Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the 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