A050509 - OEIS

%I #46 Feb 17 2022 03:56:14

%S 1,10,36,88,175,306,490,736,1053,1450,1936,2520,3211,4018,4950,6016,

%T 7225,8586,10108,11800,13671,15730,17986,20448,23125,26026,29160,

%U 32536,36163,40050,44206,48640,53361,58378,63700,69336,75295,81586,88218

%N House numbers (version 2): a(n) = (n+1)^3 + (n+1)*Sum_{i=0..n} i.

%C Also as a(n) = (1/6)*(9*n^3-3*n^2), n>0: structured pentagonal prism numbers (Cf. A100177 - structured prisms; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004

%C Number of inequivalent tetrahedral edge colorings using at most n+1 colors so that no color appears only once. - _David Nacin_, Feb 22 2017

%H Vincenzo Librandi, <a href="/A050509/b050509.txt">Table of n, a(n) for n = 0..5000</a>

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

%F a(n) = A000578(n+1) + (n+1)*A000217(n).

%F a(n) = (1/2) *(3*n+2)*(n+1)^2.

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=10, a(2)=36, a(3)=88. - _Harvey P. Dale_, Jun 26 2011

%F G.f.: (1+6*x+2*x^2)/(1-x)^4. - _Colin Barker_, Jun 08 2012

%F a(n) = Sum_{i=0..n} (n+1)*(3*i+1). - _Bruno Berselli_, Sep 08 2015

%F Sum_{n>=0} 1/a(n) = 9*log(3) - sqrt(3)*Pi - Pi^2/3 = 1.15624437161388... . - _Vaclav Kotesovec_, Oct 04 2016

%e         *     *

%e a(2) = * * + * * = 10.

%e        * *   * *

%t Table[((1+n)^2*(2+3n))/2,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,10,36,88},40] (* _Harvey P. Dale_, Jun 26 2011 *)

%o (Magma) [(3*n+2)*(n+1)^2/2: n in [0..40]]; // _Vincenzo Librandi_, Jul 19 2011

%o (PARI) a(n)=(1/2)*(3*n+2)*(n+1)^2 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A000217, A000578, A051662.

%Y Cf. similar sequences, with the formula (k*n-k+2)*n^2/2, listed in A262000.

%K nonn,nice,easy

%O 0,2

%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 28 1999