A152011 -id:A152011 - cp's OEIS frontend

A346831 Table read by rows, coefficients of the characteristic polynomials of the tangent matrices.

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

Peter Luschny, Sep 11 2021

The tangent matrix M(n, k) is an N X N matrix defined with h = floor((N+1)/2) as:
   M[n - k, k + 1] = if n < h then 1 otherwise -1,
   M[N - n + k + 1, N - k] = if n < N - h then -1 otherwise 1,
   for n in [1..N-1] and for k in [0..n-1], and 0 in the main antidiagonal.
The name 'tangent matrix' derives from M(n, k) = signum(tan(Pi*(n + k)/(N + 1))) whenever the right side of this equation is defined.

			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;

Cf. A135670, A152011, A346837 (generalized tangent matrix).

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 *)

The rows with even index equal those of A135670.

The determinants of tangent matrices with even dimension are A152011.

A305492 a(n) = ((1 + y)^n - (1 - y)^n)/y with y = sqrt(8).

Original entry on oeis.org

0, 2, 4, 22, 72, 298, 1100, 4286, 16272, 62546, 238996, 915814, 3504600, 13419898, 51371996, 196683278, 752970528, 2882724002, 11036241700, 42251551414, 161756794728, 619274449354, 2370846461804, 9076614069086, 34749153370800

Offset: 0

Peter Luschny, Jun 02 2018

			Array ((1+y)^n - (1-y)^n)/y with y = sqrt(k).
[k\n]
[1]   1, 2, 4,  8, 16, 32,   64,  128,    256,   512,   1024, ...
[2]   0, 2, 4, 10, 24, 58,  140,  338,    816,  1970,   4756, ...
[3]   0, 2, 4, 12, 32, 88,  240,  656,   1792,  4896,  13376, ...
[4]   0, 2, 4, 14, 40, 122, 364,  1094,  3280,  9842,  29524, ...
[5]   0, 2, 4, 16, 48, 160, 512,  1664,  5376, 17408,  56320, ...
[6]   0, 2, 4, 18, 56, 202, 684,  2378,  8176, 28242,  97364, ...
[7]   0, 2, 4, 20, 64, 248, 880,  3248, 11776, 43040, 156736, ...
[8]   0, 2, 4, 22, 72, 298, 1100, 4286, 16272, 62546, 238996, ...
[9]   0, 2, 4, 24, 80, 352, 1344, 5504, 21760, 87552, 349184, ...

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,7).

Let f(n, y) = ((1 + y)^n - (1 - y)^n)/y.
f(n,      1 ) = A000079(n);
f(n, sqrt(2)) = A163271(n+1);
f(n, sqrt(3)) = A028860(n+2);
f(n,      2 ) = A152011(n) for n>0;
f(n, sqrt(5)) = A103435(n);
f(n, sqrt(6)) = A083694(n);
f(n, sqrt(7)) = A274520(n);
f(n, sqrt(8)) = a(n);
f(n,      3 ) = A192382(n+1);
Cf. A305491.
Equals 2 * A015519.

egf :=  (n,x) -> 2*exp(x)*sinh(sqrt(n)*x)/sqrt(n):
ser := series(egf(8,x), x, 26):
seq(n!*coeff(ser,x, n), n=0..24);

Table[Simplify[((1 + Sqrt[8])^n - (1 - Sqrt[8])^n)/ Sqrt[8]], {n, 0, 24}]

concat(0, Vec(2*x / (1 - 2*x - 7*x^2) + O(x^40))) \\ Colin Barker, Jun 05 2018

E.g.f.: 2*exp(x)*sinh(sqrt(n)*x)/sqrt(n) for n = 8.
From Colin Barker, Jun 02 2018: (Start)
G.f.: 2*x / (1 - 2*x - 7*x^2).
a(n) = 2*a(n-1) + 7*a(n-2) for n>1.
(End)

A228815 Symmetric triangle, read by rows, related to Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 5, 5, 2, 3, 10, 14, 10, 3, 5, 20, 36, 36, 20, 5, 8, 38, 83, 106, 83, 38, 8, 13, 71, 182, 281, 281, 182, 71, 13, 21, 130, 382, 690, 834, 690, 382, 130, 21, 34, 235, 778, 1606, 2268, 2268, 1606, 778, 235, 34, 55, 420, 1546, 3586, 5780, 6750

Offset: 0

Philippe Deléham, Oct 30 2013

Triangles satisfying the same recurrence: A091533, A091562, A185081, A205575, A209137, A209138.

