A260918 Number of squares of all sizes in polyominoes obtained by union of two pyramidal figures (A092498) with intersection equals A002623.
0, 1, 5, 15, 33, 60, 100, 154, 224, 313, 423, 555, 713, 898, 1112, 1358, 1638, 1953, 2307, 2701, 3137, 3618, 4146, 4722, 5350, 6031, 6767, 7561, 8415, 9330, 10310, 11356, 12470, 13655, 14913, 16245, 17655, 19144, 20714, 22368, 24108, 25935, 27853, 29863
Offset: 0
Examples
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.
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Luce ETIENNE, Illustration of initial terms
- Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1).
Programs
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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
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Mathematica
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 *) -
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
Formula
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))).
a(n) = (52*n^3+90*n^2+20*n-3*(32*floor((n+1)/3)+3*(1-(-1)^n)))/144.
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
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
Extensions
Two repeated terms deleted by Colin Barker, Aug 08 2015
Comments