|
%I #73 Feb 12 2024 02:28:41
%S 0,5,11,18,26,35,45,56,68,81,95,110,126,143,161,180,200,221,243,266,
%T 290,315,341,368,396,425,455,486,518,551,585,620,656,693,731,770,810,
%U 851,893,936,980,1025,1071,1118,1166,1215,1265,1316,1368,1421,1475
%N a(n) = n*(n+9)/2.
%C Numbers m >= 0 such that 8m+81 is a square. - _Bruce J. Nicholson_, Jul 29 2017
%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 193.
%H G. C. Greubel, <a href="/A056000/b056000.txt">Table of n, a(n) for n = 0..5000</a>
%H Leo Tavares, <a href="/A056000/a056000.jpg">Illustration: Triangular Pairs</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = A000217(n+4) - 10.
%F G.f.: x(5-4x)/(1-x)^3.
%F From _Zerinvary Lajos_, Oct 01 2006: (Start)
%F a(n) = A000096(n) + 3*n.
%F a(n) = A055999(n) + n.
%F a(n) = A056115(n) - n.
%F (End)
%F a(n) = binomial(n,2) - 4*n, n >= 9. - _Zerinvary Lajos_, Nov 25 2006
%F a(n) = A126890(n,4) for n > 3. - _Reinhard Zumkeller_, Dec 30 2006
%F a(n) = A028569(n)/2. - _Zerinvary Lajos_, Feb 12 2007
%F If we define f(n,i,a) = Sum_{k=0..(n-i)} binomial(n,k)*stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then a(n) = -f(n,n-1,5), for n >= 1. - _Milan Janjic_, Dec 20 2008
%F a(n) = n + a(n-1) + 4. - _Vincenzo Librandi_, Aug 07 2010
%F a(n) = Sum_{k=1..n} (k+4). - _Gary Detlefs_, Aug 10 2010
%F Sum_{n>=1} 1/a(n) = 7129/11340. - _R. J. Mathar_, Jul 14 2012
%F a(n) = 5n - floor(n/2) + floor(n^2/2). - _Wesley Ivan Hurt_, Jun 15 2013
%F E.g.f.: (1/2)*(x^2 + 10*x)*exp(x). - _G. C. Greubel_, Jul 17 2017
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/9 - 1879/11340. - _Amiram Eldar_, Jul 03 2020
%F a(n) = A000217(n+1) + A008585(n) - 1. - _Leo Tavares_, Sep 22 2022
%F From _Amiram Eldar_, Feb 12 2024: (Start)
%F Product_{n>=1} (1 - 1/a(n)) = -567*cos(sqrt(89)*Pi/2)/(220*Pi).
%F Product_{n>=1} (1 + 1/a(n)) = 35*cos(sqrt(73)*Pi/2)/(4*Pi). (End)
%t Table[n (n + 9)/2, {n, 0, 50}] (* or *)
%t FoldList[#1 + #2 + 4 &, Range[0, 50]] (* or *)
%t Table[PolygonalNumber[n + 4] - 10, {n, 0, 50}] (* or *)
%t CoefficientList[Series[x (5 - 4 x)/(1 - x)^3, {x, 0, 50}], x] (* _Michael De Vlieger_, Jul 30 2017 *)
%o (PARI) a(n)=n*(n+9)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A000096, A055998, A055999, A001477.
%Y Column m=2 of (1, 5)-Pascal triangle A096940.
%Y Cf. numbers of the form n*(d*n+10-d)/2 indexed in A140090.
%Y Cf. A000217, A008585.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, Jun 16 2000
%E More terms from _James A. Sellers_, Jul 04 2000
|
