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Cf. also A123975, A074909, A185700, A071762, A075157, A075158, A243353, A243354, A242424, A243051.

Other identities:

A227183(a(n)) = A227183(n). [This operation doesn't change the total sum of the partition.]

a(n) = A243354(A242424(A243353(n))).

a(n) = A075158(A243051(1+A075157(n))-1).

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Antti Karttunen, <a href="/A226062/b226062_1.txt">Table of n, a(n) for n = 0..8191</a>

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Antti Karttunen, <a href="/A129594/a129594_1.txt">Python-program for computing this and related sequences.</a>

(Python)

def A226062(n):

'''Bulgarian solitaire operation applied to the partition encoded in the runlengths of binary expansion of n.'''

return(ascpart_to_binexp(bulgarian_operation(binexp_to_ascpart(n))))

def bulgarian_operation(ap):

'''Apply Bulgarian Solitaire operation to unordered partition: decrement one from each part and add a new part of the same size as there were originally parts.'''

if([] == ap): return(ap)

np = [len(ap)]+[(part-1) for part in ap if part != 1]

np.sort() # Sort back into nondecreasing order.

return(np)

# Compare the next definition with the Python-prodecure for A227183:

def binexp_to_ascpart(n):

'''Convert n according to its binary expansion to a unique unordered partition, listed in nondecreasing order.'''

ap = []

b = n%2

i = 1

while (n != 0):

n >>= 1

if ((n%2) == b):

i += 1

else:

b = n%2

ap += [i]

return(ap)

def ascpart_to_binexp(ap):

'''Convert an unordered partition (listed in nondecreasing order) to an integer which encodes that partition.'''

n = 0

prevpart = 1

parity = (len(ap)%2)

p2 = 1

for part in ap:

runlen = (part - prevpart)+1

if (0 != parity): n += (((2**runlen)-1) * p2)

p2 <<= runlen

prevpart = part

parity = 1-parity

return(n)

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