cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A037099 Sequence A037094 shown in octal.

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

Views

Author

Antti Karttunen, Jan 29 1999

Keywords

Crossrefs

Formula

a(n) = A007094(A037094(n)).

Extensions

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

Views

Author

Antti Karttunen, Jan 28 1999

Keywords

Examples

			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).
		

Crossrefs

Same sequence in octal: A037098. Cf. also: A102370, A000045, A037094-A037095, A036284.

Formula

a(n) := Sum(bit_n(A000045(n+i), i)*(2^i), i=0..inf) [ bit_n := (x, n) -> `mod`(floor(x/(2^n)), 2); ]
In practice, n can be used as an upper limit instead of infinity.

Extensions

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

Views

Author

Antti Karttunen, Jan 28 1999

Keywords

Examples

			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).
		

Crossrefs

Programs

  • Maple
    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
  • PARI
    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
    
  • PARI
    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

Formula

a(n) = A339601(A000244(n)). - Antti Karttunen, Dec 09 2020

Extensions

Entry revised Dec 29 2007
More terms from Sean A. Irvine, Dec 08 2020
Showing 1-3 of 3 results.