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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). 0

%I

%S 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,

%T 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,

%U 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

%N 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).

%C 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).

%C 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.

%C 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.

%H A. Luzon and M. A. Morón, <a href="http://arxiv.org/abs/0904.2672">Recurrence relations for polynomial sequences via Riordan matrices</a>, arXiv:0904.2672 [math.CO]

%e Using T^m =cos(m*t) as a notational shortcut, the expansions start

%e ; B_1(2 cos t)-2*cos 1 t = 0

%e 1 ; B_2(2 cos t)-2*cos 2 t = 1

%e 0 2 ; B_3(2 cos t)-2*cos 3 t = 2*T

%e 1 0 2 ; B_4(2 cos t)-2*cos 4 t = 1+2*T^2

%e 0 2 0 2 ; B_5(2 cos t)-2*cos 5 t = 2*T+2*T^3

%e 1 0 2 0 2 ; B_6(2 cos t)-2*cos 6 t = 1+2*T^2+2*T^4

%e 0 2 0 2 0 2 ; B_7(2 cos t)-2*cos 7 t = 2*T+2*T^3+2*T^5

%e 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

%e 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

%e 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

%e 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

%t centralPolygonalQ[n_] := Resolve[Exists[k, k>0, n == k^2-k+1], Integers];

%t b[n_] := If[n == 0 || centralPolygonalQ[n], 1, 2];

%t a[n_] := b[n-1];

%t Table[a[n], {n, 2, 106}] (* _Jean-François Alcover_, Oct 31 2018, after _R. J. Mathar_ *)

%K nonn,tabl

%O 2,2

%A _Haydar Rahmanov_, May 05 2009

%E Definition clarified, publication title corrected, sequence extended by _R. J. Mathar_, Dec 07 2009

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Last modified August 11 15:12 EDT 2020. Contains 336428 sequences. (Running on oeis4.)