A293846
Numbers such that k is the altitude of a Heronian triangle with sides m-13, m, m+13.
Original entry on oeis.org
9, 24, 39, 60, 105, 156, 231, 396, 585, 864, 1479, 2184, 3225, 5520, 8151, 12036, 20601, 30420, 44919, 76884, 113529, 167640, 286935, 423696, 625641, 1070856, 1581255, 2334924, 3996489, 5901324, 8714055, 14915100, 22024041, 32521296, 55663911, 82194840
Offset: 0
If the sides are 15, 28, 41 the triangle has the altitude 9 and is a part of the Pythagorean triangle with the sides 9, 40, 41, so 9 is a term.
-
CoefficientList[ Series[ 3(3x^4 +8x^3 +13x^2 +8x +3)/(x^6 -4x^3 +1), {x, 0, 35}], x] (* or *)
LinearRecurrence[{0, 0, 4, 0, 0, -1}, 3 {3, 8, 13, 20, 35, 52}, 36] (* Robert G. Wilson v, Dec 27 2017 *)
-
Vec(3*(3 + 8*x + 13*x^2 + 8*x^3 + 3*x^4) / (1 - 4*x^3 + x^6) + O(x^40)) \\ Colin Barker, Dec 27 2017
A185331
Riordan array ((1-x+x^2)/(1+x^2), x/(1+x^2)).
Original entry on oeis.org
1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 0, 2, -2, -1, 1, -1, 1, 3, -3, -1, 1, 0, -3, 3, 4, -4, -1, 1, 1, -1, -6, 6, 5, -5, -1, 1, 0, 4, -4, -10, 10, 6, -6, -1, 1, -1, 1, 10, -10, -15, 15, 7, -7, -1, 1, 0, -5, 5, 20, -20, -21, 21, 8, -8, -1, 1
Offset: 0
Triangle begins:
1;
-1, 1;
0, -1, 1;
1, -1, -1, 1;
0, 2, -2, -1, 1;
-1, 1, 3, -3, -1, 1;
0, -3, 3, 4, -4, -1, 1;
1, -1, -6, 6, 5, -5, -1, 1;
0, 4, -4, -10, 10, 6, -6, -1, 1;
-1, 1, 10, -10, -15, 15, 7, -7, -1, 1;
0, -5, 5, 20, -20, -21, 21, 8, -8, -1, 1;
1, -1, -15, 15, 35, -35, -28, 28, 9, -9, -1, 1;
-
CoefficientList[Series[CoefficientList[Series[(1 - x + x^2)/(1 - y*x + x^2), {x, 0, 10}], x], {y, 0, 10}], y] // Flatten (* G. C. Greubel, Jun 27 2017 *)
A249578
List of triples (r,s,t): the matrix M = [[4,12,9][2,7,6][1,4,4]] is raised to successive powers, then (r,s,t) are the square roots of M[3,1], M[1,1], M[1,3] respectively.
Original entry on oeis.org
0, 1, 0, 1, 2, 3, 4, 7, 12, 15, 26, 45, 56, 97, 168, 209, 362, 627, 780, 1351, 2340, 2911, 5042, 8733, 10864, 18817, 32592, 40545, 70226, 121635, 151316, 262087, 453948, 564719, 978122, 1694157, 2107560, 3650401, 6322680
Offset: 0
M^0 = the 3 X 3 identity matrix = [[1,0,0][0,1,0][0,0,1]]. M[3,1] = 0; M[1,1] = 1; M[1,3] = 0. So the first triple is r = a(0) = 0; s = a(1) = 1; t = a(2) = 0.
M^1 = [[4,12,9][2,7,6][1,4,4]], so r = a(3) = 1; s = a(4) = 2; t = a(5) = 3.
-
I:=[0,1,0,1,2,3]; [n le 6 select I[n] else 4*Self(n-3)-Self(n-6): n in [1..40]]; // Vincenzo Librandi, Nov 04 2014
-
CoefficientList[Series[x (3 x^4 - 2 x^3 + x^2 + 1) / (x^6 - 4 x^3 + 1), {x, 0, 70}], x] (* Vincenzo Librandi, Nov 04 2014 *)
LinearRecurrence[{0,0,4,0,0,-1},{0,1,0,1,2,3},40] (* Harvey P. Dale, Jan 17 2017 *)
-
concat(0, Vec(x*(3*x^4-2*x^3+x^2+1)/(x^6-4*x^3+1) + O(x^100))) \\ Colin Barker, Nov 04 2014
Comments