A160372
The 4-tuple (2, ((2*n+1)^2-1)/2, ((2*n+3)^2-1)/2, a(n)), where a(n) = 4*(3 + 20n + 42n^2 + 32n^3 + 8n^4), has Diophantus's property that the product of any two distinct terms plus one is a square.
Original entry on oeis.org
420, 2380, 7812, 19404, 40612, 75660, 129540, 208012, 317604, 465612, 660100, 909900, 1224612, 1614604, 2091012, 2665740, 3351460, 4161612, 5110404, 6212812, 7484580, 8942220, 10603012, 12485004, 14607012, 16988620, 19650180, 22612812, 25898404, 29529612, 33529860
Offset: 1
For n=2, we get (2,12,24,2380), and 2*12+1 = 25 = 5^2, 2*24+1 = 49 = 7^2, 2*2380+1 = 4761 = 69^2, 12*24+1 = 289 = 17^2, 12*2380+1 = 28561 = 169^2, 4*2380+1 = 57121 = 239^2.
- Paolo Xausa, Table of n, a(n) for n = 1..10000
- Lenny Jones, A polynomial Approach to a Diophantine Problem, Math. Mag. 72 (1999), 52-55.
- Eric Weisstein's World of Mathematics, Diophantus Property.
- Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
A176097
Degree of the hyperdeterminant of the cubic format (k+1) X (k+1) X (k+1).
Original entry on oeis.org
1, 4, 36, 272, 2150, 16992, 134848, 1072192, 8536914, 68036600, 542607560, 4329671040, 34561892560, 275979195520, 2204266118400, 17609217372416, 140698273234634, 1124340854572296, 8985828520591912, 71822662173752800
Offset: 0
For k=1, the hyperdeterminant of the matrix (a_ijk) (for 0 <= i,j,k <= 1) is (a_000 * a_111)^2 + (a001 * a110)^2 + (a_010 * a_101)^2 + (a_011 * a_100)^2 -2(a_000 * a_001 * a_110 * a_111 + a_000 * a_010 * a_101 * a_111 + a_000 * a_011 * a_100 * a_111 + a_001 * a_010 * a_101 * a_110 + a_001 * a_011 * a_110 * a_100 + a_010 * a_011 * a_101 * a_100) + 4(a_000 * a_011 * a_101 * a_110 + a_001 * a_010 * a_100 * a_111) (see Gelfand, Kapranov & Zelevinsky, pp. 2 and 448.) [Corrected by _Petros Hadjicostas_, Sep 12 2019]
- I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhauser, 2008, p. 456 (Ch. 14, Corollary 2.9).
- Arthur Cayley, On the theory of linear transformations, The Cambridge Mathematical Journal, Vol. IV, No. XXIII, February 1845, pp. 193-209. [Accessible only in the USA through the Hathi Trust Digital Library.]
- Arthur Cayley, On the theory of linear transformations, The collected mathematical papers of Arthur Cayley, Cambridge University Press (1889-1897), pp. 80-94. [Accessible through the University of Michigan Historical Math Collection; click on pp. 80 through 94.]
- Arthur Cayley, On linear transformations, Cambridge and Dublin Mathematical Journal, Vol. I, 1846, pp. 104-122. [Accessible only in the USA through the Hathi Trust Digital Library.]
- Arthur Cayley, On linear transformations, The collected mathematical papers of Arthur Cayley, Cambridge University Press (1889-1897), pp. 95-112. [Accessible through the University of Michigan Historical Math Collection; click on pp. 95 through 112.]
- I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Hyperdeterminants, Advances in Mathematics 96(2) (1992), 226-263; see Corollary 3.9 (p. 246).
- David G. Glynn, The modular counterparts of Cayley's hyperdeterminants, Bulletin of the Australian Mathematical Society 57(3) (1998), 479-492.
- Giorgio Ottaviani, Luca Sodomaco, and Emuanuele Ventura, Asymptotics of degrees and ED degrees of Segre products, arXiv:2008.11670 [math.AG], 2020.
- Ludwig Schläfli, Über die Resultante eines Systemes mehrerer algebraischen Gleichungen, ein Beitrag zur Theorie der Elimination, Denkschr. der Kaiserlicher Akad. der Wiss. math-naturwiss. Klasse, 4 Band, 1852.
- Eric Weisstein's World of Mathematics, Hyperdeterminant.
- Wikipedia, Hyperdeterminants.
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a:= k-> add((j+k+1)! /(j!)^3 /(k-2*j)! *2^(k-2*j), j=0..floor(k/2)): seq(a(n), n=0..20);
# Second program:
a := proc(n) option remember; if n = 0 then return 1 elif n = 1 then return 4 fi;
(a(n-1)*(21*n^3-10*n^2-9*n+6)+a(n-2)*(24*n^3+16*n^2))/((3*n-1)*n^2) end:
seq(a(n), n=0..19); # Peter Luschny, Sep 12 2019
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Table[Sum[(j + n + 1)!*2^(n - 2*j)/(j!^3*(n - 2*j)!), {j, 0, n/2}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 12 2019 *)
A160451
a(n) = (4/3)*u*(u^3+6*u^2+8*u-3) where u=floor((3*n+5)/2).
Original entry on oeis.org
1008, 2080, 6440, 10208, 22360, 31416, 57408, 75208, 122816, 153680, 232408, 281520, 402600, 476008, 652400, 757016, 1003408, 1147008, 1479816, 1671040, 2108408, 2356760, 2918560, 3234408, 3942240, 4336816, 5214008, 5699408, 6771016, 7360200, 8653008, 9359800
Offset: 1
For n=1 we get the 4-tuple (3,5,16,1008), and 3*5+1=16=4^2, 3*16+1=49=7^2, 3*1008+1=3025=55^2, 5*16+1=81=9^2, 5*1008+1=5041=71^2, 16*1008+1=16129=127^2.
- Lenny Jones, A polynomial Approach to a Diophantine Problem, Math. Mag. 72 (1999) 52-55.
- Eric Weisstein's World of Mathematics, Diophantus Property.
- Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).
-
Table[u=Floor[(3n+5)/2];4/3 u(u^3+6u^2+8u-3),{n,30}] (* or *) LinearRecurrence[{1,4,-4,-6,6,4,-4,-1,1},{1008,2080,6440,10208,22360,31416,57408,75208,122816},30] (* Harvey P. Dale, Nov 19 2013 *)
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