A245180 -id:A245180 - cp's OEIS frontend

A245540 Partial sums of A245180.

1, 2, 5, 6, 14, 17, 31, 32, 40, 48, 72, 75, 99, 113, 165, 166, 174, 182, 206, 214, 278, 302, 414, 417, 441, 465, 537, 551, 663, 715, 931, 932, 940, 948, 972, 980, 1044, 1068, 1180, 1188, 1252, 1316, 1508, 1532, 1724, 1836, 2252, 2255, 2279, 2303, 2375, 2399, 2591, 2663, 2999, 3013, 3125, 3237

Offset: 1

N. J. A. Sloane, Jul 26 2014

Cf. A245180, A160239, A245542.

b[n_] := b[n] = Which[n == 1, 1, Mod[n, 2] == 0, b[n/2], Mod[n, 4] == 3, 2b[(n-1)/2] + b[n-2], True, 8b[(n-1)/4]];
Accumulate[Array[b, 58]] (* Jean-François Alcover, Oct 01 2018 *)

a(n) = (A245542(n) - 1)/8. - Omar E. Pol, Mar 07 2015

A245181 Numbers appearing in A245180.

Original entry on oeis.org

1, 3, 8, 14, 24, 52, 64, 72, 112, 192, 216, 336, 416, 512, 576, 848, 896, 1248, 1536, 1568, 1728, 2688, 3328, 3424, 4096, 4608, 5184, 5824, 6784, 7168, 8064, 9984, 12288, 12544, 13632, 13824, 20352, 21504, 21632, 24192, 26624, 27392, 29952, 32768, 36864, 37632, 41472, 46592, 54272, 54656

Offset: 1

N. J. A. Sloane, Jul 17 2014

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..325

Cf. A245180. A083424 is a subsequence.

A245190 Products of successive blocks of 2^n terms of A245180.

Original entry on oeis.org

1, 3, 336, 80510976, 4632013631072036192256, 9259244309598002961669050091730162686136971834955923456

Offset: 0

N. J. A. Sloane, Jul 21 2014

nonn

			a(2) = 1*8*3*14 = 336.

Cf. A245180.

A160239 Number of "ON" cells in a 2-dimensional cellular automaton ("Fredkin's Replicator") evolving according to the rule that a cell is ON in a given generation if and only if there was an odd number of ON cells among the eight nearest neighbors in the preceding generation, starting with one ON cell.

Original entry on oeis.org

1, 8, 8, 24, 8, 64, 24, 112, 8, 64, 64, 192, 24, 192, 112, 416, 8, 64, 64, 192, 64, 512, 192, 896, 24, 192, 192, 576, 112, 896, 416, 1728, 8, 64, 64, 192, 64, 512, 192, 896, 64, 512, 512, 1536, 192, 1536, 896, 3328, 24, 192, 192, 576, 192, 1536, 576, 2688, 112, 896, 896, 2688, 416, 3328, 1728, 6784

Offset: 0

John W. Layman, May 05 2009

This is the odd-rule cellular automaton defined by OddRule 757 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - N. J. A. Sloane, Feb 25 2015

The partial sums are in A245542, in which the structure also looks like an irregular stepped pyramid. - Omar E. Pol, Jan 29 2015

			From _Omar E. Pol_, Jul 22 2014 (Start):
Written as an irregular triangle in which row lengths is A011782 the sequence begins:
1;
8;
8, 24;
8, 64, 24, 112;
8, 64, 64, 192, 24, 192, 112, 416;
8, 64, 64, 192, 64, 512, 192, 896, 24, 192, 192, 576, 112, 896, 416, 1728;
8, 64, 64, 192, 64, 512, 192, 896, 64, 512, 512, 1536, 192, 1536, 896, 3328, 24, 192, 192, 576, 192, 1536, 576, 2688, 112, 896, 896, 2688, 416, 3328, 1728, 6784;
(End)
Right border gives A246030. - _Omar E. Pol_, Jan 29 2015 [This is simply a restatement of the theorem that this sequence is the Run Length Transform of A246030. - _N. J. A. Sloane_, Jan 29 2015]
.
From _Omar E. Pol_, Mar 18 2015 (Start):
Also, the sequence can be written as an irregular tetrahedron as shown below:
1;
..
8;
..
8;
24;
.........
8,    64;
24;
112;
...................
8,    64,  64, 192;
24,  192;
112;
416;
.....................................
8,    64,  64, 192, 64, 512,192, 896;
24,  192, 192, 576;
112, 896;
416;
1728;
.......................................................................
8,    64,  64, 192, 64, 512,192, 896,64,512,512,1536,192,1536,896,3328;
24,  192, 192, 576,192,1536,576,2688;
112, 896, 896,2688;
416,3328;
1728;
6784;
...
Apart from the initial 1, we have that T(s,r,k) = T(s+1,r,k). On the other hand, it appears that the configuration of ON cells of T(s,r,k) is also the central part of the configuration of ON cells of T(s+1,r+1,k).
(End)

