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A368876
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a(n) is the number of ways n can be reached by the following method: we start with 1, then add or multiply alternately, and each operand must be 3 or 4 or 5.
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1
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1, 0, 1, 2, 2, 2, 2, 3, 2, 1, 0, 1, 0, 0, 2, 2, 1, 3, 2, 4, 5, 2, 4, 9, 5, 2, 7, 8, 6, 6, 4, 7, 5, 3, 8, 5, 3, 2, 4, 8, 2, 0, 4, 4, 7, 0, 0, 4, 3, 4, 2, 1, 2, 3, 2, 1, 3, 1, 1, 5, 2, 2, 6, 4, 3, 5, 4, 5, 7, 2, 3, 10, 5, 4, 10, 7, 5, 7, 7, 10, 9, 2, 5, 17, 9, 5
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OFFSET
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1,4
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COMMENTS
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We may start either with multiplication or summation. After summation the next step will be multiplication or vice versa.
The only zeros in this sequence are at a(n) where n=2, 11, 13, 14, 42, 46, 47, 142, 146, 442; the only ones in this sequence are at a(n) where n=1, 3, 10, 12, 17, 52, 56, 58, 59, 182.
Proof: (Start) Let k be any number greater than 1334. If k == 0 (mod 3) subtract 3, if k == 1 subtract 4, if k == 2 subtract 5, then divide by 3. Repeat this process until k < 1335. Obviously we will get some number between 443 and 1334. By computation it is known that all these numbers can be reached, so all numbers > 442 can be reached, thus the zeros in this sequence can be verified.
Similarly, if k > 1334 can only be reached in one way, some number between 443 and 1334 can be reached in no more than one way, which is a contradiction, thus the ones in this sequence can also be verified. (End)
The graphs represent a fluctuating upward trend. This is caused by an "outlier" at a(2) = 0, whose effect is amplified with recursion.
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LINKS
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EXAMPLE
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There are 3 ways reaching 8: 1*3+5=8, 1*4+4=8 and 1*5+3=8, so a(8)=3.
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MAPLE
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b:= proc(n, t) option remember; `if`(n=1, 1, add(`if`(t=1 and i<n,
b(n-i, 1-t), `if`(t=0 and irem(n, i)=0, b(n/i, 1-t), 0)), i=3..5))
end:
a:= n-> `if`(n=1, 1, add(b(n, i), i=0..1)):
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PROG
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(PARI) for (n=1, #a=vector(#m=vector(86)), if (n==1, a[n] = m[n] = 1, if (n-3>0, a[n] += m[n-3]; ); if (n-4>0, a[n] += m[n-4]; ); if (n-5>0, a[n] += m[n-5]; ); if (n%3==0, m[n] += a[n/3]; ); if (n%4==0, m[n] += a[n/4]; ); if (n%5==0, m[n] += a[n/5]; ); ); print1(a[n]+m[n]-(n==1)", "); );
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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