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C. Pekora and Trampoline
time limit per test
2 seconds
memory limit per test
256 megabytes
standard input
standard output

There is a trampoline park with $$$n$$$ trampolines in a line. The $$$i$$$-th of which has strength $$$S_i$$$.

Pekora can jump on trampolines in multiple passes. She starts the pass by jumping on any trampoline of her choice.

If at the moment Pekora jumps on trampoline $$$i$$$, the trampoline will launch her to position $$$i + S_i$$$, and $$$S_i$$$ will become equal to $$$\max(S_i-1,1)$$$. In other words, $$$S_i$$$ will decrease by $$$1$$$, except of the case $$$S_i=1$$$, when $$$S_i$$$ will remain equal to $$$1$$$.

If there is no trampoline in position $$$i + S_i$$$, then this pass is over. Otherwise, Pekora will continue the pass by jumping from the trampoline at position $$$i + S_i$$$ by the same rule as above.

Pekora can't stop jumping during the pass until she lands at the position larger than $$$n$$$ (in which there is no trampoline). Poor Pekora!

Pekora is a naughty rabbit and wants to ruin the trampoline park by reducing all $$$S_i$$$ to $$$1$$$. What is the minimum number of passes she needs to reduce all $$$S_i$$$ to $$$1$$$?


The first line contains a single integer $$$t$$$ ($$$1 \le t \le 500$$$) — the number of test cases.

The first line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 5000$$$) — the number of trampolines.

The second line of each test case contains $$$n$$$ integers $$$S_1, S_2, \dots, S_n$$$ ($$$1 \le S_i \le 10^9$$$), where $$$S_i$$$ is the strength of the $$$i$$$-th trampoline.

It's guaranteed that the sum of $$$n$$$ over all test cases doesn't exceed $$$5000$$$.


For each test case, output a single integer — the minimum number of passes Pekora needs to do to reduce all $$$S_i$$$ to $$$1$$$.

1 4 2 2 2 2 2
2 3
1 1 1 1 1

For the first test case, here is an optimal series of passes Pekora can take. (The bolded numbers are the positions that Pekora jumps into during these passes.)

  • $$$[1,4,\textbf{2},2,\textbf{2},2,\textbf{2}]$$$
  • $$$[1,\textbf{4},1,2,1,\textbf{2},1]$$$
  • $$$[1,\textbf{3},1,2,\textbf{1},\textbf{1},\textbf{1}]$$$
  • $$$[1,\textbf{2},1,\textbf{2},1,\textbf{1},\textbf{1}]$$$

For the second test case, the optimal series of passes is show below.

  • $$$[\textbf{2},3]$$$
  • $$$[1,\textbf{3}]$$$
  • $$$[1,\textbf{2}]$$$

For the third test case, all $$$S_i$$$ are already equal to $$$1$$$.