Sums of Padovan numbers equal to sums of powers of plastic number

The Padovan numbers A000931 at OEIS are extended with negative indices in the obvious way and then plotted. For each positive integer, the red and green dots show a set of Padovan numbers that add up to that integer. If the Padovan numbers are p_i+...+p_j, then it is required that b^i+...+b^j also sum to the same integer, where the base b is the plastic number (see Wikipedia). The number of summands is shown in the video since these numbers were used to create the video Tuba Fantasy and the more popular Tuba Fantasy (Slow Version).

The red/green/blue color coding distinguishes the negative, positive and zero indexed Padovan numbers. With this distinction, each sum of can be partitioned into three groups of summands. It can be seen that the negative indexed Padovan numbers add up to form larger portion of each sum. The ratio negative/total appears to be headed for a limit value.

Most of the negative indexed Padovan numbers make a positive contribution to the sum. In fact, the negative digit with the largest absolute value always yields a positive number. But the
negative index with the second largest absolute value (the second red dot from the left) sometimes dips below the line, yielding a negative value. This happens at 8, 23 and 42 and would continue 120 285 549 1495 3568 7048 12673 19427 27824 27832 27843 40233 40244 107526 173016 173019 (Not in OEIS).

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Science & Technology

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• math