3,3,3,7,6,5,4,7,7,7,5,5,7,7,7,6,7,6,7,6,7,7,7,7,7,7,7,
%T A265705 7,7,15,14,13,12,11,10,9,8,15,15,15,13,13,11,11,9,9,15,15,15,14,15,14,
%U A265705 11,10,11,10,15,14,15,15,15,15,15,11,11,11,11,15
%N A265705 Triangle read by rows: T(n,k) = k IMPL n, 0 <= k <= n, bitwise logical IMPL.
%H A265705 Reinhard Zumkeller, <a href="/A265705/b265705.txt">Rows n = 0..255 of triangle, flattened</a>
%H A265705 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Implies.html">Implies</a>
%F A265705 T(n,0) = T(n,n) = A003817(n).
%F A265705 T(2*n,n) = A265716(n).
%F A265705 Let m = A089633(n): T(m,k) = T(m,m-k), k = 0..m.
%F A265705 Let m = A158582(n): T(m,k) != T(m,m-k) for at least one k <= n.
%F A265705 Let m = A247648(n): T(2*m,m) = 2*m.
%F A265705 For n > 0: A029578(n+2) = number of odd terms in row n; no even terms in odd-indexed rows.
%F A265705 A265885(n) = T(prime(n),n).
%F A265705 A053644(n) = smallest k such that row k contains n.
%e A265705 . 10 | 1010 12 | 1100
%e A265705 . 4 | 100 6 | 110
%e A265705 . ----------+----- ----------+-----
%e A265705 . 4 IMPL 10 | 1011 -> T(10,4)=11 6 IMPL 12 | 1101 -> T(12,6)=13
%e A265705 .
%e A265705 First 16 rows of the triangle, where non-symmetrical rows are marked, see comment concerning A158582 and A089633:
%e A265705 . 0: 0
%e A265705 . 1: 1 1
%e A265705 . 2: 3 2 3
%e A265705 . 3: 3 3 3 3
%e A265705 . 4: 7 6 5 4 7 X
%e A265705 . 5: 7 7 5 5 7 7
%e A265705 . 6: 7 6 7 6 7 6 7
%e A265705 . 7: 7 7 7 7 7 7 7 7
%e A265705 . 8: 15 14 13 12 11 10 9 8 15 X
%e A265705 . 9: 15 15 13 13 11 11 9 9 15 15 X
%e A265705 . 10: 15 14 15 14 11 10 11 10 15 14 15 X
%e A265705 . 11: 15 15 15 15 11 11 11 11 15 15 15 15
%e A265705 . 12: 15 14 13 12 15 14 13 12 15 14 13 12 15 X
%e A265705 . 13: 15 15 13 13 15 15 13 13 15 15 13 13 15 15
%e A265705 . 14: 15 14 15 14 15 14 15 14 15 14 15 14 15 14 15
%e A265705 . 15: 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 .
%p A265705 A265705 := (n, k) -> Bits:-Implies(k, n):
%p A265705 seq(seq(A265705(n, k), k=0..n), n=0..11); # _Peter Luschny_, Sep 23 2019
%t A265705 T[n_, k_] := If[n == 0, 0, BitOr[2^Length[IntegerDigits[n, 2]]-1-k, n]];
%t A265705 Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 25 2021, after _David A. Corneth_'s PARI code *)
%o A265705 (Haskell)
%o A265705 a265705_tabl = map a265705_row [0..]
%o A265705 a265705_row n = map (a265705 n) [0..n]
%o A265705 a265705 n k = k `bimpl` n where
%o A265705 bimpl 0 0 = 0
%o A265705 bimpl p q = 2 * bimpl p' q' + if u <= v then 1 else 0
%o A265705 where (p', u) = divMod p 2; (q', v) = divMod q 2
%o A265705 (PARI) T(n, k) = if(n==0,return(0)); bitor((2<<logint(n,2))-1-k,n) \\ _David A. Corneth_, Sep 24 2021
%o A265705 (Julia)
%o A265705 using IntegerSequences
%o A265705 for n in 0:15 println(n == 0 ? [0] : [Bits("IMP", k, n) for k in 0:n]) end # _Peter Luschny_, Sep 25 2021
%Y A265705 Cf. A003817, A007088, A029578, A089633, A158582, A247648, A265716 (central terms), A265736 (row sums).
%Y A265705 Cf. A053644, A265885, A327490.
%Y A265705 Other triangles: A080099 (AND), A080098 (OR), A051933 (XOR), A102037 (CNIMPL).
%K A265705 nonn,easy,tabl,look
%O A265705 0,4
%A A265705 _Reinhard Zumkeller_, Dec 15 2015