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a(n) = binomial(n+1, 2) + 13.
2

%I #37 Sep 12 2024 15:27:37

%S 13,14,16,19,23,28,34,41,49,58,68,79,91,104,118,133,149,166,184,203,

%T 223,244,266,289,313,338,364,391,419,448,478,509,541,574,608,643,679,

%U 716,754,793,833,874,916,959,1003,1048,1094,1141,1189,1238,1288,1339,1391

%N a(n) = binomial(n+1, 2) + 13.

%C Numbers m such that 8*m - 103 is a square. - _Bruce J. Nicholson_, Jul 26 2017

%C For n>2, a(n+14) is the number of ways to tile an equilateral triangle of side length 2*n+1 with smaller equilateral triangles of side length n and side length 1. For example, with n=3, there are a(17)=166 ways to tile an equilateral triangle of side length 7 with smaller ones of sides 3 and 1, including the one way with 49 triangles of sides 1 and the seven ways with four triangles of sides 3 (and thirteen triangles of sides 1). - _Ahmed ElKhatib_ and _Greg Dresden_, Sep 02 2024

%H G. C. Greubel, <a href="/A173036/b173036.txt">Table of n, a(n) for n = 0..500</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) + 13.

%F From _G. C. Greubel_, Feb 19 2021: (Start)

%F G.f.: (13 -25*x +13*x^2)/(1-x)^3.

%F E.g.f.: (26 +2*x +x^2)*exp(x)/2. (End)

%F Sum_{n>=0} 1/a(n) = 2*Pi*tanh(sqrt(103)*Pi/2)/sqrt(103). - _Amiram Eldar_, Dec 13 2022

%p A173036:= n-> 13 + binomial(n+1,2); seq(A173036(n), n=0..60) # _G. C. Greubel_, Feb 19 2021

%t Table[n*(n+1)/2 +13, {n,0,5!}]

%o (PARI) a(n)=n*(n+1)/2 + 13 \\ _Charles R Greathouse IV_, Jun 16 2017

%o (Sage) [13 + binomial(n+1,2) for n in (0..60)] # _G. C. Greubel_, Feb 19 2021

%o (Magma) [13 + Binomial(n+1,2): n in [0..60]]; // _G. C. Greubel_, Feb 19 2021

%Y Cf. A000217.

%K nonn,easy,less

%O 0,1

%A _Vladimir Joseph Stephan Orlovsky_, Feb 07 2010

%E Title and offset changed by _G. C. Greubel_, Feb 19 2021