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Cf. A261159.
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Inspired by an analogous sequence for the French language, published by E. Angelini on Facebook, see A261159.
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3, 0, 1, 4, 32, 2, 2, 2, 2, 2, 2, 10, 10, 10, 10, 10, 10, 10, 2, 2, 2, 8, 2, 2, 8, 2, 2, 8, 2, 2, 8, 2, 2, 8, 2, 2, 8, 2, 5, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 7, 10, 7
3, 0, 1, 4, 32, 2, 2, 2, 2, 2, 2, 10, 10, 10, 10, 10, 10, 10, 2, 2, 2, 8, 2, 2, 8, 2, 2, 8, 2, 2, 8, 2, 2, 8, 2, 2, 8, 2, 5, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 7, 10, 7
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List the length of chunks of letters different from 'e' when the sequence is spelled out in English. (Lexicographically : this is the lexicographically first such sequence of nonnegative integers.)
Spelling out the sequence in English yields: three zero one thrEE zEro onE four thirty-two two two two two two two ten ten ten tentEn tEn tEn tEn...
Counting the lengths of chunks of letters different from 'eE' ("thr", "", "z", "roon", "fourthirtytwotwotwotwotwotwotwot", "nt", "nt", ...) gives: 3, 0, 1, 4, 32, 2, 2, 2, 2, 2, 2, 10, 10, ..., which is the sequence itself.