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A172005
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Shortest SNUSP representation of a number using only + and @
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5
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1, 2, 3, 4, 4, 5, 5, 5, 6, 7, 6, 7, 6, 7, 7, 8, 7, 7, 8, 8, 7, 8, 8, 9, 9, 8, 8, 8, 8, 9, 9, 10, 9, 8, 9, 10, 9, 9, 10, 11, 9, 10, 9, 10, 9, 10, 9, 10, 11, 11, 11, 12, 10, 11, 9, 10, 10, 10, 10, 10, 10, 11, 11, 12, 10, 11, 11, 10, 10, 11, 11, 12, 10, 11, 11, 10, 11, 11, 12, 11, 12, 13, 11, 12
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OFFSET
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1,2
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COMMENTS
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SNUSP is a programming language where each command is an individual letter. The two of concern here are + and @ and #. + increments the current data value, @ is a "subroutine call" and # is a "return". When an @ is encountered, a record of the location is put on a stack and execution continues. When a # is encountered, if there is a return point on the stack, the execution continues at that a single character beyond that return point. If there is no return point on the stack, execution terminates.
Thus "@@++#" would put the first two "@" return points on the stack, increment data twice, return from the second "@" to the last "+", increment the data once more, return from the first "@" to the first "+", increment the data two more times and finally terminate when it hits the "#" with no return points on the stack. The data is always initialized to zero so this effectively puts 5 into the data. In order to place a particular value into the data, there is a minimal string of these characters. The i-th element of the sequence gives the minimal number of characters (excluding the "#" which is always the last character) to produce an SNUSP program which sets the data to i. The string above is a minimal string to produce 5 and have four characters before the # so the 5th item in the sequence is 4.
All sequences for values >= 3 end in @++ or +++, both of which are equivalent so there are an even number of sequences for every value, half of which end in +++ and half of which end in @++. There are several variations to this sequence which I'll also enter in. They include allowing "-" to decrement the data, the number of minimal sequences and numbers which require a "-" in their minimal expression.
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LINKS
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EXAMPLE
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To produce 10, there are 4 minimal sequences, each of length 7 (as always, excluding the #): +@+++++# ++@@+++# +@++@++# ++@@@++# Thus a(10)=7.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Darrell Plank (jar_czar(AT)msn.com), Jan 22 2010
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STATUS
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approved
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