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

0,3

Comments

The resulting polyforms are n*(3*n-1)/2-polyominoes.

Also they are 6*n-gons with n>1.

Schäfli's notation for figure corresponding to a(1): 4.

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).

Formula

a(n) = A258440(n)- A000212(n+1).

a(n) = (1/8)*(Sum{j=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

a(n) = (52*n^3+90*n^2+20*n-3*(32*floor((n+1)/3)+3*(1-(-1)^n)))/144.

Example

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.

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 *)

Prog

(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
(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

Crossrefs

Cf. A002623, A092498, A173196, A000212, A258440.

Sequence in context: A155013 A134887 A228599 * A212983 A055004 A332355

Adjacent sequences: A260915 A260916 A260917 * A260919 A260920 A260921

Keyword

nonn,easy

Author

Luce ETIENNE, Aug 04 2015

Extensions

Two repeated terms deleted by Colin Barker, Aug 08 2015

Status

approved