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
Charles R Greathouse IV, Compact illustration of generations 0..17
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, Enlarged illustration of first 16 generations (pdf).
N. J. A. Sloane, Illustration for a(15) = 416 (png).
N. J. A. Sloane, Illustration for a(15) = 416 (pdf).
N. J. A. Sloane, Illustration for a(31) = 1728.
N. J. A. Sloane, Illustration for a(63) = 6784.
N. J. A. Sloane, Illustration for a(127) = 27392 (tiff).
N. J. A. Sloane, Illustration for a(127) = 27392 (png).
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Alexander Yu. Vlasov, Modelling reliability of reversible circuits with 2D second-order cellular automata, arXiv:2312.13034 [nlin.CG], 2023. See page 13.
Index entries for sequences related to cellular automata

Cf. A122108, A147562, A164032, A245180 (gives a(n)/8, n>=2).

Cf. also A245542 (Partial sums), A245543, A083424, A245562, A246030, A254731 (an "even-rule" version).

import Data.List (transpose)
a160239 n = a160239_list !! n
a160239_list = 1 : (concat $
   transpose [a8, hs, zipWith (+) (map (* 2) hs) a8, tail a160239_list])
   where a8 = map (* 8) a160239_list;
         hs = h a160239_list; h (_:x:xs) = x : h xs
-- Reinhard Zumkeller, Feb 13 2015

# From N. J. A. Sloane, Jan 19 2015:
f:=proc(n) option remember;
if n=0 then RETURN(1);
elif n mod 2 = 0 then RETURN(f(n/2))
elif n mod 4 = 1 then RETURN(8*f((n-1)/4))
else RETURN(f(n-2)+2*f((n-1)/2)); fi;
end;
[seq(f(n),n=0..255)];

A160239[n_] :=
CellularAutomaton[{52428, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, {{n}}][[1]] // Total@*Total (* Charles R Greathouse IV, Aug 21 2014 *)
ArrayPlot /@ CellularAutomaton[{52428, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, 30] (* Charles R Greathouse IV, Aug 21 2014 *)

A160239=[];a(n)={if(n>#A160239,A160239=concat(A160239,vector(n-#A160239)),n||return(1);A160239[n]&&return(A160239[n]));A160239[n]=if(bittest(n,0),if(bittest(n,1),a(n-2)+2*a(n\2),a(n\4)*8),a(n\2))} \\ M. F. Hasler, May 10 2016

a(0) = 1; a(2t)=a(t), a(4t+1)=8*a(t), a(4t+3)=2*a(2t+1)+8*a(t) for t >= 0. (Conjectured by Hrothgar, Jul 11 2014; proved by N. J. A. Sloane, Oct 04 2014.)
For n >= 2, a(n) = 8^r * Product_{lengths i of runs of 1 in binary expansion of n} R(i), where r is the number of runs of 1 in the binary expansion of n and R(i) = A083424(i-1) = (5*4^(i-1)+(-2)^(i-1))/6. Note that row i of the table in A245562 lists the lengths of runs of 1 in binary expansion of i. Example: n=7 = 111 in binary, so r=1, i=3, R(3) = A083424(2) = 14, and so a(7) = 8^1*14 = 112. That is, this sequence is the Run Length Transform of A246030. - N. J. A. Sloane, Oct 04 2014
The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product). - N. J. A. Sloane, Aug 25 2014

Offset changed to 1 by Hrothgar, Jul 11 2014

Offset reverted to 0 by N. J. A. Sloane, Jan 19 2015

A245196 Write n>=1 as either n=2^k-2^r with 0 <= r <= k-1, in which case a(2^k-2^r)=wt(k-r-1), or as n=2^k-2^r+j with 2 <= r <= k-1, 1 <= j < 2^r-1, in which case a(2^k-2^r+j)=a(j)*wt(k-r-1) (where wt(i) = A000120(i)).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0

Offset: 1

N. J. A. Sloane, Jul 25 2014

Other sequences defined by a recurrence of this class (see the Formula and Maple sections) include A245180, A245195, A048896, A245536, A038374.

