Search: a006583 -id:a006583
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A003986
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Table T(n,k) = n OR k read by antidiagonals.
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+10
82
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0, 1, 1, 2, 1, 2, 3, 3, 3, 3, 4, 3, 2, 3, 4, 5, 5, 3, 3, 5, 5, 6, 5, 6, 3, 6, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 7, 6, 7, 4, 7, 6, 7, 8, 9, 9, 7, 7, 5, 5, 7, 7, 9, 9, 10, 9, 10, 7, 6, 5, 6, 7, 10, 9, 10, 11, 11, 11, 11, 7, 7, 7, 7, 11, 11, 11, 11, 12, 11, 10, 11, 12, 7, 6, 7, 12, 11, 10, 11, 12, 13, 13, 11
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,4
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LINKS
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FORMULA
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EXAMPLE
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The upper left corner of the array starts in row x=0 with columns y>=0 as:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, ...
2, 3, 2, 3, 6, 7, 6, 7, 10, 11, 10, 11, 14, ...
3, 3, 3, 3, 7, 7, 7, 7, 11, 11, 11, 11, 15, ...
4, 5, 6, 7, 4, 5, 6, 7, 12, 13, 14, 15, 12, ...
5, 5, 7, 7, 5, 5, 7, 7, 13, 13, 15, 15, 13, ...
6, 7, 6, 7, 6, 7, 6, 7, 14, 15, 14, 15, 14, ...
7, 7, 7, 7, 7, 7, 7, 7, 15, 15, 15, 15, 15, ...
8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, ...
9, 9, 11, 11, 13, 13, 15, 15, 9, 9, 11, 11, 13, ...
10, 11, 10, 11, 14, 15, 14, 15, 10, 11, 10, 11, 14, ...
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MAPLE
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read("transforms") ;
A003986 := proc(x, y) ORnos(x, y) ; end proc:
for d from 0 to 12 do for x from 0 to d do printf("%d, ", A003986(x, d-x)) ; end do: end do: # R. J. Mathar, May 28 2011
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MATHEMATICA
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Table[BitOr[k, n - k], {n, 0, 20}, {k, 0, n}] //Flatten (* Indranil Ghosh, Apr 01 2017 *)
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PROG
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(Haskell)
import Data.Bits ((.|.))
a003986 n k = (n - k) .|. k :: Int
a003986_row n = map (a003986 n) [0..n]
a003986_tabl = map a003986_row [0..]
(PARI)
tabl(nn) = {for(n=0, nn, for(k=0, n, print1(bitor(k, n - k), ", "); ); print(); ); };
(Python)
for n in range(21):
print([k|(n - k) for k in range(n + 1)])
(C)
#include <stdio.h>
int main()
{
int n, k;
for (n=0; n<=20; n++){
for(k=0; k<=n; k++){
printf("%d, ", (k|(n - k)));
}
printf("\n");
}
return 0;
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A328566
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a(n) is the sum of the elements of the set O_n = {(n-k) OR k, k = 0..n} (where OR denotes the bitwise OR operator).
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+10
4
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0, 0, 1, 3, 3, 9, 8, 14, 7, 25, 21, 37, 18, 46, 31, 45, 15, 65, 54, 96, 45, 119, 79, 115, 38, 130, 97, 159, 65, 155, 94, 124, 31, 161, 135, 243, 112, 304, 199, 289, 93, 331, 246, 404, 163, 393, 237, 313, 78, 338, 267, 461, 199, 517, 326, 456, 133, 443, 317, 505
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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-1,4
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COMMENTS
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The number of elements of the set O_n appears to be A002487(n+1); a(-1) = 0 as O_{-1} is the empty set.
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LINKS
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FORMULA
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MAPLE
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a:= n-> add(i, i={seq(Bits[Or](n-k, k), k=0..n)}):
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PROG
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(PARI) a(n) = vecsum(Set(apply(k -> bitor(k, n-k), [0..n])))
(Python)
def A328566(n): return sum(set(k|n-k for k in range((n>>1)+1))) # Chai Wah Wu, May 07 2023
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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A099027
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a(n) = Sum_{k=0..n} n-k AND NOT k.
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+10
2
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0, 1, 2, 6, 6, 11, 16, 28, 24, 29, 34, 50, 54, 71, 88, 120, 104, 105, 106, 126, 126, 147, 168, 212, 208, 229, 250, 298, 318, 367, 416, 496, 448, 433, 418, 438, 422, 443, 464, 524, 504, 525, 546, 610, 630, 695, 760, 872, 840, 857, 874, 942, 958, 1027, 1096
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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Antidiagonal sums of array A099026.
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LINKS
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FORMULA
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Recurrence: a(0) = 0, a(2n) = 2a(n) + 2a(n-1), a(2n+1) = 4a(n) + n+1. [corrected by Peter J. Taylor, May 30 2024]
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PROG
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(PARI) a(n) = sum(k=0, n, bitand(n-k, bitneg(k))); \\ Michel Marcus, Oct 30 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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