cp's OEIS Frontend

A082375 Irregular triangle read by rows: row n begins with n and decreases by 2 until 0 or 1 is reached, for n >= 0.

%I A082375 #45 Mar 01 2025 12:12:41
%S A082375 0,1,2,0,3,1,4,2,0,5,3,1,6,4,2,0,7,5,3,1,8,6,4,2,0,9,7,5,3,1,10,8,6,4,
%T A082375 2,0,11,9,7,5,3,1,12,10,8,6,4,2,0,13,11,9,7,5,3,1,14,12,10,8,6,4,2,0,
%U A082375 15,13,11,9,7,5,3,1,16,14,12,10,8,6,4,2,0,17,15,13,11,9,7,5,3,1,18,16,14
%N A082375 Irregular triangle read by rows: row n begins with n and decreases by 2 until 0 or 1 is reached, for n >= 0.
%C A082375 As a sequence, a(n) = A025644(n+1) for n <= 142.
%C A082375 The length of row n is given by A008619(n) = 1 + floor(n/2).
%C A082375 From _Wolfdieter Lang_, Feb 17 2020: (Start)
%C A082375 This table T(n, m) can be used for the conversion identity
%C A082375 2*cos(Pi*k/N) = 2*sin((Pi/(2*N))*(N - 2*k)) = 2*sin((Pi/(2*N))*T(N-2, k-1)), here for N = n+2 >= 2, and k = m + 1 = 1, 2, ..., floor(N/2).
%C A082375 2*cos((Pi/N)*k) = R(k, rho(N)), where R is a monic Chebyshev polynomial from A127672 and rho(N) = 2*cos(Pi/N), gives part of the roots of the polynomial S(N-1, x), for k = 1, 2, ..., floor(N/2), with the Chebyshev S polynomials from A049310.
%C A082375 2*sin((Pi/(2*N))*q) = d^{(2*N)}_q/r, for q = 1, 2, ..., N, with the length ratio (q-th diagonal)/r, where r is the radius of the circle circumscribing a regular (2*N)-gon. The counting q starts with the diagonal d^{(2*N)}_1 = s(2*N) (in units of r), the side of the (2*N)-gon. The next diagonal is d^{(2*N)}_2 = rho(2*N)*s(2*N) (in units of r).
%C A082375 For the instances N = 4 (n = 2) and 5 (n = 3), we have:
%C A082375   N = n+2 = 4:
%C A082375     k = m+1 = 1, 2*cos(Pi*1/4) = 2*sin(Pi*2/8) = sqrt(2);
%C A082375     k = 2, 2*cos(Pi*2/4) = 2*sin(Pi*0/8) = 0.
%C A082375   N = 5 (n=3):
%C A082375     k=1 (m=0), 2*cos(Pi*1/5) = 2*sin(Pi*3/10) = (1 + sqrt(5))/2 = rho(5) = A001622;
%C A082375     k=2: 2*cos(Pi*2/5) = 2*sin(Pi*1/10) = rho(5) - 1. (End)
%C A082375 If b > 0 and c > 0 are the integer coefficients of a monic quadratic x^2 + b*x + c, it has integer roots if its discriminant d^2 = b^2 - 4c is a perfect square. This sequence is the values of d for increasing b sorted by b then c. The first pair of (b, c) = (2, 1) and has d = a(0) = 0. The n-th pair of (b, c) = (A027434(n), A350634(n)) and has d = a(n-1). - _Frank M Jackson_, Jan 20 2024
%C A082375 This sequence is related to an instance of Clark Kimberling's generic dispersion arrays; in this case the leader sequence is the square numbers A000290 (without 0), and the follower sequence is the nonsquare numbers A000037. This sequence gives the 0-origin column index of n in the resulting dispersion array. - _Allan C. Wechsler_, Feb 26 2025
%H A082375 Harvey P. Dale, <a href="/A082375/b082375.txt">Table of n, a(n) for n = 0..1000</a>
%H A082375 Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/intersp.html">Interspersions and Dispersions</a>.
%H A082375 Clark Kimberling, <a href="http://dx.doi.org/10.1090/S0002-9939-1993-1111434-0">Interspersions and dispersions</a>, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
%F A082375 T(n, m) = n - 2*m, m = 0, 1, ..., floor(n/2), n >= 0 (see the name and programs). - _Wolfdieter Lang_, Feb 17 2020
%F A082375 a(n) = A199474(n+1) - A122197(n+1). - _Wesley Ivan Hurt_, Jan 09 2022
%F A082375 a(n) = sqrt((A027434(n+1))^2 - 4*A350634(n+1)). - _Frank M Jackson_, Jan 20 2024
%e A082375 The irregular triangle T(n, m) begins:
%e A082375   n\m  0 1 2 3 4 5 ...
%e A082375   0:   0
%e A082375   1:   1
%e A082375   2:   2 0
%e A082375   3:   3 1
%e A082375   4:   4 2 0
%e A082375   5:   5 3 1
%e A082375   6:   6 4 2 0
%e A082375   7:   7 5 3 1
%e A082375   8:   8 6 4 2 0
%e A082375   9:   9 7 5 3 1
%e A082375   10: 10 8 6 4 2 0
%e A082375   ...
%t A082375 Flatten[Table[Range[n,0,-2],{n,0,20}]] (* _Harvey P. Dale_, Apr 03 2019 *)
%t A082375 lst = {}; Do[If[IntegerQ[d=Sqrt[b^2-4c]], AppendTo[lst, d]], {b, 1, 20}, {c, 1, b^2/4}]; lst (* _Frank M Jackson_, Jan 20 2024 *)
%o A082375 (PARI) a(n)=local(m); if(n<0,0,m=sqrtint(1+4*n); m-1-(1+4*n-m^2)\2)
%Y A082375 Cf. A025644, A008619, A008619, A027434, A122197, A199474, A350634.
%K A082375 nonn,tabf
%O A082375 0,3
%A A082375 _Michael Somos_, Apr 09 2003