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A122751
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Number of essentially different semi-magic squares of order 3 with semimagic sum n.
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0
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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; internal format)
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OFFSET
| 3,2
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REFERENCES
| Submitted for publication: "Zum Abzahlen semimagischer Quadrate" [Apparently unpublished. - R. J. Mathar, Nov 13 2011]
P. A. MacMahon, Combinatory Analysis, Vol II; Chelsea, New York, 1960.
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LINKS
| More information [?Broken link]
Index to sequences with linear recurrences with constant coefficients, signature (3,-2,-2,4,-4,2,2,-3,1).
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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
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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]
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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:
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CROSSREFS
| Sequence in context: A200084 A184704 A005998 * A152944 A018437 A120739
Adjacent sequences: A122748 A122749 A122750 * A122752 A122753 A122754
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KEYWORD
| nonn
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AUTHOR
| Christoph Gerber (christoph.gerber(AT)phbern.ch), Jun 25 2007
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