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A260918 Number of squares of all sizes in polyominoes obtained by union of two pyramidal figures (A092498) with intersection equals A002623.

%I A260918 #37 Mar 04 2025 02:46:53
%S A260918 0,1,5,15,33,60,100,154,224,313,423,555,713,898,1112,1358,1638,1953,
%T A260918 2307,2701,3137,3618,4146,4722,5350,6031,6767,7561,8415,9330,10310,
%U A260918 11356,12470,13655,14913,16245,17655,19144,20714,22368,24108,25935,27853,29863
%N A260918 Number of squares of all sizes in polyominoes obtained by union of two pyramidal figures (A092498) with intersection equals A002623.
%C A260918 The resulting polyforms are n*(3*n-1)/2-polyominoes.
%C A260918 Also they are 6*n-gons with n>1.
%C A260918 Schäfli's notation for figure corresponding to a(1): 4.
%H A260918 Colin Barker, <a href="/A260918/b260918.txt">Table of n, a(n) for n = 0..1000</a>
%H A260918 Luce ETIENNE, <a href="/A260918/a260918.pdf">Illustration of initial terms</a>
%H A260918 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,-1,0,2,-1).
%F A260918 a(n) = A258440(n) - A000212(n+1).
%F A260918 a(n) = (1/8)*((Sum_{i=0..floor(2*n/3)} (4*n+1-6*i-(-1)^i)*(4*n-1-6*i+(-1)^i)) - (Sum_{j=0..(2*n-1+(-1)^n)/4} (2*n+1-(-1)^n-4*j)*(2*n+1+(-1)^n-4*j))).
%F A260918 a(n) = (52*n^3+90*n^2+20*n-3*(32*floor((n+1)/3)+3*(1-(-1)^n)))/144.
%F A260918 G.f.: x*(4*x^3+5*x^2+3*x+1) / ((x-1)^4*(x+1)*(x^2+x+1)). - _Colin Barker_, Aug 08 2015
%F A260918 E.g.f.: (3*exp(x)*x*(65 + x*(123 + 26*x)) + 32*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2) - 27*sinh(x))/216. - _Stefano Spezia_, Nov 15 2024
%e A260918 a(1)=1, a(2)=5, a(3)=12+3=15, a(4)=22+9+2=33, a(5)=35+18+7=60, a(6)=51+30+15+4=100.
%t A260918 Table[(52 n^3 + 90 n^2 + 20 n - 3 (32 Floor[(n + 1) / 3] + 3 (1 - (-1)^n))) / 144, {n, 0, 45}] (* _Vincenzo Librandi_, Aug 12 2015 *)
%o A260918 (PARI) concat(0, Vec(x*(4*x^3+5*x^2+3*x+1)/((x-1)^4*(x+1)*(x^2+x+1)) + O(x^100))) \\ _Colin Barker_, Aug 08 2015
%o A260918 (Magma) [(52*n^3+90*n^2+20*n-3*(32*Floor((n+1)/3)+3*(1-(-1)^n)))/144: n in [0..50]]; // _Vincenzo Librandi_, Aug 12 2015
%Y A260918 Cf. A002623, A092498, A173196, A000212, A258440.
%K A260918 nonn,easy
%O A260918 0,3
%A A260918 _Luce ETIENNE_, Aug 04 2015
%E A260918 Two repeated terms deleted by _Colin Barker_, Aug 08 2015