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A350311 Replace 2^k in the binary expansion of n with A000930(k+2), Narayana's cows sequence.

%I A350311 #23 Mar 11 2025 08:23:59
%S A350311 0,1,2,3,3,4,5,6,4,5,6,7,7,8,9,10,6,7,8,9,9,10,11,12,10,11,12,13,13,
%T A350311 14,15,16,9,10,11,12,12,13,14,15,13,14,15,16,16,17,18,19,15,16,17,18,
%U A350311 18,19,20,21,19,20,21,22,22,23,24,25,13,14,15,16,16,17
%N A350311 Replace 2^k in the binary expansion of n with A000930(k+2), Narayana's cows sequence.
%C A350311 A048715(n) = m, if and only if a(n) = m and for all k > n a(k) > m.
%H A350311 Rémy Sigrist, <a href="/A350311/b350311.txt">Table of n, a(n) for n = 0..8191</a>
%p A350311 b:= (n, i, j, k)->`if`(n=0, 0, k*irem(n, 2, 'q')+b(q, j, k, i+k)):
%p A350311 a:= n-> b(n, 1$3):
%p A350311 seq(a(n), n=0..100);  # _Alois P. Heinz_, Jan 26 2022
%o A350311 (Python)
%o A350311 def Interpretation(n):
%o A350311     f0, f1, f2, r = 1, 1, 1, 0
%o A350311     while n > 0:
%o A350311         if n%2 == 1:
%o A350311             r = r+f0
%o A350311         n, f0, f1, f2 = n//2, f0+f2, f0, f1
%o A350311     return r
%o A350311 n = 0
%o A350311 while n <= 69:
%o A350311     print(Interpretation(n), end = ", ")
%o A350311     n += 1
%o A350311 (PARI) my(p=Mod('x,'x^3-'x^2-1)); a(n) = vecsum(Vec(lift(subst(Pol(binary(n))*'x^2,'x,p)))); \\ _Kevin Ryde_, Dec 26 2021
%Y A350311 Cf. A000930, A048715, A350215, A350312.
%Y A350311 Cf. A022290 (analog for Fibonacci numbers).
%K A350311 nonn,base
%O A350311 0,3
%A A350311 _A.H.M. Smeets_, Dec 24 2021