A281381 a(n) = n*(n + 1)*(4*n + 5)/2.

0, 9, 39, 102, 210, 375, 609, 924, 1332, 1845, 2475, 3234, 4134, 5187, 6405, 7800, 9384, 11169, 13167, 15390, 17850, 20559, 23529, 26772, 30300, 34125, 38259, 42714, 47502, 52635, 58125, 63984, 70224, 76857, 83895, 91350, 99234, 107559, 116337, 125580, 135300, 145509, 156219, 167442, 179190, 191475

Offset: 0

Author

Peter M. Chema, Jan 21 2017

Keywords

• nonn
• easy

Comments

Shares its digital root, zero together with period 9: repeat [3, 3, 3, 6, 6, 6, 9, 9, 9] with A027480.

Final digits cycle a length period 20: repeat [0, 9, 9, 2, 0, 5, 9, 4, 2, 5, 5, 4, 4, 7, 5, 0, 4, 9, 7, 0].

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Partial sums of A195319.

Cf. A006002, A007742, A016061, A027480, A281258.

Programs

Magma

[n*(n+1)*(4*n+5)/2 : n in [0..50]]; // Wesley Ivan Hurt, Aug 30 2022

Mathematica

Table[n (n + 1) (4 n + 5)/2, {n, 0, 45}] (* or *)
CoefficientList[Series[3 x (3 + x)/(1 - x)^4, {x, 0, 45}], x] (* Michael De Vlieger, Jan 21 2017 *)

PARI

concat(0, Vec(3*x*(3 + x) / (1 - x)^4 + O(x^50))) \\ Colin Barker, Jan 21 2017

PARI

a(n) = n*(n + 1)*(4*n + 5)/2 \\ Charles R Greathouse IV, Feb 01 2017

Formula

a(n) = 2*n^3 + 9*n^2/2 + 5*n/2.

a(n) = 3*A016061(n).

a(n) = A006002(n+1)*(n) - A006002(n)*(n-1).

a(n) = A007742(n)*(n - 1)/2.

From Colin Barker, Jan 21 2017: (Start)

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

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. (End)

From Stefano Spezia, Aug 30 2022: (Start)

E.g.f.: exp(x)*x*(18 + 21*x + 4*x^2)/2.

Sum_{n>0} 1/a(n) = 2*(20*log(8) + 10*Pi - 71)/25 = 0.1603805895595720759728288896228498341201... . (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = 4*sqrt(2)*Pi/5 + 4*(3+sqrt(2))*log(2)/5 - 8*sqrt(2)*log(2-sqrt(2))/5 - 178/25. - Amiram Eldar, Sep 22 2022