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COMMENTS
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This is looking one step past the well-known rules for the last digit of successive powers: Powers of integers ending in digit 2 always repeat 2486 in the last digit pattern, powers of integers ending in digit 3 always repeat 3971, of integers ending in 4 repeat 46, of integers ending in 1, 5, and 6 repeat themselves, of integers ending in 7 repeat 7931, of integers ending in 8 repeat 8426, and of integers ending in 9 repeat 91.
Is there a pattern in the repeating patterns in the penultimate digits? Possibly 99 patterns, for x = 01 to 99?
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EXAMPLE
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Powers of 2:
2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096...
Second-to-the-last digits, skipping the one-digit powers: 1,3,6,2,5,1,2,4,9,9,8,6...
Find repeating pattern and concatenate digits: 13625124998637487500
10 does not repeat its penultimate digit (1), so a(10)=0.
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