Original entry on oeis.org
0, 7, 35, 162, 1713, 7157, 34534, 162417, 1712101, 7157706, 34534213, 162417333, 1733115660, 7147321437, 34532167225, 362414612202, 1633316153423, 7127276460567, 34502517467034, 362420550141507
Offset: 0
Entry revised Dec 29 2007
A037093
"Sloping binary representation" of Fibonacci numbers, slope = +1.
Original entry on oeis.org
0, 1, 3, 14, 57, 229, 916, 7761, 29567, 117474, 469113, 3973641, 15138352, 60146777, 240187355, 2070207870, 7733090689, 30791909229, 260408711716, 991495872825, 3942106110215, 15739612088946, 133333733918417
Offset: 0
When Fibonacci numbers are written in binary (see A004685), under each other as:
0000000 (0)
0000001 (1)
0000001 (1)
0000010 (2)
0000011 (3)
0000101 (5)
0001000 (8)
0001101 (13)
0010101 (21)
0100010 (34)
0110111 (55)
1011001 (89)
and one starts collecting their bits from column-0 to SW-direction (from the least to the most significant end), one gets 000... (0), ...00001 (1), ...00011 (3), ...001110 (14), etc. (See A102370 for similar transformation done on nonnegative integers).
Entry revised Dec 29 2007
A037095
"Sloping binary representation" of powers of 3 (A000244), slope = -1.
Original entry on oeis.org
1, 1, 3, 1, 3, 9, 11, 17, 19, 25, 123, 65, 195, 169, 171, 753, 435, 249, 2267, 4065, 8163, 841, 843, 31313, 29651, 39769, 38331, 30081, 160643, 49769, 53867, 563377, 700659, 1611961, 760731, 1207073, 5668771, 5566345, 11844619, 8699025, 10386067, 55868313
Offset: 0
When powers of 3 are written in binary (see A004656), under each other as:
000000000001 (1)
000000000011 (3)
000000001001 (9)
000000011011 (27)
000001010001 (81)
000011110011 (243)
001011011001 (729)
100010001011 (2187)
and one collects their bits from the column-0 to NW-direction (from the least to the most significant end), one gets 1 (1), 01 (1), 011 (3), 0001 (1), 00011 (3), 001001 (9), etc. (See A105033 for similar transformation done on nonnegative integers, A001477).
-
A037095:= n-> add(bit_n(3^(n-i), i)*(2^i), i=0..n):
bit_n := (x, n) -> `mod`(floor(x/(2^n)), 2):
seq(A037095(n), n=0..41);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, (p->
expand((p-(p mod 2))*x/2)+3^n)(b(n-1)))
end:
a:= n-> subs(x=2, b(n) mod 2):
seq(a(n), n=0..42); # Alois P. Heinz, Dec 10 2020
-
A339601(n) = { my(m=1, s=0); while(n>=m, s += bitand(m,n); m <<= 1; n \= 3); (s); };
A037095(n) = A339601(3^n); \\ Antti Karttunen, Dec 09 2020
-
BINSLOPE(f) = n -> sum(i=0,n,bitand(2^(n-i),f(i))); \\ General transformation for these kinds of sequences.
A037095 = BINSLOPE(n -> 3^n); \\ And its application to A000244. - Antti Karttunen, Dec 09 2020
Entry revised Dec 29 2007
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