A269457 a(n) = 5*(n + 1)*(n + 4)/2.

10, 25, 45, 70, 100, 135, 175, 220, 270, 325, 385, 450, 520, 595, 675, 760, 850, 945, 1045, 1150, 1260, 1375, 1495, 1620, 1750, 1885, 2025, 2170, 2320, 2475, 2635, 2800, 2970, 3145, 3325, 3510, 3700, 3895, 4095, 4300, 4510, 4725, 4945, 5170, 5400, 5635

Offset: 0

Author

Ilya Gutkovskiy, Feb 27 2016

Keywords

• nonn
• easy

Comments

More generally, the ordinary generating function for the sequences of the form k*(n + 1)*(n - 1 + k)/2 is (k*(k - 1)/2 + (k*(3 - k)/2)*x)/(1 - x)^3 (see links section).

Examples

			a(0) = 0 + 1 + 2 + 3 + 4 = 10;
a(1) = 0 + 1 + 2 + 3 + 4 + 1 + 2 + 3 + 4 + 5 = 25;
a(2) = 0 + 1 + 2 + 3 + 4 + 1 + 2 + 3 + 4 + 5 + 2 + 3 + 4 + 5 + 6 = 45, etc.
		

Links

• Ilya Gutkovskiy, Sequences of the form k*(n + 1)*(n - 1 + k)/2.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000096, A008587, A028895, A040002, A045943, A054000, A055998, A067728.

Programs

Magma

[5*(n+1)*(n+4)/2: n in [0..50]]; // Vincenzo Librandi, Feb 28 2016

Mathematica

Table[5 (n + 1) ((n + 4)/2), {n, 0, 45}]
Table[Sum[5 (k + 2), {k, 0, n}], {n, 0, 45}]
LinearRecurrence[{3, -3, 1}, {10, 25, 45}, 46]

PARI

a(n) = 5*(n + 1)*(n + 4)/2; \\ Michel Marcus, Feb 29 2016

PARI

Vec(5*(2-x)/(1-x)^3 + O(x^100)) \\ Altug Alkan, Mar 04 2016

Formula

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

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

a(n) = Sum_{k=0..n} 5*(k + 2) = Sum_{k=0..n} A008587(k + 2).

Sum_{n>=0} 1/a(n) = 11/45 = 0.24444444444... = A040002.

a(n) = 5*A000096(n+1).

a(n) = A055998(2*n+2) + A055998(n+1). - Bruno Berselli, Sep 23 2016

E.g.f.: 5*exp(x)*(4 + 6*x + x^2)/2. - Elmo R. Oliveira, Dec 24 2024