cp's OEIS Frontend

A231425 The Schramm triangle: T(n,k) = f(gcd(n,k)), where f = Dirichlet inverse of Euler totient.

%I A231425 #31 Aug 01 2019 00:12:52
%S A231425 1,1,-1,1,1,-2,1,-1,1,-1,1,1,1,1,-4,1,-1,-2,-1,1,2,1,1,1,1,1,1,-6,1,
%T A231425 -1,1,-1,1,-1,1,-1,1,1,-2,1,1,-2,1,1,-2,1,-1,1,-1,-4,-1,1,-1,1,4,1,1,
%U A231425 1,1,1,1,1,1,1,1,-10,1,-1,-2,-1,1,2,1,-1,-2,-1,1,2
%N A231425 The Schramm triangle: T(n,k) = f(gcd(n,k)), where f = Dirichlet inverse of Euler totient.
%C A231425 When taking matrix powers of the reversed triangle it might be more interesting to start with the first term T(1,1) set to 0.
%C A231425 Three fundamental number theoretic sequences are found from this triangle. The first is the Dirichlet inverse of the Euler totient which are the entries themselves. The Greatest Common Divisor-Fourier transform described by Wolfgang Schramm gives the Möbius function times n =1, -2, -3, 0, ... = A055615, as follows:
%C A231425 1*cos(-2*k*Pi/n) = 1
%C A231425 1*cos(-2*k*Pi/n) -1*cos(-2*k*Pi/n) = -2
%C A231425 1*cos(-2*k*Pi/n) +1*cos(-2*k*Pi/n) -2*cos(-2*k*Pi/n) = -3
%C A231425 The two components in this GCD-Fourier triangle both sum to the sequence 1,0,0,0,0, ... A000007.
%C A231425 1 = 1
%C A231425 1 -1 = 0
%C A231425 1 +1 -2 = 0
%C A231425 ...
%C A231425 cos(-2*k*Pi/n) = 1
%C A231425 cos(-2*k*Pi/n), cos(-2*k*Pi/n) = 0
%C A231425 cos(-2*k*Pi/n), cos(-2*k*Pi/n), cos(-2*k*Pi/n) = 0
%C A231425 ...
%C A231425 This latter Fourier transform like triangle is also called the chaotic set by some authors.
%C A231425 The third arithmetic sequence is the von Mangoldt function that can be computed as sums with periods equal to rows in this triangle:
%C A231425 1
%C A231425 log(2) = Sum_{n>=0} (1/(n+1) -1/(n+2))
%C A231425 log(3) = Sum_{n>=0} (1/(n+1) +1/(n+2) -2/(n+3))
%C A231425 log(2) = Sum_{n>=0} (1/(n+1) -1/(n+2) +1/(n+3) -1/(n+4))
%C A231425 log(5) = Sum_{n>=0} (1/(n+1) +1/(n+2) +1/(n+3) +1/(n+4) -4/(n+5))
%C A231425 log(1) = Sum_{n>=0} (1/(n+1) -1/(n+2) -2/(n+3) -1/(n+4) +1/(n+5) +2/(n+6))
%C A231425 ...
%C A231425 Also the matrix inverse of the reversal of this number triangle gives the all-ones sequence in the first column. Therefore this number triangle is a companion to A054524.
%C A231425 A subset and also a companion to this triangle in terms of Greatest Common Divisor Fourier transform is A054521, since from A054521 one gets the Mobius function while from this triangle one gets the Möbius function elementwise multiplied by the natural numbers.
%C A231425 The special polynomial found in A199514 is also the solution to the row-wise equations of the symmetric polynomial described in A199514 times the chaotic set or Greatest Common Divisor Fourier transform, so that A199514 is the solution.
%H A231425 G. C. Greubel, <a href="/A231425/b231425.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A231425 T(n,k) = A023900(gcd(n,k)) for n >= k.
%t A231425 Clear[nn, t, n, k]; nn = 12; t[n_, 1] = 1; t[1, k_] = 1;
%t A231425 t[n_, k_] := t[n, k] = If[n >= k, -Sum[t[n - i, k], {i, 1, k - 1}], -Sum[t[k - i, n], {i, 1, n - 1}]]; Flatten[Table[Table[t[n, k], {k, 1, n}], {n, 1, nn}]]
%Y A231425 Cf. A191898, A014963, A008683, A023900.
%K A231425 sign,tabl
%O A231425 1,6
%A A231425 _Mats Granvik_, Nov 19 2013