A354785
Numbers of the form 3*2^k or 9*2^k.
Original entry on oeis.org
3, 6, 9, 12, 18, 24, 36, 48, 72, 96, 144, 192, 288, 384, 576, 768, 1152, 1536, 2304, 3072, 4608, 6144, 9216, 12288, 18432, 24576, 36864, 49152, 73728, 98304, 147456, 196608, 294912, 393216, 589824, 786432, 1179648, 1572864, 2359296, 3145728, 4718592, 6291456, 9437184, 12582912, 18874368, 25165824, 37748736, 50331648
Offset: 1
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with
A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent:
A029744 (s(n));
A052955 (s(n)-1),
A027383 (s(n)-2),
A354788 (s(n)-3),
A347789 (s(n)-4),
A209721 (s(n)+1),
A209722 (s(n)+2),
A343177 (s(n)+3),
A209723 (s(n)+4);
A060482,
A136252 (minor differences from
A354788 at the start);
A354785 (3*s(n)),
A354789 (3*s(n)-7). The first differences of
A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences:
A016116,
A060546,
A117575,
A131572,
A152166,
A158780,
A163403,
A320770. The bisections of
A029744 are
A000079 and
A007283.
-
seq[max_] := Union[Table[3*2^n, {n, 0, Floor[Log2[max/3]]}], Table[9*2^n, {n, 0, Floor[Log2[max/9]]}]]; seq[10^8] (* Amiram Eldar, Jan 16 2024 *)
A160444
Expansion of g.f.: x^2*(1 + x - x^2)/(1 - 2*x^2 - 2*x^4).
Original entry on oeis.org
0, 1, 1, 1, 2, 4, 6, 10, 16, 28, 44, 76, 120, 208, 328, 568, 896, 1552, 2448, 4240, 6688, 11584, 18272, 31648, 49920, 86464, 136384, 236224, 372608, 645376, 1017984, 1763200, 2781184, 4817152, 7598336, 13160704, 20759040, 35955712, 56714752
Offset: 1
Willibald Limbrunner (w.limbrunner(AT)gmx.de), May 14 2009
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- W. Beinert, Villardscher Teilungskanon, Lexikon der Typographie
- W. Limbrunner, Das Quadrat, ein Wunder der Geometrie. (in German)
- Willibald Limbrunner, Family of sequences for k
- M-T. Zenner, Villard de Honnecourt and Euclidean Geoometry, Nexus Network Journal 4 (2002) 65-78.
- Index entries for linear recurrences with constant coefficients, signature (0,2,0,2).
-
I:=[0,1,1,1]; [n le 4 select I[n] else 2*(Self(n-2) +Self(n-4)): n in [1..40]]; // G. C. Greubel, Feb 18 2023
-
LinearRecurrence[{0,2,0,2}, {0,1,1,1}, 40] (* G. C. Greubel, Feb 18 2023 *)
-
@CachedFunction
def a(n): # a = A160444
if (n<5): return ((n+1)//3)
else: return 2*(a(n-2) + a(n-4))
[a(n) for n in range(1, 41)] # G. C. Greubel, Feb 18 2023
A206474
Riordan array ((1+x-x^2)/(1-x^2), x/(1-x^2)).
Original entry on oeis.org
1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 2, 1, 1, 1, 1, 3, 3, 1, 1, 0, 3, 3, 4, 4, 1, 1, 1, 1, 6, 6, 5, 5, 1, 1, 0, 4, 4, 10, 10, 6, 6, 1, 1, 1, 1, 10, 10, 15, 15, 7, 7, 1, 1, 0, 5, 5, 20, 20, 21, 21, 8, 8, 1, 1, 1, 1, 15, 15, 35, 35, 28, 28, 9, 9, 1, 1
Offset: 0
Triangle begins :
1
1, 1
0, 1, 1
1, 1, 1, 1
0, 2, 2, 1, 1
1, 1, 3, 3, 1, 1
0, 3, 3, 4, 4, 1, 1
1, 1, 6, 6, 5, 5, 1, 1
0, 4, 4, 10, 10, 6, 6, 1, 1
1, 1, 10, 10, 15, 15, 7, 7, 1, 1
0, 5, 5, 20, 20, 21, 21, 8, 8, 1, 1
1, 1, 15, 15, 35, 35, 28, 28, 9, 9, 1, 1
-
t[1, 0] = 1; t[2, 0] = 0; t[n_, n_] = 1; t[n_ /; n >= 0, k_ /; k >= 0] /; k <= n := t[n, k] = t[n-1, k-1] + t[n-2, k]; t[n_, k_] = 0; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 28 2013 *)
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