A347315 a(n) = sum of row beginning with n when inventory sequence A342585 is written as an irregular triangle.

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

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

Michael De Vlieger, Table of n, a(n) for n = 0..1000

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
print(aupton(52)) # Michael S. Branicky, Nov 14 2021

Crossrefs

Cf. A342585.

Sequence in context: A217687 A212459 A099056 * A046691 A098167 A081689

Adjacent sequences: A347312 A347313 A347314 * A347316 A347317 A347318

Keyword

nonn

Author

N. J. A. Sloane, Sep 09 2021

Extensions

More terms from Alois P. Heinz, Sep 09 2021

Status

approved