A348966 Variation on the Inventory Sequence A342585: record the number of occurrences of the pair sum of all adjacent terms until 0 is recorded, then restart the count from 0. Start with a(0) = 0. See the Comments.

0, 0, 1, 1, 1, 0, 1, 3, 2, 0, 1, 4, 3, 0, 1, 5, 3, 1, 2, 2, 1, 1, 1, 0, 1, 7, 5, 3, 3, 2, 2, 1, 3, 0, 1, 8, 5, 5, 5, 3, 2, 1, 4, 1, 2, 0, 1, 9, 6, 7, 5, 6, 2, 1, 5, 1, 3, 1, 2, 2, 0, 1, 10, 7, 9, 8, 6, 4, 1, 5, 1, 4, 2, 2, 2, 1, 1, 1, 2, 0, 1, 11, 10, 11, 10, 8, 7, 1, 6, 1, 4, 2, 3, 2, 1, 2, 1, 2

Offset

0,8

Comments

This sequence is a variation of A342585. Here we record the number of previous occurrences of the pair sum of all adjacent terms until 0 is recorded, after which the pair sum count restarts at 0. For example the terms 0,0,1,1,1 contain one pair that sums to 0 (0,0), one pair that sums to 1 (0,1), and two pairs that sum to 2 (1,1 and 1,1). See the Examples below.

After 20 million terms the largest term is a(19997365) = 512758, which counts the occurrences of pairs that sum to 15, while there are 13766 terms between zeros. It is likely the most common sum increases to arbitrarily large values as n->infinity.

Links

Table of n, a(n) for n=0..97.

Scott R. Shannon, Image of the first 1 million terms.

Example

a(1) = 0 as there have been no pairs so far in the sequence.
a(2) = 1 as there has been one pair that sums to 0: a(0) + a(1).
a(3) = 1 as there has been one pair that sums to 1: a(1) + a(2).
a(4) = 1 as there has been one pair that sums to 2: a(2) + a(3).
a(5) = 0 as there have been no pairs that sum to 3. The count now resets to 0.
a(6) = 1 as there has been one pair that sums to 0: a(0) + a(1).
a(7) = 3 as there have been three pairs that sum to 1: a(1) + a(2), a(4) + a(5), a(5) + a(6).

Prog

(Python)
from collections import Counter
def aupton(terms):
    num, alst, inventory = 0, [0, 0], Counter([0])
    for n in range(2, terms+1):
        c = inventory[num]
        num = 0 if c == 0 else num + 1
        alst.append(c)
        inventory.update([alst[-2] + alst[-1]])
    return alst
print(aupton(97)) # Michael S. Branicky, Nov 05 2021

Crossrefs

Cf. A342585, A348967 (pair differences), A000045.

Sequence in context: A286223 A341163 A329278 * A008783 A139144 A360866

Adjacent sequences: A348963 A348964 A348965 * A348967 A348968 A348969

Keyword

nonn,look

Author

Scott R. Shannon, Nov 05 2021

Status

approved