A269237 a(n) = (n + 1)^2*(5*n^2 + 10*n + 2)/2.

1, 34, 189, 616, 1525, 3186, 5929, 10144, 16281, 24850, 36421, 51624, 71149, 95746, 126225, 163456, 208369, 261954, 325261, 399400, 485541, 584914, 698809, 828576, 975625, 1141426, 1327509, 1535464, 1766941, 2023650, 2307361, 2619904, 2963169, 3339106, 3749725, 4197096

Offset: 0

Author

Ilya Gutkovskiy, Apr 09 2016

Keywords

• nonn
• easy

Comments

Partial sums of centered dodecahedral numbers (A005904).

Links

• OEIS Wiki, Centered Platonic numbers
• Eric Weisstein's World of Mathematics, Platonic Solid
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Crossrefs

Cf. A005904, A006566, A008354, A014820, A037270, A132366.

Programs

Magma

[(n + 1)^2*(5*n^2 + 10*n + 2)/2 : n in [0..50]]; // Wesley Ivan Hurt, Oct 15 2017

Maple

A269237:=n->(n + 1)^2*(5*n^2 + 10*n + 2)/2: seq(A269237(n), n=0..50); # Wesley Ivan Hurt, Oct 15 2017

Mathematica

Table[(n + 1)^2 ((5 n^2 + 10 n + 2)/2), {n, 0, 35}]
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 34, 189, 616, 1525}, 36]

PARI

x='x+O('x^99); Vec((1+29*x+29*x^2+x^3)/(1-x)^5) \\ Altug Alkan, Apr 10 2016

Formula

G.f.: (1 + 29*x + 29*x^2 + x^3)/(1 - x)^5.

E.g.f.: exp(x)*(2 + 66*x + 122*x^2 + 50*x^3 + 5*x^4)/2.

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

Sum_{n>=0} 1/a(n) = (5 - Pi^2 - sqrt(15)*Pi*cot(sqrt(3/5)*Pi))/9 = 1.0377796966... . - Vaclav Kotesovec, Apr 10 2016