			Triangle begins :
0
1, 1
1, 2, 1
2, 5, 5, 2
3, 10, 14, 10, 3
5, 20, 36, 36, 20, 5
8, 38, 83, 106, 83, 38, 8
13, 71, 182, 281, 281, 182, 71, 13
21, 130, 382, 690, 834, 690, 382, 130, 21
34, 235, 778, 1606, 2268, 2268, 1606, 778, 235, 34
55, 420, 1546, 3586, 5780, 6750, 5780, 3586, 1546, 420, 55

Cf. A000045 (1st column), A001629 (2nd column), A008998, A152011, A261055 (3rd column).

G.f.: x*(1+y)/(1-x-x*y-x^2-x^2*y-x^2*y^2).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 0, T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
Sum_{k = 0..n} T(n,k)*x^k = A000045(n), 2*A015518(n), 3*A015524(n), 4*A200069(n) for x = 0, 1, 2, 3 respectively.
Sum_{k = 0..floor(n/2)} T(n-k,k) = A008998(n+1).

A136423 Floor((x^n - (1-x)^n)/2 +.5) where x = (sqrt(4)+1)/2 = 3/2.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 13, 19, 29, 43, 65, 97, 146, 219, 328, 493, 739, 1108, 1663, 2494, 3741, 5611, 8417, 12626, 18938, 28408, 42611, 63917, 95876, 143813, 215720, 323580, 485370, 728055, 1092082, 1638123, 2457185, 3685777, 5528666, 8292999

Offset: 1

Cino Hilliard, Apr 01 2008

nonn

This is analogous to the closed form of the formula for the n-th Fibonacci number. Even before truncation, these numbers are rational and the decimal part always ends in 5. For x=(sqrt(4)+1)/2=3/2, a(n)/a(n-1) -> x.

g(n,r) = for(m=1,n,print1(fib(m,r)",")) fib(n,r) = x=(sqrt(r)+1)/2;floor((x^n-(1-x)^n)/sqrt(r)+.5)

The general form of x is (sqrt(r)+1)/2, r=1,2,3..

a(n) = floor(b(n)/2^n) where b(n) = 2^(n-1)+A152011(n). - R. J. Mathar, Sep 10 2016

A169629 Array T(n,k) read by antidiagonals: T(n,k) = Sum_{v=1..n, v odd} binomial(n,v)*k^v.

Original entry on oeis.org

1, 2, 2, 4, 4, 3, 8, 14, 6, 4, 16, 40, 36, 8, 5, 32, 122, 120, 76, 10, 6, 64, 364, 528, 272, 140, 12, 7, 128, 1094, 2016, 1684, 520, 234, 14, 8, 256, 3280, 8256, 7448, 4400, 888, 364, 16, 9, 512, 9842, 32640, 40156, 21280, 9966, 1400, 536, 18, 10

Offset: 1

Roger L. Bagula, Mar 03 2010

Antidiagonal sums are: 1, 4, 11, 32, 105, 366, 1387, ...

			   1,    2,    3,     4,      5,      6,       7, ...
   2,    4,    6,     8,     10,     12,      14, ...
   4,   14,   36,    76,    140,    234,     364, ...
   8,   40,  120,   272,    520,    888,    1400, ...
  16,  122,  528,  1684,   4400,   9966,   20272, ...
  32,  364, 2016,  7448,  21280,  51012,  107744, ...
  64, 1094, 8256, 40156, 148160, 450834, 1188544, ...

Cf. A152011.

Cf. A005843 (2nd line), A079908 (3rd line), A105374 (4th line).

tabl(nn) = {for (n=1, nn, for (k=1, nn, print1(sum(v=1, n, (v%2)*binomial(n, v)*k^v), ", ");); print(););} \\ Michel Marcus, Jul 22 2015

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A346831 Table read by rows, coefficients of the characteristic polynomials of the tangent matrices.

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A305492 a(n) = ((1 + y)^n - (1 - y)^n)/y with y = sqrt(8).

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A228815 Symmetric triangle, read by rows, related to Fibonacci numbers.

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A136423 Floor((x^n - (1-x)^n)/2 +.5) where x = (sqrt(4)+1)/2 = 3/2.

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A169629 Array T(n,k) read by antidiagonals: T(n,k) = Sum_{v=1..n, v odd} binomial(n,v)*k^v.

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