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This problem has a recursive theme, but not in the usual way. I hope you find it interesting.
Clive, thanks a lot! Problem is online now, minor highlighting was added to numbers etc but hopefully nothing is damaged in this elaborate description! It reminded me of the recent story from Mathias, about red-blue eggs, though there seem to be nothing common beyond colors. Perhaps William and Easter Bunnies are neighbors :)
Rodion, I couldn't believe how rapidly the problem appeared on the site. This must be a record! The presentation looks fine. Thanks again for the speedy posting.
Thanks for the new problem. I think conceptually it is closer to Zeckendorf Lite.
Clive, you are too kind :) actually it's more of coincidence - I'm currently looking for new job (happens to me yearly) and opened mailbox to which some HR sent some invitation to yet another interview - and, voila, instead your letter have fallen into, like magic, even before your notification! Regretfully the day was a bit chaotic and I haven't yet approached the problem, very sorry, but I'll try to rehabilitate soon!
I think conceptually it is closer to Zeckendorf Lite.
Hm-m-m, thanks for the insight, I'll pretend I missed this useful hint :)))