A160242 Triangle A(n,m) read by rows: a quarter of the Fourier coefficient [cos(m*t)] of the shifted Boubaker polynomial B_n(2*cos t)-2*cos(n*t).

1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset

2,2

Comments

Starting from the polynomials B_n(x) defined in A137276 and A135929, we insert x=2*cos(t), and define the Fourier coefficients A(n,m) by B_n(2*cos t)-2*cos(n*t) = 4*sum(m=0,..,n-2) A(n,m)*cos(m*t).

A(n,m) is not an integer for n=0, so the table starts at n=1. Furthermore, A(n,m)=0 if n-m is odd, these regular zeros are skipped as usual, so effectively the first table entry appears at n=2.

Simpler definition from R. J. Mathar, Apr 15 2010: a(n)=1 if n =0 or n in A002061, otherwise a(n)=2. So this is a kind of characteristic function of the central polygonal numbers A002061.

Links

Table of n, a(n) for n=2..106.

A. Luzon and M. A. Morón, Recurrence relations for polynomial sequences via Riordan matrices, arXiv:0904.2672 [math.CO]

Example

Using T^m =cos(m*t) as a notational shortcut, the expansions start
; B_1(2 cos t)-2*cos 1 t = 0
1 ; B_2(2 cos t)-2*cos 2 t = 1
0 2 ; B_3(2 cos t)-2*cos 3 t = 2*T
1 0 2 ; B_4(2 cos t)-2*cos 4 t = 1+2*T^2
0 2 0 2 ; B_5(2 cos t)-2*cos 5 t = 2*T+2*T^3
1 0 2 0 2 ; B_6(2 cos t)-2*cos 6 t = 1+2*T^2+2*T^4
0 2 0 2 0 2 ; B_7(2 cos t)-2*cos 7 t = 2*T+2*T^3+2*T^5
1 0 2 0 2 0 2 ; B_8(2 cos t)-2*cos 8 t = 1+2*T^2+2*T^4+2*T^6
0 2 0 2 0 2 0 2 ; B_9(2 cos t)-2*cos 9 t = 2*T+2*T^3+2*T^5+2*T^7
1 0 2 0 2 0 2 0 2 ; B_10(2 cos t)-2*cos 10 t = 1+2*T^2+2*T^4+2*T^6+2*T^8
0 2 0 2 0 2 0 2 0 2 ; B_11(2 cos t)-2*cos 11 t = 2*T^3+2*T^5+2*T^7+2*T^9+2*T

Mathematica

centralPolygonalQ[n_] := Resolve[Exists[k, k>0, n == k^2-k+1], Integers];
b[n_] := If[n == 0 || centralPolygonalQ[n], 1, 2];
a[n_] := b[n-1];
Table[a[n], {n, 2, 106}] (* Jean-François Alcover, Oct 31 2018, after R. J. Mathar *)

Crossrefs

Sequence in context: A214860 A263649 A229904 * A043529 A201219 A254315

Adjacent sequences: A160239 A160240 A160241 * A160243 A160244 A160245

Keyword

nonn,tabl

Author

Haydar Rahmanov, May 05 2009

Extensions

Definition clarified, publication title corrected, sequence extended by R. J. Mathar, Dec 07 2009

Status

approved