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

1,2

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

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).

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

Example

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

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 *)

Prog

(GAP) List([1..40], n-> n^2*(n^2-2*n+5)/4) # Muniru A Asiru, Feb 22 2019
(PARI) vector(40, n, n^2*(n^2-2*n+5)/4) \\ G. C. Greubel, Feb 22 2019
(Magma) [n^2*(n^2-2*n+5)/4: n in [1..40]]; // 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

Crossrefs

Cf. A092098, A332953.

Sequence in context: A257055 A036893 A332953 * A006479 A127983 A226903

Adjacent sequences: A125638 A125639 A125640 * A125642 A125643 A125644

Keyword

nonn,easy

Author

Gary W. Adamson, Nov 28 2006

Extensions

Edited by Emeric Deutsch, Dec 27 2006

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

Status

approved