A125641 Square of the (3,1)-entry of the 3 X 3 matrix M^n, where M = [1,0,0; 1,1,0; 1,i,1].

1, 5, 18, 52, 125, 261, 490, 848, 1377, 2125, 3146, 4500, 6253, 8477, 11250, 14656, 18785, 23733, 29602, 36500, 44541, 53845, 64538, 76752, 90625, 106301, 123930, 143668, 165677, 190125, 217186, 247040, 279873, 315877, 355250, 398196, 444925

Offset: 1

Author

Gary W. Adamson, Nov 28 2006

Keywords

• nonn
• easy

Comments

Conjecture [False!]: Draw the segments joining every lattice point on axis X with every lattice point on axis Y for 1 <= x <= n and 1 <= y <= n. The number of regions formed with these segments and axis X and Y is a(n). - César Eliud Lozada, Feb 14 2013

The above conjecture appears to be wrong. The number of regions formed by this construction is given in A332953, which differs from this sequence for n > 5. - Scott R. Shannon, Mar 04 2020

Examples

			a(5)=25 because M^5 = [1,0,0; 5,1,0; 5+10i, 5i, 1] and |5+10i|^2 = 125.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• César Eliud Lozada, Counting regions [Warning: Although the drawings here appear to be correct for n <= 5, the generalization to higher n fails - see Comment above and A332953. - _N. J. A. Sloane_, Mar 04 2020]
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A092098, A332953.

Programs

GAP

List([1..40],n-> n^2*(n^2-2*n+5)/4); # Muniru A Asiru, Feb 22 2019

Magma

[n^2*(n^2-2*n+5)/4: n in [1..40]]; // G. C. Greubel, Feb 22 2019

Maple

b[1]:=1: b[2]:=2+I: b[3]:=3+3*I: for n from 4 to 45 do b[n]:=3*b[n-1]-3*b[n-2]+b[n-3] od: seq(abs(b[j])^2,j=1..45);
with(linalg): M[1]:=matrix(3,3,[1,0,0,1,1,0,1,I,1]): for n from 2 to 45 do M[n]:=multiply(M[1],M[n-1]) od: seq(abs(M[j][3,1])^2,j=1..45);
seq(sum((binomial(n,m))^2,m=1..2),n=1..37); # Zerinvary Lajos, Jun 19 2008
# alternative Maple program:
a:= n-> abs((<<1|0|0>, <1|1|0>, <1|I|1>>^n)[3,1])^2:
seq(a(n), n=1..40);  # Alois P. Heinz, Mar 09 2020

Mathematica

Table[n^2(n^2-2n+5)/4,{n,40}] (* Vincenzo Librandi, Feb 14 2012 *)

PARI

vector(40, n, n^2*(n^2-2*n+5)/4) \\ G. C. Greubel, Feb 22 2019

Sage

[n^2*(n^2-2*n+5)/4 for n in (1..40)] # G. C. Greubel, Feb 22 2019

Formula

a(n) = |b(n)|^2, where b(n) = 3b(n-1) - 3b(n-2) + b(n-3) for n >= 4; b(1)=1, b(2)=2+i, b(3)=3+3i (the recurrence relation follows from the minimal polynomial t^3 - 3t^2 + 3t - 1 of the matrix M).

a(n) = n^2*(n^2 - 2*n + 5)/4. - T. D. Noe, Feb 09 2007

O.g.f.: x*(1 + 3*x^2 + 2*x^3)/(1-x)^5. - R. J. Mathar, Dec 05 2007

a(n) = binomial(n,2)^2 + n^2, n > 1. - Gary Detlefs, Nov 23 2011

E.g.f.: x*(4 +6*x +4*x^2 +x^3)*exp(x)/4. - G. C. Greubel, Feb 22 2019

Extensions

Edited by Emeric Deutsch, Dec 27 2006

Definition revised by N. J. A. Sloane, Mar 05 2020