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A214684
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a(1)=1, a(2)=1, and, for n>2, a(n)=(a(n-1)+a(n-2))/5^k, where 5^k is the highest power of 5 dividing a(n-1)+a(n-2).
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6
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1, 1, 2, 3, 1, 4, 1, 1, 2, 3, 1, 4, 1, 1, 2, 3, 1, 4, 1, 1, 2, 3, 1, 4, 1, 1, 2, 3, 1, 4, 1, 1, 2, 3, 1, 4, 1, 1, 2, 3, 1, 4, 1, 1, 2, 3, 1, 4, 1, 1, 2, 3, 1, 4, 1, 1, 2, 3, 1, 4, 1, 1, 2, 3, 1, 4, 1, 1, 2, 3, 1, 4, 1, 1, 2, 3, 1, 4, 1, 1
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OFFSET
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1,3
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COMMENTS
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This sequence is periodic with period 1,1,2,3,1,4 of length 6.
It appears that for most choices of a(1), a(2), and divisor b^k (replacing 5^k), the resulting sequence is not periodic.
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LINKS
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FORMULA
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G.f.: x*(1+x+2*x^2+3*x^3+x^4+4*x^5)/((1-x)*(1+x)*(1-x+x^2)*(1+x+x^2)) . - Colin Barker, Jul 08 2014
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MATHEMATICA
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CoefficientList[Series[(4*x^5 + x^4 + 3*x^3 + 2*x^2 + x + 1)/((1 - x)*(x + 1)*(x^2 - x + 1)*(x^2 + x + 1)), {x, 0, 100}], x] (* Wesley Ivan Hurt, Jul 08 2014 *)
LinearRecurrence[{0, 0, 0, 0, 0, 1}, {1, 1, 2, 3, 1, 4}, 80] (* Ray Chandler, Aug 25 2015 *)
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PROG
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(PARI) lista(nn) = {va = vector(nn); va[1] = 1; va[2] = 1; for (n=3, nn, sump = va[n-1] + va[n-2]; va[n] = sump/5^(valuation(sump, 5)); ); va; } \\ Michel Marcus, Jul 08 2014
(PARI) Vec(-x*(4*x^5+x^4+3*x^3+2*x^2+x+1)/((x-1)*(x+1)*(x^2-x+1)*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Jul 08 2014
(Magma) I:=[1, 1, 2, 3, 1, 4]; [n le 6 select I[n] else Self(n-6): n in [1..100]]; // G. C. Greubel, Mar 08 2024
(SageMath)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x*(1+x+2*x^2+3*x^3+x^4+4*x^5)/(1-x^6) ).list()
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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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