A297675 - OEIS

%I #11 Apr 17 2022 08:21:25

%S 3,12,24,39,57,78,102,129,159,192,228,267,309,354,402,453,507,564,624,

%T 687,753,822,894,969,1047,1128,1212,1299,1389,1482,1578,1677,1779,

%U 1884,1992,2103,2217,2334,2454,2577,2703,2832,2964,3099,3237,3378,3522,3669,3819,3972,4128

%N a(n) = 3*(n^2+n-4)/2.

%C Also the number of chords in the n-triangular grid graph for n >=2.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphChord.html">Graph Chord</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularGridGraph.html">Triangular Grid Graph</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-1).

%F G.f.: 3*x^2*(-1 - x + x^2)/(-1 + x)^3.

%F Sum_{n>=2} 1/a(n) = 2*Pi*tan(sqrt(17)*Pi/2)/(3*sqrt(17)) + 1/2. - _Amiram Eldar_, Apr 17 2022

%t Table[3 (n^2 + n - 4)/2, {n, 2, 20}]

%t LinearRecurrence[{3, -3, 1}, {3, 12, 24}, 20]

%t CoefficientList[Series[3 (-1 - x + x^2)/(-1 + x)^3, {x, 0, 20}], x]

%o (PARI) a(n) = 3*(n^2+n-4)/2 \\ _Felix Fröhlich_, Jan 03 2018

%o (PARI) Vec(3*x^2*(x^2-x-1)/(x-1)^3 + O(x^40)) \\ _Felix Fröhlich_, Jan 03 2018

%K nonn,easy

%O 2,1

%A _Eric W. Weisstein_, Jan 03 2018