A224997
Floor(1/f(x^(1/n))) for x = 17, where f computes the fractional part.
Original entry on oeis.org
8, 1, 32, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25
Offset: 2
A224998
Floor(1/f(x^(1/n))) for x = Pi, where f computes the fractional part.
Original entry on oeis.org
7, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 21, 22, 23, 23, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 37, 37, 38, 39, 40, 41, 42, 43, 44, 44, 45, 46, 47, 48, 49, 50, 51, 51, 52, 53, 54, 55, 56, 57, 58, 58
Offset: 1
A305319
Triangle T(n,k) read by rows: coefficients in order of decreasing exponents of characteristic polynomial P_n(t) of the matrix M(i,j) = [(i+j>n) or (i+j)=n-1], 1 <= i,j <= n.
Original entry on oeis.org
1, 1, -1, 1, -1, -1, 1, -3, 1, 1, 1, -2, -4, 1, 1, 1, -4, -1, 6, 1, -1, 1, -3, -8, 3, 9, 1, -1, 1, -5, -4, 15, 5, -11, -1, 1, 1, -4, -13, 8, 27, -3, -14, 1, 1, 1, -6, -8, 29, 15, -42, -6, 18, -1, -1, 1, -5, -19, 17, 60, -19, -63, 9, 21, -1, -1, 1, -7, -13, 49, 35, -110, -29, 93, 6, -25, -1, 1, 1, -6, -26, 31, 114, -58, -189, 45, 129, -10, -30, -1, 1
Offset: 0
P(0) = 1;
P(1) = t - 1;
P(2) = t^2 - t - 1;
P(3) = t^3 - 3*t^2 + t + 1;
P(4) = t^4 - 2*t^3 - 4*t^2 + t + 1;
...
Triangle starts:
n\k [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[0] 1
[1] 1, -1;
[2] 1, -1, -1;
[3] 1, -3, 1, 1;
[4] 1, -2, -4, 1, 1;
[5] 1, -4, -1, 6, 1, -1;
[6] 1, -3, -8, 3, 9, 1, -1;
[7] 1, -5, -4, 15, 5, -11, -1, 1;
[8] 1, -4, -13, 8, 27, -3, -14, 1, 1;
[9] 1, -6, -8, 29, 15, -42, -6, 18, -1, -1;
[10] 1, -5, -19, 17, 60, -19, -63, 9, 21, -1, -1;
[11] 1, -7, -13, 49, 35, -110, -29, 93, 6, -25, -1, 1;
[12] 1, -6, -26, 31, 114, -58, -189, 45, 29, -10, -30, -1, 1;
...
For n=7 the n X n matrix M (dots for zeros):
[. . . . 1 . 1]
[. . . 1 . 1 1]
[. . 1 . 1 1 1]
[. 1 . 1 1 1 1]
[1 . 1 1 1 1 1]
[. 1 1 1 1 1 1]
[1 1 1 1 1 1 1]
has characteristic polynomial P(7) = det(t*I-M) = t^7 - 5*t^6 - 4*t^5 + 15*t^4 + 5*t^3 - 11*t^2 - t + 1 (which is irreducible over Q: an elementary check shows that P(7)(25) = 4849680601 is a prime and 25 >= 17 = 2 + max(abs([1,-5,-4,15,5,-11,-1,1]))).
-
P(n, t='t) = charpoly(matrix(n, n, i, j, (i+j > n) || (i+j)==n-1), t);
seq(N, t='t) = {
my(a=vector(N)); for (n=1, 4, a[n] = subst(P(n,'t), 't, t));
for (n=5, N,
a[n] += (2*t + 3*(-1)^(n%2))*a[n-1] - (t^2-4)*a[n-2];
a[n] += -(2*t + 3*(-1)^(n%2))*a[n-3] - a[n-4]);
a;
};
concat(1, concat(apply(p->Vec(p), seq(12))))
\\ test: N=100; vector(N, n, P(n)) == seq(N)
A063093
Dimension of the space of weight 2n cusp forms for Gamma_0( 25 ).
Original entry on oeis.org
0, 5, 9, 15, 19, 25, 29, 35, 39, 45, 49, 55, 59, 65, 69, 75, 79, 85, 89, 95, 99, 105, 109, 115, 119, 125, 129, 135, 139, 145, 149, 155, 159, 165, 169, 175, 179, 185, 189, 195, 199, 205, 209, 215, 219, 225, 229, 235, 239, 245
Offset: 1
-
Rest@ CoefficientList[Series[x^2*(5 + 4 x + x^2)/((1 - x)^2*(1 + x)), {x, 0, 50}], x] (* Michael De Vlieger, Aug 26 2016 *)
LinearRecurrence[{1,1,-1},{0,5,9,15},50] (* Harvey P. Dale, Apr 09 2019 *)
Comments