It's n-onacci this time!

\(14\) and \(197\) are somewhat special numbers. How?

  • \(14:\) Number of digits in \(14\) is \(2\). Trying to make a n-onacci sequence with \(n=2\) \[1,4,(1+4=)5,(4+5=)9,(5+9=)14...\] Thus, 14 lies in 2-onacci sequence started from its digits.

  • 197: Number of digits in \(197\) is \(3\). Repeating the above steps with \(n=3\), \[1,9,7,(1+9+7=)17,(9+7+17=)33,(7+17+33=)57 \\ ,(17+33+57=)107,(33+57+107)=197...\] Thus, 197 lies in 3-onacci sequence started from its digits.

Such special numbers are called Keith Numbers, with \(14\) being the smallest number that satisfy such property.

Calculate the sum of the 20th and 40th Keith Numbers.

Details and Assumptions:

  • \(\text{n-onacci}\) is a name given by me only.

  • \(\text{Fibonacci sequence}\) is an example of 2-onacci sequence starting with \([0,1]\).

  • \(\text{Tribonacci sequence}\) is an example of 3-onacci sequence starting with \([0,0,1]\).

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