A100164 Structured rhombic triacontahedral numbers (vertex structure 11).

1, 32, 143, 384, 805, 1456, 2387, 3648, 5289, 7360, 9911, 12992, 16653, 20944, 25915, 31616, 38097, 45408, 53599, 62720, 72821, 83952, 96163, 109504, 124025, 139776, 156807, 175168, 194909, 216080, 238731, 262912, 288673, 316064, 345135, 375936, 408517, 442928

Offset: 1

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

• easy
• nonn

Comments

Also structured triakis icosahedral numbers (vertex structure 11) (cf. A100172 = alternate vertex).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A100165 (alternate vertex), A100145 for more on structured polyhedral numbers.

Cf. A001622, A100172.

Programs

Magma

[(1/6)*(50*n^3-60*n^2+16*n): n in [1..40]]; // Vincenzo Librandi, Jul 25 2011

Mathematica

a[n_] := (n*(5*n - 2)*(5*n - 4))/3; Array[a, 30] (* Amiram Eldar, Sep 20 2022 *)

Formula

a(n) = (1/6)*(50*n^3 - 60*n^2 + 16*n) = (1/3)*n*(5*n-2)*(5*n-4).

From Jaume Oliver Lafont, Sep 08 2009: (Start)

a(n) = (5*(n-1) + 1)*(5*(n-1) + 3)*(5*(n-1) + 5)/15.

G.f.: x*(1 + 28*x + 21*x^2)/(1-x)^4. (End)

Sum_{n>=1} 1/a(n) = 3*sqrt((25-2*sqrt(5))/5)*Pi/16 + 9*sqrt(5)*log(phi)/16 - 15*log(5)/32, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 20 2022