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A152950 a(n) = 3 + n*(n-1)/2.

%I A152950 #47 Nov 27 2024 15:50:10
%S A152950 3,4,6,9,13,18,24,31,39,48,58,69,81,94,108,123,139,156,174,193,213,
%T A152950 234,256,279,303,328,354,381,409,438,468,499,531,564,598,633,669,706,
%U A152950 744,783,823,864,906,949,993,1038,1084,1131,1179,1228,1278,1329,1381,1434,1488
%N A152950 a(n) = 3 + n*(n-1)/2.
%C A152950 a(1)=3; then add 1 to the first number, then 2, 3, 4, ... and so on.
%C A152950 Numbers m such that 8*m - 23 is a square. - _Bruce J. Nicholson_, Jul 25 2017
%H A152950 Michael De Vlieger, <a href="/A152950/b152950.txt">Table of n, a(n) for n = 1..10000</a>
%H A152950 Ângela Mestre and José Agapito, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Mestre/mestre2.html">Square Matrices Generated by Sequences of Riordan Arrays</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
%H A152950 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A152950 a(n) = A152949(n+1) = 3 + A000217(n-1). - _R. J. Mathar_, Jan 03 2009
%F A152950 a(n) = 3 + C(n,2), n >= 1. - _Zerinvary Lajos_, Mar 12 2009
%F A152950 a(n) = a(n-1) + n - 1 (with a(1)=3). - _Vincenzo Librandi_, Nov 27 2010
%F A152950 Sum_{n>=1} 1/a(n) = 2*Pi*tanh(sqrt(23)*Pi/2)/sqrt(23). - _Amiram Eldar_, Dec 13 2022
%F A152950 From _Elmo R. Oliveira_, Nov 18 2024: (Start)
%F A152950 G.f.: x*(3 - 5*x + 3*x^2)/(1-x)^3.
%F A152950 E.g.f.: exp(x)*(3 + x^2/2) - 3.
%F A152950 a(n) = A027691(n-1)/2.
%F A152950 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
%p A152950 A152950:=n->3 + n*(n-1)/2; seq(A152950(n), n=1..100); # _Wesley Ivan Hurt_, Jan 28 2014
%t A152950 s=3;lst={3};Do[s+=n;AppendTo[lst,s],{n,1,5!}];lst
%t A152950 Table[3 + n*(n-1)/2, {n, 100}] (* _Wesley Ivan Hurt_, Jan 28 2014 *)
%o A152950 (Sage) [3+binomial(n,2) for n in range(1, 55)] # _Zerinvary Lajos_, Mar 12 2009
%o A152950 (PARI) a(n)=3+n*(n-1)/2 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A152950 (Magma) [3+n*(n-1)/2 : n in [1..50]]; // _Wesley Ivan Hurt_, Mar 25 2020
%Y A152950 Cf. A000124, A000217, A027691, A152947, A152948, A152949.
%K A152950 nonn,easy
%O A152950 1,1
%A A152950 _Vladimir Joseph Stephan Orlovsky_, Dec 15 2008