			May be arranged into blocks of lengths 1,2,4,8,...:
0,
0, 1,
0, 0, 1, 1,
0, 0, 0, 0, 1, 0, 1, 2,
0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 2, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 0, 1, 2,
...

Cf. A000120, A160239, A245180, A245195, A014081, A048896, A038374, A245536, A245537, A245538, A245539, A048896, A245547.

Maple code for this sequence:
wt := proc(n) local w, m, i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end:
G:=[seq(wt(n),n=0..30)];
m:=1;
f:=proc(n) option remember; global m,G; local k,r,j,np;
   k:=1+floor(log[2](n)); np:=2^k-n;
   if np=1 then r:=0; j:=0; else r:=1+floor(log[2](np-1)); j:=2^r-np; fi;
   if j=0 then G[k-r]; else m*G[k-r]*f(j); fi;
end;
[seq(f(n),n=1..120)];
# Maple code for the general recurrence:
G:=[seq(wt(n),n=0..30)]; # replace this by a list G=[G(0), G(1), G(2), ...], remembering that you have to tell Maple G[1] to get G(0), G[2] to get G(1), etc.
m:=1; # replace this by the correct multiplier
f:=proc(n) option remember; global m,G; local k,r,j,np;
   k:=1+floor(log[2](n)); np:=2^k-n;
   if np=1 then r:=0; j:=0; else r:=1+floor(log[2](np-1)); j:=2^r-np; fi;
   if j=0 then G[k-r-1+1]; else m*G[k-r-1+1]*f(j); fi;
end;
[seq(f(n),n=1..120)];
# If G(n) = wt(n) and m=1 we get the present sequence
# If G(n) = A083424(n) and m=1 we get A245537
# If G(n) = A083424(n) and m=2 we get A245538
# If G(n) = A083424(n) and m=4 we get A245539
# If G(n) = A083424(n) and m=8 we get A245180 (and presumably A160239)
# If G(n) = n (n>=0) and m=1 we get A245536
# If G(n) = n+1 (n>=0) and m=1 we get A038374
# If G(n) = (n+1)(n+2)/2 (n>=0) and m=1 we get A245541
# If G(n) = (n+1)(n+2)/2 (n>=0) and m=2 we get A245547
# If G(n) = 2^n (n>=0) and m=1 we get A245195 (= 2^A014081)
# If G(n) = 2^n (n>=0) and m=2 we get A048896

This is an example of a class of sequences defined by the following recurrence.
We first choose a sequence G = [G(0), G(1), G(2), G(3), ...], which are the terms that will appear at the ends of the blocks: a(2^k-1) = G(k-1), and we also choose a parameter m (the "multiplier"). Then the recurrence (this defines a(1), a(2), a(3), ...) is:
a(2^k-2^r)=G(k-r-1) if 0 <= r <= k-1, a(2^k-2^r+j)=m*a(j)*G(k-r-1) if 2 <= r <= k-1, 1 <= j < 2^r-1.
To help apply the recurrence, here are the values of k,r,j for the first few values of n (if n=2^k-2^r we set j=0, although it is not used):
n:  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
k:  1  2  2  3  3  3  3  4  4  4  4  4  4  4  4
r:  0  1  0  2  2  1  0  3  3  3  3  2  2  1  0
j:  0  0  0  0  1  0  0  0  1  2  3  0  1  0  0
--------------------------------------------------
n: 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
k:  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5
r:  4  4  4  4  4  4  4  4  3  3  3  3  2  2  1  0
j:  0  1  2  3  4  5  6  7  0  1  2  3  0  1  0  0
--------------------------------------------------
In the present example G(n) = wt(n) and m=1.

