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A152744 7 times pentagonal numbers: a(n) = 7*n*(3*n-1)/2. 4

%I #36 Feb 27 2022 05:46:30

%S 0,7,35,84,154,245,357,490,644,819,1015,1232,1470,1729,2009,2310,2632,

%T 2975,3339,3724,4130,4557,5005,5474,5964,6475,7007,7560,8134,8729,

%U 9345,9982,10640,11319,12019,12740,13482,14245,15029,15834,16660,17507,18375,19264

%N 7 times pentagonal numbers: a(n) = 7*n*(3*n-1)/2.

%H Ivan Panchenko, <a href="/A152744/b152744.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) = (21*n^2 - 7*n)/2 = A000326(n)*7.

%F a(n) = a(n-1) + 21*n - 14 (with a(0)=0). - _Vincenzo Librandi_, Nov 26 2010

%F G.f.: 7*x*(1+2*x)/(1-x)^3. - _Colin Barker_, Feb 14 2012

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - _Harvey P. Dale_, Aug 08 2013

%F a(n) = Sum_{i = 2..8} P(i,n), where P(i,m) = m*((i-2)*m-(i-4))/2. - _Bruno Berselli_, Jul 04 2018

%F E.g.f.: 7*x*(2+3*x)/2. - _G. C. Greubel_, Sep 01 2018

%F From _Amiram Eldar_, Feb 27 2022: (Start)

%F Sum_{n>=1} 1/a(n) = (9*log(3) - sqrt(3)*Pi)/21.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(Pi*sqrt(3) - 6*log(2))/21. (End)

%t Table[7n (3n-1)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,7,35},50] (* _Harvey P. Dale_, Aug 08 2013 *)

%o (PARI) a(n)=7*n*(3*n-1)/2 \\ _Charles R Greathouse IV_, Jun 17 2017

%o (Magma) [7*n*(3*n-1)/2: n in [0..50]]; // _G. C. Greubel_, Sep 01 2018

%Y Cf. A000326, A014642, A152743.

%Y Similar sequences are listed in A316466.

%K nonn,easy

%O 0,2

%A _Omar E. Pol_, Dec 12 2008

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Last modified March 29 05:48 EDT 2024. Contains 371265 sequences. (Running on oeis4.)