A331943 a(n) = n^2 + 1 - ceiling((n + 2)/3).

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

Offset: 1

Author

Hugo Pfoertner, Feb 10 2020

Keywords

• easy
• nonn

Comments

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.

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Crossrefs

Cf. A001008, A001620, A002805, A008620, A331777, A331778.

Programs

Mathematica

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 *)

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)),", "))

Formula

From Colin Barker, Feb 10 2020: (Start)

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