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A331943
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a(n) = n^2 + 1 - ceiling((n + 2)/3).
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2
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1, 3, 8, 15, 23, 34, 47, 61, 78, 97, 117, 140, 165, 191, 220, 251, 283, 318, 355, 393, 434, 477, 521, 568, 617, 667, 720, 775, 831, 890, 951, 1013, 1078, 1145, 1213, 1284, 1357, 1431, 1508, 1587, 1667, 1750, 1835, 1921, 2010, 2101, 2193, 2288, 2385, 2483, 2584
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OFFSET
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1,2
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COMMENTS
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Related to expansion of exp(2*(H_k-gamma))/k^2 in powers of 1/k as given by A331777/A331778.
The agreement with the results of the PARI code needs an explanation. All numerators corresponding to the computed denominators are 1.
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LINKS
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FORMULA
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G.f.: x*(1 + x + 3*x^2 + x^3) / ((1 - x)^3*(1 + x + x^2)).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>5.
(End)
E.g.f.: (1/9)*(3*exp(x)*x*(2 + 3*x) + 2*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2)). - Stefano Spezia, Feb 14 2020
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MATHEMATICA
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Table[n^2+1-Ceiling[(n+2)/3], {n, 60}] (* or *) LinearRecurrence[{2, -1, 1, -2, 1}, {1, 3, 8, 15, 23}, 60] (* Harvey P. Dale, Aug 30 2021 *)
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PROG
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(PARI) H(n)=sum(j=1, n, 1/j);
A(k)=exp(2*(H(k)-Euler))/k^2;
for(k=1, 51, r=(1/k)*(A(k)-1); print1(denominator(bestappr(r, k*k)), ", "))
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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