A103772
Larger of two sides in a (k,k,k-1)-integer-sided triangle with integer area.
Original entry on oeis.org
1, 17, 241, 3361, 46817, 652081, 9082321, 126500417, 1761923521, 24540428881, 341804080817, 4760716702561, 66308229755041, 923554499868017, 12863454768397201, 179164812257692801, 2495443916839302017, 34757050023492535441, 484103256412056194161
Offset: 1
- Colin Barker, Table of n, a(n) for n = 1..850
- J. B. Cosgrave, The Gauss-Factorial Motzkin connection (Maple worksheet, change suffix to .mw)
- J. B. Cosgrave and K. Dilcher, An Introduction to Gauss Factorials, The American Mathematical Monthly, 118 (Nov. 2011), 812-829.
- Project Euler, Problem 94: Almost Equilateral Triangles.
- Index entries for linear recurrences with constant coefficients, signature (15,-15,1).
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I:=[1,17]; [n le 2 select I[n] else 14*Self(n-1)-Self(n-2)+4: n in [1..20]]; // Vincenzo Librandi, Mar 05 2016
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a[1] = 1; a[2] = 17; a[3] = 241; a[n_] := a[n] = 15a[n - 1] - 15a[n - 2] + a[n - 3]; Table[ a[n] - 1, {n, 17}] (* Robert G. Wilson v, Mar 24 2005 *)
LinearRecurrence[{15,-15,1},{1,17,241},20] (* Harvey P. Dale, Jan 02 2016 *)
RecurrenceTable[{a[1] == 1, a[2] == 17, a[n] == 14 a[n-1] - a[n-2] + 4}, a, {n, 20}] (* Vincenzo Librandi, Mar 05 2016 *)
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Vec(x*(1+x)^2/((1-x)*(1-14*x+x^2)) + O(x^25)) \\ Colin Barker, Mar 05 2016
A051047
For n > 5, a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3); initial terms are 1, 3, 8, 120, 1680.
Original entry on oeis.org
1, 3, 8, 120, 1680, 23408, 326040, 4541160, 63250208, 880961760, 12270214440, 170902040408, 2380358351280, 33154114877520, 461777249934008, 6431727384198600, 89582406128846400, 1247721958419651008, 17378525011746267720, 242051628206028097080
Offset: 1
- Colin Barker, Table of n, a(n) for n = 1..850
- Andrej Dujella and Attila Petho, Generalization of a theorem of Baker and Davenport
- B. W. Jones, A Variation of a Problem of Davenport and Diophantus, Quart. J. Math. (Oxford) Ser. (2) 27, 349-353, 1976.
- Index entries for linear recurrences with constant coefficients, signature (15,-15,1).
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I:=[1,3,8,120,1680]; [n le 5 select I[n] else 14*Self(n-1)-Self(n-2)+8: n in [1..20]]; // Vincenzo Librandi, Mar 05 2016
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With[{x = 1},
Join[{x, x + 2},
RecurrenceTable[{c[-1] == c[0] == 0,
c[k] == (4 x^2 + 8 x + 2) c[k - 1] - c[k - 2] + 4 (x + 1)}, c, {k, 1, 12}]]]
LinearRecurrence[{15, -15, 1}, {1, 3, 8, 120, 1680}, 22] (* Charles R Greathouse IV, Oct 31 2011 *)
Join[{1, 3}, RecurrenceTable[{a[1] == 8, a[2] == 120, a[n] == 14 a[n-1] - a[n-2] + 8}, a, {n, 20}]] (* Vincenzo Librandi, Mar 05 2016 *)
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Vec((3*x^4-44*x^3+22*x^2+12*x-1)/(x^3-15*x^2+15*x-1)+O(x^99)) \\ Charles R Greathouse IV, Oct 31 2011
Entry revised by
N. J. A. Sloane, Oct 25 2009, following correspondence with Eric Weisstein
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 *)
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