|
%I #89 Feb 12 2024 02:28:36
%S 0,4,9,15,22,30,39,49,60,72,85,99,114,130,147,165,184,204,225,247,270,
%T 294,319,345,372,400,429,459,490,522,555,589,624,660,697,735,774,814,
%U 855,897,940,984,1029,1075,1122,1170,1219,1269,1320,1372,1425,1479
%N a(n) = n*(n + 7)/2.
%C If X is an n-set and Y a fixed (n-4)-subset of X then a(n-3) is equal to the number of 2-subsets of X intersecting Y. - _Milan Janjic_, Aug 15 2007
%C Numbers m >= 0 such that 8m+49 is a square. - _Bruce J. Nicholson_, Jul 28 2017
%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 193.
%H G. C. Greubel, <a href="/A055999/b055999.txt">Table of n, a(n) for n = 0..5000</a>
%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H Leo Tavares, <a href="/A055999/a055999.jpg">Diamond illustration</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: x*(4-3*x)/(1-x)^3.
%F a(n) = A126890(n,3) for n>2. - _Reinhard Zumkeller_, Dec 30 2006
%F a(n) = A028563(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-1,4), for n>=1. - _Milan Janjic_, Dec 20 2008
%F a(n) = n + a(n-1) + 3 (with a(0)=0). - _Vincenzo Librandi_, Aug 07 2010
%F a(n) = Sum_{k=1..n} (k+3). - _Gary Detlefs_, Aug 10 2010
%F Sum_{n>=1} 1/a(n) = 363/490. - _R. J. Mathar_, Jul 14 2012
%F a(n) = 4n - floor(n/2) + floor(n^2/2). - _Wesley Ivan Hurt_, Jun 15 2013
%F a(n) = Sum_{i=4..n+3} i. - _Wesley Ivan Hurt_, Jun 28 2013
%F E.g.f.: (1/2)*x*(x+8)*exp(x). - _G. C. Greubel_, Jul 13 2017
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/7 - 319/1470. - _Amiram Eldar_, Jan 10 2021
%F a(n) = A000290(n+1) - A000217(n-2). - _Leo Tavares_, Jan 28 2023
%F From _Amiram Eldar_, Feb 12 2024: (Start)
%F Product_{n>=1} (1 - 1/a(n)) = 15*cos(sqrt(57)*Pi/2)/(8*Pi).
%F Product_{n>=1} (1 + 1/a(n)) = -63*cos(sqrt(41)*Pi/2)/(8*Pi). (End)
%t Table[n*(n + 7)/2, {n, 0, 50}] (* _G. C. Greubel_, Jul 13 2017 *)
%o (PARI) a(n)=n*(n+7)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Equals A000217(n+3) - 6.
%Y Cf. A000096, A055998, A074171, A056000, A001477, A002522, A028563, A126890.
%Y Third column (m=2) of (1, 4)-Pascal triangle A095666.
%Y Cf. A000290.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, Jun 16 2000
%E More terms from _James A. Sellers_, Jul 04 2000
|
