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A343376
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a(0) = 0, a(1) = 1. For n >= 2, a(n) = a(n-1)/(n-1) if n-1 divides a(n-1); otherwise, a(n) = a(n-1) + a(n-2).
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5
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0, 1, 1, 2, 3, 5, 1, 6, 7, 13, 20, 2, 22, 24, 46, 70, 116, 186, 302, 488, 790, 1278, 2068, 94, 2162, 2256, 4418, 6674, 11092, 17766, 28858, 46624, 1504, 47, 1551, 1598, 3149, 4747, 7896, 12643, 20539, 33182, 53721, 86903, 2021, 88924, 90945, 179869, 3827, 183696
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OFFSET
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0,4
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COMMENTS
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a(n) = A000045(n) for n = 0,1,2,3,4,5 and a(n) = A139047(n-1) for n = 1,2,3,...,10. Also, a(n) = n for n = 0,1,5 and a(n) < n for n = 2,3,4,6,7,8,11.
The values of n for which n divides a(n) are given in A343456 and the corresponding quotients are given in A343457.
If r and s are consecutive terms from A343456, then a(r+1) will divide the terms a(r) through a(s). Proof: Clearly, a(r+1) divides itself. Since a(r+1) = a(r)/r implies that r = a(r)/a(r+1), a(r+1) divides a(r). So, a(r+2) = a(r+1)+a(r) is also divisible by a(r+1). Similarly, the remaining terms a(r+3) = a(r+2)+a(r+1) through a(s) = a(s-1)+a(s-2) will also be divisible by a(r+1). QED
Example: a(r) = a(43) = 86903 = 2021*43 = a(44)*(1*43+0)
a(44) = 2021 = 2021* 1 = a(44)*(0*43+1)
a(45) = 88924 = 2021*44 = a(44)*(1*43+1)
a(46) = 90945 = 2021*45 = a(44)*(1*43+2)
a(s) = a(47) = 179869 = 2021*89 = a(44)*(2*43+3).
Notice that the values b and c in a(43+k) = a(44)*(b*43+c) for k = 0,1,2,3,4 follow the extended Fibonacci sequence A212804. In general, any run of terms a(r) through a(s) between successive divisions will be given by a(r+k) = a(r+1)*(A212804(k)*r+A212804(k+1)) for k = 0,1,2,...,s-r.
2 divides a(10) through a(32), 3 divides a(134) through a(144), 5 and 7 divide a(116) through a(140), and 11 divides a(89) through a(297). Conjecture: For any prime p, there exist r and s = k*p from A343456 for which p divides a(r) through a(s).
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LINKS
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EXAMPLE
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a(2) = a(1)/1 = 1, a(3) = a(2)+a(1) = 2, a(4) = a(3)+a(2) = 3, a(5) = a(4)+a(3) = 5, a(6) = a(5)/5 = 1, a(7) = a(6)+a(5) = 6, ...
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MAPLE
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a:= proc(n) option remember; local q; `if`(n<2, n,
`if`(irem(a(n-1), n-1, 'q')=0, q, a(n-1)+a(n-2)))
end:
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MATHEMATICA
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Block[{a = {0, 1}}, Do[AppendTo[a, If[Mod[#, i] == 0, #/i, # + a[[-2]] ]] &@ a[[-1]], {i, 48}]; a] (* Michael De Vlieger, Apr 22 2021 *)
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PROG
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(PARI) lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, if (va[n-1] % (n-1), va[n] = va[n-1] + va[n-2], va[n] = va[n-1]/(n-1)); ); concat(0, va); } \\ Michel Marcus, Apr 20 2021
(Python)
a = [0, 1]
print("0 0")
print("1 1")
for n in range (2, 101):
if a[n-1] % (n - 1) == 0: a.append(a[n-1]//(n-1))
else: a.append(a[n-1] + a[n-2])
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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