cp's OEIS Frontend

A317298 a(n) = (1/2)*(1 + (-1)^n + 2*n + 4*n^2).

%I A317298 #43 Sep 08 2022 08:46:22
%S A317298 1,3,11,21,37,55,79,105,137,171,211,253,301,351,407,465,529,595,667,
%T A317298 741,821,903,991,1081,1177,1275,1379,1485,1597,1711,1831,1953,2081,
%U A317298 2211,2347,2485,2629,2775,2927,3081,3241,3403,3571,3741,3917,4095,4279,4465,4657
%N A317298 a(n) = (1/2)*(1 + (-1)^n + 2*n + 4*n^2).
%C A317298 For n > 0, first differences of A304487.
%C A317298 All the terms of this sequence are odd numbers.
%H A317298 Stefano Spezia, <a href="/A317298/b317298.txt">Table of n, a(n) for n = 0..10000</a>
%H A317298 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A317298 a(n) = (1/2)*(A033999(n) + A005408(n) + 4*A000290(n)).
%F A317298 a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 3.
%F A317298 a(2*n) = A188135(n).
%F A317298 a(2*n-1) = A033567(n), for n > 0.
%F A317298 O.g.f.: -(1 + x + 5*x^2 + x^3)/(-1 + x)^3*(1 + x).
%F A317298 E.g.f.: (1/2)*exp(-x)*(1 + exp(2*x)*(1 + 6*x + 4*x^2)).
%F A317298 Sum_{n>0} 1/a(n) = (1/4)*(Pi - log(4)) + i*(polygamma(0, 1/8 - i*sqrt(7)/8) - polygamma(0, 1/8 + i*sqrt(7)/8))/(2*sqrt(7)) = 1.603596691017309384564895..., where i is the imaginary unit. - _Stefano Spezia_, Feb 10 2019
%F A317298 a(n) = 1 + 2*(n^2 + floor(n/2)). - _Stefano Spezia_, Dec 08 2021
%p A317298 a:=n->(1/2)*(1 + (-1)^n + 2*n + 4*n^2): seq(a(n), n=0..50);
%t A317298 a[n_]:=(1/2)*(1 + (-1)^n + 2*n + 4*n^2); Array[a, 50, 0]
%o A317298 (GAP) Flat(List([0..50], n->(1/2)*(1 + (-1)^n + 2*n + 4*n^2)));
%o A317298 (Magma) [(1/2)*(1+(-1)^n+2*n+4*n^2): n in [0..50]];
%o A317298 (Maxima) makelist((1/2)*(1+(-1)^n+2*n+4*n^2), n, 0, 50);
%o A317298 (PARI) a(n) = (1/2)*(1+(-1)^n+2*n+4*n^2);
%o A317298 (Python) [(1+(-1)**n+2*n+4*n**2)/2 for n in range(0,50)]
%Y A317298 Cf. A000290, A004526, A005408, A033567, A033999, A188135, A304487.
%Y A317298 Cf. A306362 (prime numbers subsequence).
%K A317298 nonn,easy
%O A317298 0,2
%A A317298 _Stefano Spezia_, Jan 22 2019