A245195 a(n) = 2^A014081(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 2, 4, 8, 1, 1, 1, 2, 1, 1, 2, 4, 2, 2, 2, 4, 4, 4, 8, 16, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 2, 4, 8, 2, 2, 2, 4, 2, 2, 4, 8, 4, 4, 4, 8, 8, 8, 16, 32, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 2, 4, 8, 1, 1, 1, 2, 1, 1, 2, 4, 2, 2, 2, 4, 4, 4, 8, 16, 2, 2, 2, 4, 2

Offset: 0

N. J. A. Sloane, Jul 24 2014

This sequence provides a bridge between A245180 (and, presumably, A160239) and A014081.
See A245196 for more about this class of sequences.
Run length transform of A011782: 1,1,2,4,8,16,32,64,... - Chai Wah Wu, Oct 19 2016

Chai Wah Wu and Robert Israel, Table of n, a(n) for n = 0..10000
Robert Israel, Proof that A277560 is the same as A245195 [This will be modified to reflect the fact that the two sequences have now been merged]
Chai Wah Wu, Sums of products of binomial coefficients mod 2 and run length transforms of sequences, arXiv:1610.06166 [math.CO], 2016.

Cf. A014081, A245180, A160239, A048896, A245196, A005725, A005940, A011782, A106737, A162510.

# This Maple program applies more generally to a sequence where the recurrence across a block is as follows. The parameters to be set are the sequence G(0), G(1), G(2), ... (the final terms in the blocks), and the multiplier m.
# For n in the range 2^(k-1) <= n < 2^k, write n = 2^k-2^r+j, with 0 <= r <= k-1 and 0 <= j < 2^(r-1), and j=0 if r=0. Then
# (if j=0) a(2^k-2^r) = G(k-r-1),
# (if j>0) a(2^k-2^r+j) = m*G(k-r-1)*a(j).
# Since Maple gives its lists an offset of 1, it is necessary to add 1 to the arguments of G.
# For the present sequence, G(n)=2^n and m=1.
G:=[seq(2^n,n=0..30)];
m:=1;
f:=proc(n) option remember; global m,G; local k,r,j,np;
if n <= 2 then G[0+1] elif n=3 then G[1+1]
elif n=4 then G[0+1] elif n=5 then m*G[0+1] elif n=6 then G[1+1] elif n=7 then G[2+1]
else
   k:=1+floor(log[2](n)); np:=2^k-n;
   if np=1 then r:=0; j:=0; else r:=1+floor(log[2](np-1)); j:=2^r-np; fi;
   if j=0 then G[k-r-1+1]; else m*G[k-r-1+1]*f(j); fi;
fi;
end;
[seq(f(n),n=1..520)]:
# Setting G(n) = A083424(n) and m = 8 gives A245180. Setting G(n) = 2^n and m = 2 gives A048896.
A245195:=n->add(binomial(n,2*k)*binomial(n,k) mod 2, k=0..floor(n/2)): seq(A245195(n), n=0..200); # Wesley Ivan Hurt, Nov 01 2016

Table[Sum[Mod[Binomial[n, 2 k] Binomial[n, k], 2], {k, 0, n}], {n, 0, 85}] (* Michael De Vlieger, Oct 21 2016 *)

a(n) = 2^hammingweight(bitand(n, n>>1)) \\ Charles R Greathouse IV, Jul 16 2016

a(n) = sum(k=0, n, binomial(n, 2*k)*binomial(n,k) % 2); \\ Michel Marcus, Oct 21 2016

from _future_ import division
def A277560(n):
    return sum(int(not (~n & 2*k) | (~n & k)) for k in range(n//2+1))

def A245195(n): return 1<<(n&(n>>1)).bit_count() # Chai Wah Wu, Feb 11 2023

The entries may be arranged into blocks of sizes 1,2,4,8,...:
B_0: 1,
B_1: 1, 2,
B_2: 1, 1, 2, 4,
B_3: 1, 1, 1, 2, 2, 2, 4, 8,
B_4: 1, 1, 1, 2, 1, 1, 2, 4, 2, 2, 2, 4, 4, 4, 8, 16,
B_5: 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 2, 4, 8, 2, 2, 2, 4, 2, 2, 4, 8, 4, 4, 4, 8, 8, 8, 16, 32,
...
Consider the block B_{k-1} containing terms a(2^(k-1)), a(2^(k-1)+1), ..., a(2^k-1). It is convenient to index the terms working backwards from the next, 2^k-th, term. For n in the range 2^(k-1) <= n < 2^k, write n = 2^k-2^r+j, with 0 <= r <= k-1 and 0 <= j < 2^(r-1), and j=0 if r=0. Then
(if j=0) a(2^k-2^r) = 2^(k-r-1),
(if j>0) a(2^k-2^r+j) = 2^(k-r-1)*a(j).
a(n) = A162510(A005940(1+n)). - Antti Karttunen, Oct 29 2016
From Robert Israel, Nov 02 2016: (Start)
a(2*k)   = a(k).
a(4*k+1) = a(k).
a(4*k+3) = 2*a(2*k+1).
G.f. g(x) satisfies g(x) = x + (2*x+1)*g(x^2) - x*g(x^4). (End)
Also, a(n) = Sum_{k=0..floor(n/2)} ((binomial(n,2k)*binomial(n,k)) mod 2). - Chai Wah Wu, Oct 19 2016 and Robert Israel, Nov 04 2016. For proof, see the article by Chai Wah Wu, Sums of products of binomial coefficients mod 2 and run length transforms of sequences, arXiv:1610.06166, or the Robert Israel link.

