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A336288 Numbers of squares formed by this procedure on n-th step: Step 1, draw a unit square. Step n, draw a unit square with center in every intersection of lines of the figure in step n-1.

%I A336288 #53 Jul 30 2025 00:57:24
%S A336288 1,10,43,116,245,446,735,1128,1641,2290,3091,4060,5213,6566,8135,9936,
%T A336288 11985,14298,16891,19780,22981,26510,30383,34616,39225,44226,49635,
%U A336288 55468,61741,68470,75671,83360,91553,100266,109515,119316,129685,140638,152191,164360,177161
%N A336288 Numbers of squares formed by this procedure on n-th step: Step 1, draw a unit square. Step n, draw a unit square with center in every intersection of lines of the figure in step n-1.
%H A336288 Kelvin Voskuijl, <a href="/A336288/b336288.txt">Table of n, a(n) for n = 1..10000</a>
%H A336288 Ilario Miriello, <a href="https://youtu.be/OK446wjO0Js">Step 1,2,3</a>, Youtube video, Jul 16 2020.
%H A336288 Ilario Miriello, <a href="/A336288/a336288.png">Illustration for a(2) and a(3)</a>
%H A336288 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A336288 a(n) = (8*n^3 - 12*n^2 + 7*n)/3.
%F A336288 From _Colin Barker_, Jul 17 2020: (Start)
%F A336288 G.f.: x*(1 + 3*x)^2 / (1 - x)^4.
%F A336288 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.
%F A336288 (End)
%F A336288 E.g.f.: exp(x)*x*(3 + 12*x + 8*x^2)/3. - _Stefano Spezia_, Jul 23 2020
%F A336288 a(n+1) - a(n) = 8*n^2 + 1 = A081585(n). - _Charlie Marion_, Mar 21 2022
%t A336288 Table[(8*n^3 - 12*n^2 + 7*n)/3, {n, 1, 50}] (* _Amiram Eldar_, Jul 16 2020 *)
%t A336288 LinearRecurrence[{4,-6,4,-1},{1,10,43,116},50] (* _Harvey P. Dale_, Sep 12 2021 *)
%o A336288 (Magma) [(8*n^3 - 12*n^2 + 7*n)/3 : n in [1..50]]; // _Wesley Ivan Hurt_, Jul 16 2020
%o A336288 (PARI) a(n) = (8*n^3 - 12*n^2 + 7*n)/3; \\ _Michel Marcus_, Jul 16 2020
%o A336288 (PARI) Vec(x*(1 + 3*x)^2 / (1 - x)^4 + O(x^40)) \\ _Colin Barker_, Jul 17 2020
%Y A336288 Cf. A081585.
%K A336288 nonn,easy,nice
%O A336288 1,2
%A A336288 _Ilario Miriello_, Jul 16 2020
%E A336288 More terms from _Michel Marcus_, Jul 16 2020