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A229896
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Sizes of logical groups of the same integer in A229895.
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1
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1, 1, 4, 1, 5, 27, 1, 7, 37, 256, 1, 9, 61, 369, 3125, 1, 11, 91, 671, 4651, 46656, 1, 13, 127, 1105, 9031, 70993, 823543, 1, 15, 169, 1695, 15961, 144495, 1273609, 16777216, 1, 17, 217, 2465, 26281, 269297, 2685817, 26269505, 387420489, 1, 19, 271, 3439
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OFFSET
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1,3
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COMMENTS
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The two ones at the start of the parent sequence represent parent and child 1-tuples in the grandparent sequence [(1) and (2) respectively], hence this sequence also starts with 1, 1 rather than 2, which would otherwise be a more sensible way to describe the pair of ones.
All other elements are effectively run-lengths of strings of the same integer in A229895.
The first occurrence of an integer, n, in the parent sequence, is the first of a run of n^n elements of value n. For later occurrences, the run length is n^k-(n-1)^k where k is the size of the k-tuple in the grandparent sequence, A229873.
The elements can be arranged into a triangle thus:
.... 1
.... 1, 4
.... 1, 5, 27
.... 1, 7, 37, 256
.... 1, 9, 61, 369, 3125
.... etc.
where the n-th line is:
.... n^1-(n-1)^1, n^2-(n-1)^2, ..., n^(k-1)-(n-1)^(k-1), n^n; 1 <= k < n
The first terms, for sufficiently large n simplifying as:
.... 1, 2n-1, 3n^2-3n+1, 4n^3-6n^2+4n-1, etc.
Row sums are first differences of A031972, and thus the cumulative sum of rows at the end of each row is A031972 itself, i.e., n*(n^n - 1)/(n-1).
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LINKS
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MAPLE
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T := proc (n, k) if k < n then n^k-(n-1)^k elif k = n then n^n else end if end proc: for n to 12 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jan 30 2017
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PROG
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(bc) /* GNU bc */ for(n=1; n<=10; n++)for(p=1; p<=n; p++){if(p==n){t=n^n}else{t=n^p-(n-1)^p}; print t, ", "}; print "...\n"
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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