A229896 Sizes of logical groups of the same integer in A229895.

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

Offset: 1

Author

Carl R. White, Oct 03 2013

Keywords

• nonn
• easy
• tabl

Comments

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).

Links

• Carl R. White, Rows 1..44 for triangle, flattened

Crossrefs

Cf. A229873, A229895, A031972.

Programs

Maple

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

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"