OFFSET
0,4
LINKS
Reinhard Zumkeller, Rows n = 0..255 of triangle, flattened
Eric Weisstein's World of Mathematics, Implies
FORMULA
T(n,0) = T(n,n) = A003817(n).
T(2*n,n) = A265716(n).
Let m = A089633(n): T(m,k) = T(m,m-k), k = 0..m.
Let m = A158582(n): T(m,k) != T(m,m-k) for at least one k <= n.
Let m = A247648(n): T(2*m,m) = 2*m.
For n > 0: A029578(n+2) = number of odd terms in row n; no even terms in odd-indexed rows.
A265885(n) = T(prime(n),n).
A053644(n) = smallest k such that row k contains n.
EXAMPLE
. 10 | 1010 12 | 1100
. 4 | 100 6 | 110
. ----------+----- ----------+-----
. 4 IMPL 10 | 1011 -> T(10,4)=11 6 IMPL 12 | 1101 -> T(12,6)=13
.
First 16 rows of the triangle, where non-symmetrical rows are marked, see comment concerning A158582 and A089633:
. 0: 0
. 1: 1 1
. 2: 3 2 3
. 3: 3 3 3 3
. 4: 7 6 5 4 7 X
. 5: 7 7 5 5 7 7
. 6: 7 6 7 6 7 6 7
. 7: 7 7 7 7 7 7 7 7
. 8: 15 14 13 12 11 10 9 8 15 X
. 9: 15 15 13 13 11 11 9 9 15 15 X
. 10: 15 14 15 14 11 10 11 10 15 14 15 X
. 11: 15 15 15 15 11 11 11 11 15 15 15 15
. 12: 15 14 13 12 15 14 13 12 15 14 13 12 15 X
. 13: 15 15 13 13 15 15 13 13 15 15 13 13 15 15
. 14: 15 14 15 14 15 14 15 14 15 14 15 14 15 14 15
. 15: 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 .
MAPLE
A265705 := (n, k) -> Bits:-Implies(k, n):
seq(seq(A265705(n, k), k=0..n), n=0..11); # Peter Luschny, Sep 23 2019
MATHEMATICA
T[n_, k_] := If[n == 0, 0, BitOr[2^Length[IntegerDigits[n, 2]]-1-k, n]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 25 2021, after David A. Corneth's PARI code *)
PROG
(Haskell)
a265705_tabl = map a265705_row [0..]
a265705_row n = map (a265705 n) [0..n]
a265705 n k = k `bimpl` n where
bimpl 0 0 = 0
bimpl p q = 2 * bimpl p' q' + if u <= v then 1 else 0
where (p', u) = divMod p 2; (q', v) = divMod q 2
(PARI) T(n, k) = if(n==0, return(0)); bitor((2<<logint(n, 2))-1-k, n) \\ David A. Corneth, Sep 24 2021
(Julia)
using IntegerSequences
for n in 0:15 println(n == 0 ? [0] : [Bits("IMP", k, n) for k in 0:n]) end # Peter Luschny, Sep 25 2021
CROSSREFS
AUTHOR
Reinhard Zumkeller, Dec 15 2015
STATUS
approved