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A246595 Run Length Transform of squares. 11
1, 1, 1, 4, 1, 1, 4, 9, 1, 1, 1, 4, 4, 4, 9, 16, 1, 1, 1, 4, 1, 1, 4, 9, 4, 4, 4, 16, 9, 9, 16, 25, 1, 1, 1, 4, 1, 1, 4, 9, 1, 1, 1, 4, 4, 4, 9, 16, 4, 4, 4, 16, 4, 4, 16, 36, 9, 9, 9, 36, 16, 16, 25, 36, 1, 1, 1, 4, 1, 1, 4, 9, 1, 1, 1, 4, 4, 4, 9, 16, 1, 1, 1, 4, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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).

LINKS

Chai Wah Wu, Table of n, a(n) for n = 0..8192

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015

FORMULA

a(n) = A227349(n)^2. - Omar E. Pol, Feb 10 2015

EXAMPLE

From Omar E. Pol, Feb 10 2015: (Start)

Written as an irregular triangle in which row lengths is A011782:

1;

1;

1,4;

1,1,4,9;

1,1,1,4,4,4,9,16;

1,1,1,4,1,1,4,9,4,4,4,16,9,9,16,25;

1,1,1,4,1,1,4,9,1,1,1,4,4,4,9,16,4,4,4,16,4,4,16,36,9,9,9,36,16,16,25,36;

...

Right border gives A253909: 1 together with the positive squares.

(End)

From Omar E. Pol, Mar 19 2015: (Start)

Also, the sequence can be written as an irregular tetrahedron T(s,r,k) as shown below:

1;

..

1;

..

1;

4;

.......

1,   1;

4;

9;

...............

1,   1,  1,  4;

4,   4;

9;

16;

.............................

1,   1,  1,  4, 1, 1,  4,  9;

4,   4,  4, 16;

9,   9;

16;

25;

......................................................

1,   1,  1,  4, 1, 1,  4,  9, 1, 1, 1, 4, 4, 4, 9, 16;

4,   4,  4, 16, 4, 4, 16, 36;

9,   9,  9, 36;

16, 16;

25;

36;

...

Apart from the initial 1, we have that T(s,r,k) = T(s+1,r,k).

(End)

MAPLE

ans:=[];

for n from 0 to 100 do lis:=[]; t1:=convert(n, base, 2); L1:=nops(t1); out1:=1; c:=0;

for i from 1 to L1 do

   if out1 = 1 and t1[i] = 1 then out1:=0; c:=c+1;

   elif out1 = 0 and t1[i] = 1 then c:=c+1;

   elif out1 = 1 and t1[i] = 0 then c:=c;

   elif out1 = 0 and t1[i] = 0 then lis:=[c, op(lis)]; out1:=1; c:=0;

   fi;

   if i = L1 and c>0 then lis:=[c, op(lis)]; fi;

                   od:

a:=mul(i^2, i in lis);

ans:=[op(ans), a];

od:

ans;

MATHEMATICA

Table[Times @@ (Length[#]^2&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 85}] (* Jean-Fran├žois Alcover, Jul 11 2017 *)

PROG

(Python)

from operator import mul

from functools import reduce

from re import split

def A246595(n):

....return reduce(mul, (len(d)**2 for d in split('0+', bin(n)[2:]) if d != '')) if n > 0 else 1 # Chai Wah Wu, Sep 07 2014

(Sage)

# For the function RLT see A246660.

A246595_list = lambda len: RLT(lambda n: n^2, len)

A246595_list(86) # Peter Luschny, Sep 07 2014

(MIT/GNU Scheme)

(define (A246595 n) (fold-left (lambda (a r) (* a r r)) 1 (bisect (reverse (binexp->runcount1list n)) (- 1 (modulo n 2)))))

;; Other functions are as in A227349 - Antti Karttunen, Sep 08 2014

CROSSREFS

Cf. A000290, A253082.

Cf. A003714 (gives the positions of ones).

Run Length Transforms of other sequences: A071053, A227349, A246588, A246596, A246660, A246661, A246674.

Sequence in context: A026998 A301626 A080061 * A209571 A269845 A124258

Adjacent sequences:  A246592 A246593 A246594 * A246596 A246597 A246598

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Sep 06 2014

STATUS

approved

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Last modified January 20 12:50 EST 2019. Contains 319330 sequences. (Running on oeis4.)