A269342 a(n) = (n + 1)*(2*n + 1)*(4*n + 9)/3.

3, 26, 85, 196, 375, 638, 1001, 1480, 2091, 2850, 3773, 4876, 6175, 7686, 9425, 11408, 13651, 16170, 18981, 22100, 25543, 29326, 33465, 37976, 42875, 48178, 53901, 60060, 66671, 73750, 81313, 89376, 97955, 107066, 116725, 126948, 137751, 149150, 161161

Offset: 0

Author

Ilya Gutkovskiy, Feb 24 2016

Keywords

• nonn
• easy

Examples

			a(0) = 0*2 + 1*3 = 3;
a(1) = 0*2 + 1*3 + 2*4 + 3*5 = 26;
a(2) = 0*2 + 1*3 + 2*4 + 3*5 + 4*6 + 5*7 = 85;
a(3) = 0*2 + 1*3 + 2*4 + 3*5 + 4*6 + 5*7 + 6*8 + 7*9 = 196;
a(4) = 0*2 + 1*3 + 2*4 + 3*5 + 4*6 + 5*7 + 6*8 + 7*9 + 8*10 + 9*11 = 375, etc.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000466, A005563, A033996, A195241.

Programs

Magma

[(n+1)*(2*n+1)*(4*n+9)/3: n in [0..50]]; // Vincenzo Librandi, Feb 25 2016

Mathematica

Table[(n + 1) (2 n + 1) (4 n + 9)/3, {n, 0, 38}]
LinearRecurrence[{4, -6, 4, -1}, {3, 26, 85, 196}, 39]
Table[Sum[8 k^2 + 12 k + 3, {k, 0, n}], {n, 0, 38}]

PARI

Vec((3 + 14*x - x^2)/(1 - x)^4 + O(x^50)) \\ Michel Marcus, Feb 25 2016

Formula

G.f.: (3 + 14*x - x^2)/(1 - x)^4.

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

a(n) = Sum_{k=0..n} (8*k^2 + 12*k + 3).

Sum_{n>=0} 1/a(n) = 3*(80*log(2) + 5*Pi - 48)/175 = 0.397024075075621559...