A135671
a(n) = ceiling(n - n^(2/3)).
Original entry on oeis.org
0, 1, 1, 2, 3, 3, 4, 4, 5, 6, 7, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 34, 35, 36, 37, 38, 39, 39, 40, 41, 42, 43, 44, 44, 45, 46
Offset: 1
A135672
a(n) = floor(n - n^(2/3)).
Original entry on oeis.org
0, 0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 14, 14, 15, 16, 17, 18, 18, 19, 20, 21, 21, 22, 23, 24, 25, 25, 26, 27, 28, 29, 29, 30, 31, 32, 33, 33, 34, 35, 36, 37, 38, 38, 39, 40, 41, 42, 43, 43, 44, 45
Offset: 1
Cf.
A135660,
A135661,
A135662,
A135663,
A135664,
A135665,
A028391,
A135668,
A135669,
A135670,
A135671.
A346831
Table read by rows, coefficients of the characteristic polynomials of the tangent matrices.
Original entry on oeis.org
1, 0, 1, -1, 0, 1, 2, -1, -2, 1, 1, 0, -6, 0, 1, 4, 9, -4, -10, 0, 1, -1, 0, 15, 0, -15, 0, 1, 14, -1, -46, 19, 34, -19, -2, 1, 1, 0, -28, 0, 70, 0, -28, 0, 1, 40, 81, -88, -196, 56, 150, -8, -36, 0, 1, -1, 0, 45, 0, -210, 0, 210, 0, -45, 0, 1
Offset: 0
Table starts:
[0] 1;
[1] 0, 1;
[2] -1, 0, 1;
[3] 2, -1, -2, 1;
[4] 1, 0, -6, 0, 1;
[5] 4, 9, -4, -10, 0, 1;
[6] -1, 0, 15, 0, -15, 0, 1;
[7] 14, -1, -46, 19, 34, -19, -2, 1;
[8] 1, 0, -28, 0, 70, 0, -28, 0, 1;
[9] 40, 81, -88, -196, 56, 150, -8, -36, 0, 1.
.
The first few tangent matrices:
1 2 3 4 5
---------------------------------------------------------------
0; -1 0; 1 -1 0; 1 -1 -1 0; 1 1 -1 -1 0;
0 1; -1 0 1; -1 -1 0 1; 1 -1 -1 0 1;
0 1 1; -1 0 1 1; -1 -1 0 1 1;
0 1 1 -1; -1 0 1 1 1;
0 1 1 1 -1;
-
using AbstractAlgebra
function TangentMatrix(N)
M = zeros(ZZ, N, N)
H = div(N + 1, 2)
for n in 1:N - 1
for k in 0:n - 1
M[n - k, k + 1] = n < H ? 1 : -1
M[N - n + k + 1, N - k] = n < N - H ? -1 : 1
end
end
M end
function A346831Row(n)
n == 0 && return [ZZ(1)]
R, x = PolynomialRing(ZZ, "x")
S = MatrixSpace(ZZ, n, n)
M = TangentMatrix(n)
c = charpoly(R, S(M))
collect(coefficients(c))
end
for n in 0:9 println(A346831Row(n)) end
-
TangentMatrix := proc(N) local M, H, n, k;
M := Matrix(N, N); H := iquo(N + 1, 2);
for n from 1 to N - 1 do for k from 0 to n - 1 do
M[n - k, k + 1] := `if`(n < H, 1, -1);
M[N - n + k + 1, N - k] := `if`(n < N - H, -1, 1);
od od; M end:
A346831Row := proc(n) if n = 0 then return 1 fi;
LinearAlgebra:-CharacteristicPolynomial(TangentMatrix(n), x);
seq(coeff(%, x, k), k = 0..n) end:
seq(A346831Row(n), n = 0..10);
-
TangentMatrix[N_] := Module[{M, H, n, k},
M = Array[0&, {N, N}]; H = Quotient[N + 1, 2];
For[n = 1, n <= N - 1, n++, For[k = 0, k <= n - 1, k++,
M[[n - k, k + 1]] = If[n < H, 1, -1];
M[[N - n + k + 1, N - k]] = If[n < N - H, -1, 1]]]; M];
A346831Row[n_] := Module[{c}, If[n == 0, Return[{1}]];
c = CharacteristicPolynomial[TangentMatrix[n], x];
(-1)^n*CoefficientList[c, x]];
Table[A346831Row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 15 2024, after Peter Luschny *)
A135673
Ceiling(n + n^(2/3)).
Original entry on oeis.org
2, 4, 6, 7, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 82, 83
Offset: 1
a(6) = 10; ceiling(6 + 6^(2/3)) = ceiling(9.30192...) = 10.
Cf.
A135660,
A135661,
A135662,
A135663,
A135664,
A135665,
A028391,
A135668,
A135669,
A135670,
A135671,
A135672.
A135674
Floor(n+n^(2/3)).
Original entry on oeis.org
2, 3, 5, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 74
Offset: 1
Cf.
A135660,
A135661,
A135662,
A135663,
A135664,
A135665,
A028391,
A135668,
A135669,
A135670,
A135671,
A135672,
A135673.
Showing 1-5 of 5 results.
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