# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a024966
Showing 1-1 of 1

%I A024966 #70 Feb 21 2023 02:15:44
%S A024966 0,7,21,42,70,105,147,196,252,315,385,462,546,637,735,840,952,1071,
%T A024966 1197,1330,1470,1617,1771,1932,2100,2275,2457,2646,2842,3045,3255,
%U A024966 3472,3696,3927,4165,4410,4662,4921,5187,5460,5740,6027,6321,6622
%N A024966 7 times triangular numbers: 7*n*(n+1)/2.
%C A024966 Sequence found by reading the line from 0, in the direction 0, 7, ... and the same line from 0, in the direction 1, 21, ..., in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the main diagonal in the spiral. - _Omar E. Pol_, Sep 09 2011
%C A024966 Also sequence found by reading the same line mentioned above in the square spiral whose vertices are the generalized enneagonal numbers A118277. Axis perpendicular to A195145 in the same spiral. - _Omar E. Pol_, Sep 18 2011
%C A024966 Sequence provides all integers m such that 56*m + 49 is a square. - _Bruno Berselli_, Oct 07 2015
%C A024966 Sum of the numbers from 3*n to 4*n. - _Wesley Ivan Hurt_, Dec 22 2015
%H A024966 Ivan Panchenko, <a href="/A024966/b024966.txt">Table of n, a(n) for n = 0..1000</a>
%H A024966 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A024966 a(n) = (7/2)*n*(n+1).
%F A024966 G.f.: 7*x/(1-x)^3.
%F A024966 a(n) = (7*n^2 + 7*n)/2 = 7*A000217(n). - _Omar E. Pol_, Dec 12 2008
%F A024966 a(n) = a(n-1) + 7*n with n > 0, a(0)=0. - _Vincenzo Librandi_, Nov 19 2010
%F A024966 a(n) = A069099(n+1) - 1. - _Omar E. Pol_, Oct 03 2011
%F A024966 a(n) = a(-n-1), a(n+2) = A193053(n+2) + 2*A193053(n+1) + A193053(n). - _Bruno Berselli_, Oct 21 2011
%F A024966 From _Philippe Deléham_, Mar 26 2013: (Start)
%F A024966 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 0, a(1) = 7, a(2) = 21.
%F A024966 a(n) = A174738(7*n+6).
%F A024966 a(n) = A179986(n) + n = A186029(n) + 2*n = A022265(n) + 3*n = A022264(n) + 4*n = A218471(n) + 5*n = A001106(n) + 6*n. (End)
%F A024966 a(n) = Sum_{i=3*n..4*n} i. - _Wesley Ivan Hurt_, Dec 22 2015
%F A024966 E.g.f.: (7/2)*x*(x+2)*exp(x). - _G. C. Greubel_, Aug 19 2017
%F A024966 From _Amiram Eldar_, Feb 25 2022: (Start)
%F A024966 Sum_{n>=1} 1/a(n) = 2/7.
%F A024966 Sum_{n>=1} (-1)^(n+1)/a(n) = (2/7)*(2*log(2) - 1). (End)
%F A024966 From _Amiram Eldar_, Feb 21 2023: (Start)
%F A024966 Product_{n>=1} (1 - 1/a(n)) = -(7/(2*Pi))*cos(sqrt(15/7)*Pi/2).
%F A024966 Product_{n>=1} (1 + 1/a(n)) = (7/(2*Pi))*cosh(Pi/(2*sqrt(7))). (End)
%p A024966 [seq(7*binomial(n,2), n=1..44)]; # _Zerinvary Lajos_, Nov 24 2006
%t A024966 7 Table[n (n + 1)/2, {n, 0, 43}] (* or *)
%t A024966 Table[Sum[i, {i, 3 n, 4 n}], {n, 0, 43}] (* or *)
%t A024966 Table[SeriesCoefficient[7 x/(1 - x)^3, {x, 0, n}], {n, 0, 43}] (* _Michael De Vlieger_, Dec 22 2015 *)
%o A024966 (Magma) [ (7*n^2 + 7*n)/2 : n in [0..50] ]; // _Wesley Ivan Hurt_, Jun 09 2014
%o A024966 (PARI) x='x+O('x^100); concat(0, Vec(7*x/(1-x)^3)) \\ _Altug Alkan_, Dec 23 2015
%Y A024966 Cf. A000217, A001106, A002378, A022264, A022265, A028895, A028896, A033996.
%Y A024966 Cf. A045943, A046092, A069099, A118277, A174738, A179986, A186029, A193053.
%Y A024966 Cf. A195019, A195020, A195145, A218471.
%K A024966 nonn,easy
%O A024966 0,2
%A A024966 Joe Keane (jgk(AT)jgk.org), Dec 11 1999

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE