A304579 a(n) = (n^2 + 1)*(n^2 + 2).

2, 6, 30, 110, 306, 702, 1406, 2550, 4290, 6806, 10302, 15006, 21170, 29070, 39006, 51302, 66306, 84390, 105950, 131406, 161202, 195806, 235710, 281430, 333506, 392502, 459006, 533630, 617010, 709806, 812702, 926406, 1051650, 1189190, 1339806, 1504302, 1683506

Offset: 0

Author

Vincenzo Librandi, May 17 2018

Keywords

• nonn
• easy

Comments

a(n) and A304578(n) are coprime for all n.

Links

• Daniele Mastrostefano and Carlo Sanna, On numbers n with polynomial image coprime with the nth term of a linear recurrence, arXiv:1805.05114. [math.NT], 2018 (see 4.2, page 7).
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A002522, A304578.

Subsequence of A002378, A045619, A279019.

Programs

Magma

[(n^2+1)*(n^2+2): n in [0..40]];

Mathematica

CoefficientList[Series[2 (1 - 2 x + 10 x^2 + 3 x^4) / (1 - x)^5, {x, 0, 35}], x] (* or *) Table[(n^2 + 1) (n^2 + 2), {n, 0, 40}]
LinearRecurrence[{5,-10,10,-5,1},{2,6,30,110,306},40] (* Harvey P. Dale, Nov 13 2022 *)

PARI

a(n) = my(k=n^2+1); k*(k+1); \\ Altug Alkan, May 17 2018

Formula

G.f.: 2*(1 - 2*x + 10*x^2 + 3*x^4)/(1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

a(n) = A002378(A002522(n)). - Altug Alkan, May 17 2018

Sum_{n>=0} 1/a(n) = 1/4 + coth(Pi)*Pi/2 - coth(sqrt(2)*Pi)*Pi/(2*sqrt(2)). - Amiram Eldar, Feb 24 2023