login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A079340 Absolute value of determinant of n X n matrix whose entries are the integers from 1 to n^2 spiraling outward, ending in a corner. 4
1, 5, 72, 1380, 31920, 861840, 26611200, 925404480, 35805369600, 1526139014400, 71066912716800, 3590219977344000, 195589552648089600, 11430978821982720000, 713448513897799680000, 47363888351558338560000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

If n == 0 or 1 (mod 4), the sign of the determinant will be independent of the orientation of the spiral. For n == 2 or 3 (mod 4), the sign will be reversed when the orientation is rotated by 1/4 or flipped on the horizontal or vertical axis. - Franklin T. Adams-Watters, Dec 31 2013

This distribution of the integers is sometimes known as Ulam's spiral, although that is sometimes reserved for when the primes are marked out in some way. - Franklin T. Adams-Watters, Dec 31 2013

LINKS

Table of n, a(n) for n=1..16.

Gaurav Bhatnagar, Christian Krattenthaler, Spiral determinants, arXiv:1704.02859 [math.CO], 2017.

FORMULA

a(n) = (2*n^2-3*n+3) (2n-2)!/(2 (n-1)!) = A096376(n-1)*A000407(n-2), n>1. - Conjectured by Dean Hickerson, Jan 30 2003. Proved in the article by Bhatnagar and Krattenthaler.

D-finite with recurrence (2*n^2-7*n+8)*a(n) -2*(2*n-3)*(2*n^2-3*n+3)*a(n-1)=0. - R. J. Mathar, May 03 2019

EXAMPLE

n=2, det=-5: {1 2 / 4 3 }.

n=3, det=72: {7 8 9 / 6 1 2 / 5 4 3 }.

n=4, det=-1380: { 7 8 9 10 / 6 1 2 11 /5 4 3 12 / 16 15 14 13 }.

n=5, det=31920: { 21 22 23 24 25 / 20 7 8 9 10 / 19 6 1 2 11 /18 5 4 3 12 / 17 16 15 14 13 }

MATHEMATICA

M[0, 0] = 1;

M[i_, j_] := If[i <= j,

If[i + j >= 0, If[i != j, M[i + 1, j] + 1, M[i, j - 1] + 1],

M[i, j + 1] + 1],

If[i + j > 1, M[i, j - 1] + 1, M[i - 1, j] + 1]

]

M[n_] := If[EvenQ[n],

Table[M[i, j], {j, n/2, -n/2 + 1, -1}, {i, -n/2 + 1, n/2}],

Table[M[i, j], {j, (n - 1)/2, -(n - 1)/2, -1}, {i, -(n - 1)/2, (n - 1)/2}]]

a[n_]:=Det[M[n]] (* Christian Krattenthaler, Apr 19 2017 *)

PROG

(Maxima) A079340(n):=if n=1 then 1 else (2*n^2-3*n+3)*(2*n-2)!/(2*(n-1)!)$

makelist(A079340(n), n, 1, 30); /* Martin Ettl, Nov 05 2012 */

CROSSREFS

Cf. A078475, A023999, A067276, A052182.

Sequence in context: A079874 A186664 A223128 * A268782 A282788 A197324

Adjacent sequences: A079337 A079338 A079339 * A079341 A079342 A079343

KEYWORD

nonn,easy

AUTHOR

Kit Vongmahadlek (kit119(AT)yahoo.com), Jan 03 2003

EXTENSIONS

Extended by Robert G. Wilson v, Jan 25 2003

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 1 08:20 EST 2023. Contains 359992 sequences. (Running on oeis4.)