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A340862
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Number of times the number n turns up in pseudo-Fibonacci sequences starting with [k, 1] (with k >= 1), excluding the starting terms.
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1
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0, 1, 2, 2, 3, 2, 3, 3, 3, 2, 4, 2, 4, 3, 3, 2, 4, 3, 3, 3, 4, 2, 5, 2, 3, 3, 3, 3, 5, 2, 3, 3, 4, 3, 4, 2, 4, 4, 3, 2, 4, 2, 4, 3, 4, 2, 5, 3, 3, 3, 3, 2, 6, 2, 4, 3, 3, 3, 4, 3, 4, 3, 4, 2, 4, 2, 3, 4, 4, 2, 4, 2, 5, 3, 3, 3, 5, 3, 3, 3, 3, 2, 5, 2, 4, 4, 3, 3
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OFFSET
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1,3
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COMMENTS
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In the first 100000 terms, this never exceeds 8. For any n > 2, a(n) will be at least 2, since k=n-1 and k=n-2 will both work.
Conjecture: for n > 2, a(n) appears to be equal to 1 + A067148(n).
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LINKS
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EXAMPLE
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For n=2, the single solution is the third term of the Fibonacci sequence (k=1), so a(2)=1.
For n=3, we observe the value as the fourth term for k=1, and the third term for k=2 for a total count of a(3) = 2.
For n=4, we have k=2 and k=3, so a(4) = 2.
For n=5, we have k=1, k=3, k=4.
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PROG
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(Python)
def get_val(n):
res = 0
for k in range(1, n):
(a, b) = (k, 1)
while b < n:
(a, b) = (b, a+b)
if b == n:
res += 1
return res
(PARI) a(n) = my(c, x, y=1); while(n>=x+=2*y, y=x-y; x-=y; if((n-y)%x==0, c++)); c; \\ Jinyuan Wang, Mar 20 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Offset changed and a(1) inserted by Jinyuan Wang, Mar 20 2021
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STATUS
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approved
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