A254473 24-hedral numbers: a(n) = (2*n + 1)*(8*n^2 + 14*n + 7).

7, 87, 335, 847, 1719, 3047, 4927, 7455, 10727, 14839, 19887, 25967, 33175, 41607, 51359, 62527, 75207, 89495, 105487, 123279, 142967, 164647, 188415, 214367, 242599, 273207, 306287, 341935, 380247, 421319, 465247, 512127, 562055, 615127, 671439, 731087

Offset: 0

Author

Luciano Ancora, Mar 26 2015

Keywords

• nonn
• easy

Comments

This sequence is very close to the A046142 sequence: a(n) is asymptotic to A046142(n) as n tends to infinity.

The formula for A046142, the Haüy rhombic dodecahedral number, is remarkably similar, (2*n-1)*(8*n^2-14*n+7), where the first factor of the dodecahedral formula has "+1" versus "-1" in the 24-hedral formula, and the second factor of the former has "-14n" versus the latter of "+14n". Note that the rhombic dodecahedron has 24 faces, further explaining the relationship. The difference of these sequences is diff(n)=72*n^2 + 14. - Peter M. Chema, Jan 09 2016

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Luciano Ancora, The 24-hedral Number
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000447, A016755, A046142.

Programs

Magma

[(2*n+1)*(8*n^2+14*n+7): n in [0..40]]; // Bruno Berselli, Mar 27 2015

Maple

seq((2*n + 1)*(8*n^2 + 14*n + 7), n=0..100); # Robert Israel, Jan 11 2016

Mathematica

Table[(2 n + 1) (8 n^2 + 14 n + 7), {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {7, 87, 335, 847}, 40]

PARI

vector(40, n, n--; (2*n+1)*(8*n^2+14*n+7)) \\ Bruno Berselli, Mar 27 2015

Sage

[(2*n+1)*(8*n^2+14*n+7) for n in (0..40)] # Bruno Berselli, Mar 27 2015

Formula

G.f.: (7 + 59*x + 29*x^2 + x^3)/(1 - x)^4.

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

a(n) = -A046142(-n) with A046142(0) = -7.

a(n) = 6*Sum_{k=0..n} (2*k+1)^2 + (2*n+1)^3. - Robert FERREOL, Nov 13 2023