Changed offset to 0, merged former entry A277560 from Chai Wah Wu (Oct 19 2016) with this sequence. - N. J. A. Sloane, Nov 05 2016

A245543 First differences of A160239.

Original entry on oeis.org

7, 0, 16, -16, 56, -40, 88, -104, 56, 0, 128, -168, 168, -80, 304, -408, 56, 0, 128, -128, 448, -320, 704, -872, 168, 0, 384, -464, 784, -480, 1312, -1720, 56, 0, 128, -128, 448, -320, 704, -832, 448, 0, 1024, -1344, 1344, -640, 2432, -3304, 168, 0, 384, -384, 1344, -960, 2112, -2576, 784, 0

Offset: 0

N. J. A. Sloane, Jul 26 2014

sign

The increase in number of ON cells in A160239 in going from one step to the next.

a(n) = A160239(n+1) - A160239(n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A160239, A245180, A245542.

a245542 n = a245542_list !! n
a245542_list = scanl1 (+) a160239_list
-- Reinhard Zumkeller, Feb 13 2015

A245537 Write n>=1 as either n=2^k-2^r with 0 <= r <= k-1, in which case a(2^k-2^r)=A083424(k-r-1), or as n=2^k-2^r+j with 2 <= r <= k-1, 1 <= j < 2^r-1, in which case a(2^k-2^r+j)=A083424(k-r-1)*a(j).

Original entry on oeis.org

1, 1, 3, 1, 1, 3, 14, 1, 1, 1, 3, 3, 3, 14, 52, 1, 1, 1, 3, 1, 1, 3, 14, 3, 3, 3, 9, 14, 14, 52, 216, 1, 1, 1, 3, 1, 1, 3, 14, 1, 1, 1, 3, 3, 3, 14, 52, 3, 3, 3, 9, 3, 3, 9, 42, 14, 14, 14, 42, 52, 52, 216, 848, 1, 1, 1, 3, 1, 1, 3, 14, 1, 1, 1, 3, 3, 3, 14, 52

Offset: 1

N. J. A. Sloane, Jul 26 2014

Similar to A245180, except the multiplier 8 in that recurrence is set here to be 1.

See A245196 for a list of other sequences produced by this type of recurrence.

			Arranged into blocks:
1,
1, 3,
1, 1, 3, 14,
1, 1, 1, 3, 3, 3, 14, 52,
1, 1, 1, 3, 1, 1, 3, 14, 3, 3, 3, 9, 14, 14, 52, 216,
1, 1, 1, 3, 1, 1, 3, 14, 1, 1, 1, 3, 3, 3, 14, 52, 3, 3, 3, 9, 3, 3, 9, 42, 14, 14, 14, 42, 52, 52, 216, 848,
...

Cf. A083424, A245180, A245196.

A245538 Write n>=1 as either n=2^k-2^r with 0 <= r <= k-1, in which case a(2^k-2^r)=A083424(k-r-1), or as n=2^k-2^r+j with 2 <= r <= k-1, 1 <= j < 2^r-1, in which case a(2^k-2^r+j)=2A083424(k-r-1)a(j).

Original entry on oeis.org

1, 1, 3, 1, 2, 3, 14, 1, 2, 2, 6, 3, 6, 14, 52, 1, 2, 2, 6, 2, 4, 6, 28, 3, 6, 6, 18, 14, 28, 52, 216, 1, 2, 2, 6, 2, 4, 6, 28, 2, 4, 4, 12, 6, 12, 28, 104, 3, 6, 6, 18, 6, 12, 18, 84, 14, 28, 28, 84, 52, 104, 216, 848, 1, 2, 2, 6, 2, 4, 6, 28, 2, 4, 4, 12, 6, 12, 28, 104

Offset: 1

N. J. A. Sloane, Jul 26 2014

Similar to A245180, except the multiplier 8 in that recurrence is set here to be 2.

