

A347315


a(n) = sum of row beginning with n when inventory sequence A342585 is written as an irregular triangle.


2



0, 2, 6, 11, 17, 24, 32, 40, 51, 63, 76, 89, 102, 116, 132, 149, 169, 188, 208, 228, 249, 272, 297, 322, 349, 377, 404, 432, 461, 494, 528, 562, 597, 632, 667, 703, 740, 778, 820, 862, 903, 945, 991, 1038, 1085, 1132, 1181, 1229, 1277, 1328, 1380, 1434, 1487
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OFFSET

0,2


COMMENTS

The inventory sequence A342585 counts, for k = 0, 1, 2, ..., the k's that have occurred so far, and if zero, restarts with k = 0. The rows end where the zeros occur.
The sequence appears to grow approximately quadratically. More precisely, b(n) = sqrt(a(n)) is roughly a straight line over increasingly large intervals, but the slope is slightly larger at the beginning and then decreasing towards the end of these intervals. For example, on [1..80] the slope is almost exactly 0.72; on [150..250] the slope is roughly 1.0, over [320..420] the slope is again 0.8, over [430..520] it is again 1.0, over [530..620] it is again 0.8; then the slope increases: b(780..1000) is again a nearly straight line with slope 1.67, etc.  M. F. Hasler, Nov 14 2021


LINKS



EXAMPLE

As an irregular triangle A342585 begins:
0;
1, 1, 0;
2, 2, 2, 0;
3, 2, 4, 1, 1, 0;
4, 4, 4, 1, 4, 0;
...
and the row sums are 0, 2, 6, 11, 17, ...


MATHEMATICA

Join[{0}, Total /@ SplitBy[Block[{c, k, m, nn = 52}, c[0] = 1; Reap[Do[k = 0; While[IntegerQ[c[k]], Set[m, c[k]]; Sow[m]; If[IntegerQ@ c[m], c[m]++, c[m] = 1]; k++]; Sow[0]; c[0]++, nn]][[1, 1]]], # == 0 &][[1 ;; 1 ;; 2]]] (* Michael De Vlieger, Oct 12 2021 *)


PROG

(PARI) A347315_vec(N, c=[], i, s)=vector(N, j, until(c[1+c[i]]++&&!c[i]j==1, while(#c<=i#c<=c[i+1], c=concat(c, 0)); s+=c[i+=1]); s+s=i=0) \\ M. F. Hasler, Nov 14 2021
(Python)
from collections import Counter
def aupton(nn):
num, inventory, rowsum, alst = 0, Counter([0]), 0, [0]
while len(alst) <= nn:
c = inventory[num]
num += 1
rowsum += c
inventory.update([c])
if c == 0:
alst.append(rowsum)
num = rowsum = 0
return alst


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



