A254473 - OEIS

%I #45 Dec 11 2023 15:28:37

%S 7,87,335,847,1719,3047,4927,7455,10727,14839,19887,25967,33175,41607,

%T 51359,62527,75207,89495,105487,123279,142967,164647,188415,214367,

%U 242599,273207,306287,341935,380247,421319,465247,512127,562055,615127,671439,731087

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

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

%C 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

%H Robert Israel, <a href="/A254473/b254473.txt">Table of n, a(n) for n = 0..10000</a>

%H Luciano Ancora, <a href="/A254473/a254473_1.pdf">The 24-hedral Number</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

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

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

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

%F a(n) = 6*Sum_{k=0..n} (2*k+1)^2 + (2*n+1)^3. - _Robert FERREOL_, Nov 13 2023

%p seq((2*n + 1)*(8*n^2 + 14*n + 7), n=0..100); # _Robert Israel_, Jan 11 2016

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

%o (PARI) vector(40, n, n--; (2*n+1)*(8*n^2+14*n+7)) \\ _Bruno Berselli_, Mar 27 2015

%o (Sage) [(2*n+1)*(8*n^2+14*n+7) for n in (0..40)] # _Bruno Berselli_, Mar 27 2015

%o (Magma) [(2*n+1)*(8*n^2+14*n+7): n in [0..40]]; // _Bruno Berselli_, Mar 27 2015

%Y Cf. A000447, A016755, A046142.

%K nonn,easy

%O 0,1

%A _Luciano Ancora_, Mar 26 2015