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A117546 Number of representations of n as a sum of distinct tribonacci numbers (A000073).

%I A117546 #12 Feb 16 2025 08:33:00
%S A117546 1,1,1,1,1,1,1,2,1,1,1,1,1,2,2,1,1,1,1,1,2,1,1,1,2,2,2,2,1,1,1,2,1,1,
%T A117546 1,1,1,2,2,1,1,1,1,1,3,2,2,2,2,2,2,3,1,1,1,1,1,2,2,1,1,1,1,1,2,1,1,1,
%U A117546 2,2,2,2,1,1,1,2,1,1,1,1,1,3,3,2,2,2,2,2,4,2,2,2,2,2,3,3,1,1,1,1,1,2,1,1,1,2
%N A117546 Number of representations of n as a sum of distinct tribonacci numbers (A000073).
%C A117546 It can be shown that, like the Fibonacci numbers, the tribonacci numbers are complete; that is, a(n)>0 for all n. There is always a representation, free of three consecutive tribonacci numbers, which is analogous to the Zeckendorf representation of Fibonacci numbers. See A003726.
%H A117546 Reinhard Zumkeller, <a href="/A117546/b117546.txt">Table of n, a(n) for n = 0..10000</a>
%H A117546 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TribonacciNumber.html">Tribonacci Number</a>.
%H A117546 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ZeckendorfRepresentation.html">Zeckendorf Representation</a>.
%e A117546 a(14)=2 because 14 is both 13+1 and 7+4+2+1.
%t A117546 tr={1,2,4,7,13,24,44,81,149}; len=tr[[ -1]]; cnt=Table[0,{len}]; Do[v=IntegerDigits[k,2,Length[tr]]; s=Dot[tr,v]; If[s<=len, cnt[[s]]++ ], {k,2^(Length[tr])-1}]; cnt
%o A117546 (Haskell)
%o A117546 a117546 = p $ drop 3 a000073_list where
%o A117546    p _  0     = 1
%o A117546    p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
%o A117546 -- _Reinhard Zumkeller_, Apr 13 2014
%Y A117546 Cf. A000119 (number of representations of n as a sum of distinct Fibonacci numbers).
%Y A117546 Cf. A240844.
%K A117546 easy,nonn
%O A117546 0,8
%A A117546 _T. D. Noe_ and _Jonathan Vos Post_, Mar 28 2006
%E A117546 a(0)=1 added and offset changed by _Reinhard Zumkeller_, Apr 13 2014