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A122751 Number of essentially different semi-magic squares of order 3 with semimagic sum n. 1
1, 2, 7, 14, 29, 49, 83, 127, 192, 273, 384, 519, 694, 902, 1162, 1466, 1835, 2260, 2765, 3340, 4011, 4767, 5637, 6609, 7714, 8939, 10318, 11837, 13532, 15388, 17444, 19684, 22149, 24822, 27747, 30906, 34345, 38045, 42055, 46355, 50996, 55957, 61292 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

REFERENCES

Christoph Gerber, "Zum Abzahlen semimagischer Quadrate" [Apparently unpublished. - R. J. Mathar, Nov 13 2011]

P. A. MacMahon, Combinatory Analysis, Vol II; Chelsea, New York, 1960.

LINKS

Harvey P. Dale, Table of n, a(n) for n = 3..1000

Christoph Gerber, More information  [?Broken link]

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,4,-4,2,2,-3,1).

FORMULA

a(n) = 1/64*n^4-1/32*n^3+1/32*n^2+d*n+e with: d:=-1/8 if n=0 or n=2 (mod 4) d:=3/32 if n=1 or n=3 (mod 4) e:=0 if n=0 (mod 4) e:=-7/64 if n=1 (mod 4) e:=1/8 if n=2 (mod 4) e:=1/64 if n=3 (mod 4).

G.f.: -x^3*(1-x+3*x^2-x^3+x^4) / ( (1+x^2)*(1+x)^2*(x-1)^5 ). - R. J. Mathar, Nov 13 2011

a(n) = (2*n*(n-1)*(n^2-n+1)-7*(2*n-1)*(-1)^n-8*(-1)^((2*n-1+(-1)^n)/4)+1)/128. - Luce ETIENNE, Oct 29 2017

EXAMPLE

a(4)=2 because there are 2 essentially different semi-magic squares of order 3 with semi-magic sum 4: [1,1,2; 1,2,1; 2,1,1] and [1,1,2; 2,1,1; 1,2,1].

MAPLE

A131292:=proc(n) local d, e: if (n mod 4) in {0, 2} then d:=-1/8 fi: if (n mod 4) in {1, 3} then d:=3/32 fi: if (n mod 4) in {0} then e:=0 fi: if (n mod 4) in {1} then e:=-7/64 fi: if (n mod 4) in {2} then e:=1/8 fi: if (n mod 4) in {3} then e:=1/64 fi: return 1/64*n^4-1/32*n^3+1/32*n^2+d*n+e: end proc:

MATHEMATICA

LinearRecurrence[{3, -2, -2, 4, -4, 2, 2, -3, 1}, {1, 2, 7, 14, 29, 49, 83, 127, 192}, 50] (* Harvey P. Dale, Jan 26 2017 *)

CROSSREFS

Sequence in context: A295963 A221317 A005998 * A152944 A018437 A286857

Adjacent sequences:  A122748 A122749 A122750 * A122752 A122753 A122754

KEYWORD

nonn

AUTHOR

Christoph Gerber (christoph.gerber(AT)phbern.ch), Jun 25 2007

STATUS

approved

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Last modified December 9 00:32 EST 2019. Contains 329871 sequences. (Running on oeis4.)