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%I #61 Feb 09 2025 12:05:17
%S 1,12,36,73,123,186,262,351,453,568,696,837,991,1158,1338,1531,1737,
%T 1956,2188,2433,2691,2962,3246,3543,3853,4176,4512,4861,5223,5598,
%U 5986,6387,6801,7228,7668,8121,8587,9066,9558,10063,10581,11112,11656,12213,12783,13366
%N Number of grid points covered by a truncated triangle drawn on the hexagonal lattice with the short sides having length n and the long sides length 2*n.
%C The shapes can be constructed using compass and straightedge. a(n) identical circles must be drawn to create a truncated triangle whose shortest side is n radius lengths. See illustrations of the initial terms in the links.
%H Harvey P. Dale, <a href="/A342914/b342914.txt">Table of n, a(n) for n = 0..1000</a>
%H Mert Aydemir, <a href="/A342914/a342914_1.png">Semi-postesque hexagon lenis</a>.
%H Mert Aydemir, <a href="/A342914/a342914_2.png">Illustration of n=1</a>.
%H Mert Aydemir, <a href="/A342914/a342914_3.png">Illustration of n=3</a>.
%H Mert Aydemir, <a href="/A342914/a342914_4.png">Illustration of n=4</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = (13*n^2 + 9*n + 2)/2.
%F a(n) = A000217(4*n+1) - 3*A000217(n). - _Andrew Howroyd_, Apr 01 2021
%F G.f.: (1 + 9*x + 3*x^2)/(1 - x)^3. - _Stefano Spezia_, Apr 01 2021
%F From _Elmo R. Oliveira_, Feb 09 2025: (Start)
%F E.g.f.: exp(x)*(2 + 22*x + 13*x^2)/2.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%e a(1) = 12, a(2) = 36:
%e * * * * *
%e * * * * * * *
%e * * * * * * * * *
%e * * * * * * * * *
%e * * * * * * *
%e * * * * * *
%e * * * * *
%t CoefficientList[Series[(1+9x+3x^2)/(1-x)^3,{x,0,50}],x] (* or *) LinearRecurrence[{3,-3,1},{1,12,36},50] (* _Harvey P. Dale_, Apr 08 2023 *)
%o (PARI) a(n) = (13*n^2+9*n+2)/2 \\ _Andrew Howroyd_, Apr 01 2021
%Y Cf. A000217, A003215 (regular hexagon).
%K nonn,easy,changed
%O 0,2
%A _Mert Aydemir_, Mar 31 2021