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Pentagonal numbers with even index.
32

%I #54 Jul 03 2020 06:18:18

%S 0,5,22,51,92,145,210,287,376,477,590,715,852,1001,1162,1335,1520,

%T 1717,1926,2147,2380,2625,2882,3151,3432,3725,4030,4347,4676,5017,

%U 5370,5735,6112,6501,6902,7315,7740,8177,8626,9087,9560,10045,10542

%N Pentagonal numbers with even index.

%C If Y is a 3-subset of an (2n+1)-set X then, for n>=4, a(n-1) is the number of 4-subsets of X having at least two elements in common with Y. - _Milan Janjic_, Dec 16 2007

%C Sequence found by reading the line (one of the diagonal axes) from 0, in the direction 0, 5,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - _Omar E. Pol_, Sep 08 2011

%C a(n) is the sum of 2*n consecutive integers starting from 2*n. - _Bruno Berselli_, Jan 16 2018

%H Harvey P. Dale, <a href="/A049452/b049452.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = n*(6*n-1).

%F G.f.: x*(5+7*x)/(1-x)^3.

%F a(n) = C(6*n,2)/3. - _Zerinvary Lajos_, Jan 02 2007

%F a(n) = A001105(n) + A033991(n) = A033428(n) + A049450(n) = A022266(n) + A000326(n). - _Zerinvary Lajos_, Jun 12 2007

%F a(n) = 12*n + a(n-1) - 7 for n>0, a(0)=0. - _Vincenzo Librandi_, Aug 06 2010

%F a(n) = 4*A000217(n) + A001107(n). - _Bruno Berselli_, Feb 11 2011

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=5, a(2)=22. - _Harvey P. Dale_, Mar 07 2012

%F E.g.f.: (6*x^2 + 5*x)*exp(x). - _G. C. Greubel_, Jul 17 2017

%F From _Amiram Eldar_, Jul 03 2020: (Start)

%F Sum_{n>=1} 1/a(n) = 2*log(2) + 3*log(3)/2 - sqrt(3)*Pi/2.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi - log(2) - 2*sqrt(3)*arccoth(sqrt(3)). (End)

%p seq(n*(6*n-1),n=0..42); # _Zerinvary Lajos_, Jun 12 2007

%t Table[n(6n-1),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,5,22},50] (* _Harvey P. Dale_, Mar 07 2012 *)

%o (PARI) a(n)=n*(6*n-1) \\ _Charles R Greathouse IV_, Sep 24 2015

%Y Cf. A000326, A033570, A049453, A001318, A033568, A185019.

%Y See index to sequences with numbers of the form n*(d*n+10-d)/2 in A140090.

%K nonn,easy

%O 0,2

%A Joe Keane (jgk(AT)jgk.org)