See A245196 for a list of other sequences produced by this type of recurrence.

			Arranged into blocks:
1,
1, 3,
1, 2, 3, 14,
1, 2, 2, 6, 3, 6, 14, 52,
1, 2, 2, 6, 2, 4, 6, 28, 3, 6, 6, 18, 14, 28, 52, 216,
1, 2, 2, 6, 2, 4, 6, 28, 2, 4, 4, 12, 6, 12, 28, 104, 3, 6, 6, 18, 6, 12, 18, 84, 14, 28, 28, 84, 52, 104, 216, 848,
...

Cf. A245196, A245180.

A245539 Write n>=1 as either n=2^k-2^r with 0 <= r <= k-1, in which case a(2^k-2^r)=A083424(k-r-1), or as n=2^k-2^r+j with 2 <= r <= k-1, 1 <= j < 2^r-1, in which case a(2^k-2^r+j)=4A083424(k-r-1)a(j).

Original entry on oeis.org

1, 1, 3, 1, 4, 3, 14, 1, 4, 4, 12, 3, 12, 14, 52, 1, 4, 4, 12, 4, 16, 12, 56, 3, 12, 12, 36, 14, 56, 52, 216, 1, 4, 4, 12, 4, 16, 12, 56, 4, 16, 16, 48, 12, 48, 56, 208, 3, 12, 12, 36, 12, 48, 36, 168, 14, 56, 56, 168, 52, 208, 216, 848

Offset: 1

N. J. A. Sloane, Jul 26 2014

Similar to A245180, except the multiplier 8 in that recurrence is set here to be 4.

See A245196 for a list of other sequences produced by this type of recurrence.

			Arranged into blocks:
1,
1, 3,
1, 4, 3, 14,
1, 4, 4, 12, 3, 12, 14, 52,
1, 4, 4, 12, 4, 16, 12, 56, 3, 12, 12, 36, 14, 56, 52, 216,
1, 4, 4, 12, 4, 16, 12, 56, 4, 16, 16, 48, 12, 48, 56, 208, 3, 12, 12, 36, 12, 48, 36, 168, 14, 56, 56, 168, 52, 208, 216, 848,
...

Cf. A245196, A245180.

cp's OEIS Frontend

A245540 Partial sums of A245180.

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A245181 Numbers appearing in A245180.

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A245190 Products of successive blocks of 2^n terms of A245180.

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A245196 Write n>=1 as either n=2^k-2^r with 0 <= r <= k-1, in which case a(2^k-2^r)=wt(k-r-1), or as n=2^k-2^r+j with 2 <= r <= k-1, 1 <= j < 2^r-1, in which case a(2^k-2^r+j)=a(j)*wt(k-r-1) (where wt(i) = A000120(i)).

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A245195 a(n) = 2^A014081(n).

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A245543 First differences of A160239.

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A245537 Write n>=1 as either n=2^k-2^r with 0 <= r <= k-1, in which case a(2^k-2^r)=A083424(k-r-1), or as n=2^k-2^r+j with 2 <= r <= k-1, 1 <= j < 2^r-1, in which case a(2^k-2^r+j)=A083424(k-r-1)*a(j).

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A245538 Write n>=1 as either n=2^k-2^r with 0 <= r <= k-1, in which case a(2^k-2^r)=A083424(k-r-1), or as n=2^k-2^r+j with 2 <= r <= k-1, 1 <= j < 2^r-1, in which case a(2^k-2^r+j)=2*A083424(k-r-1)*a(j).

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A245538 Write n>=1 as either n=2^k-2^r with 0 <= r <= k-1, in which case a(2^k-2^r)=A083424(k-r-1), or as n=2^k-2^r+j with 2 <= r <= k-1, 1 <= j < 2^r-1, in which case a(2^k-2^r+j)=2A083424(k-r-1)a(j).

A245539 Write n>=1 as either n=2^k-2^r with 0 <= r <= k-1, in which case a(2^k-2^r)=A083424(k-r-1), or as n=2^k-2^r+j with 2 <= r <= k-1, 1 <= j < 2^r-1, in which case a(2^k-2^r+j)=4A083424(k-r-1)a(j).