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A231425 The Schramm triangle. T(n,k)=a(GCD(n,k)), where a = Dirichlet inverse of Euler totient. 2
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, -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, 1, 1, 1, 1, 1, 1, 1, 1, -10, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

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.

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:

1*Cos(-2*k*Pi/n) = 1

1*Cos(-2*k*Pi/n) -1*Cos(-2*k*Pi/n) = -2

1*Cos(-2*k*Pi/n) +1*Cos(-2*k*Pi/n) -2*Cos(-2*k*Pi/n) = -3

The two components in this GCD-Fourier triangle both sum to the sequence 1,0,0,0,0... A000007.

1 = 1

1 -1 = 0

1 +1 -2 = 0

...

Cos(-2*k*Pi/n) = 1

Cos(-2*k*Pi/n), Cos(-2*k*Pi/n) = 0

Cos(-2*k*Pi/n), Cos(-2*k*Pi/n), Cos(-2*k*Pi/n) = 0

...

This latter Fourier transform like triangle is also called the chaotic set by some authors.

The third arithmetic sequence is the von Mangoldt function that can be computed as sums with periods equal to rows in this triangle:

1

Log(2) = Sum_{n>=0} (1/(n+1) -1/(n+2))

Log(3) = Sum_{n>=0} (1/(n+1) +1/(n+2) -2/(n+3))

Log(2) = Sum_{n>=0} (1/(n+1) -1/(n+2) +1/(n+3) -1/(n+4))

Log(5) = Sum_{n>=0} (1/(n+1) +1/(n+2) +1/(n+3) +1/(n+4) -4/(n+5))

Log(1) = Sum_{n>=0} (1/(n+1) -1/(n+2) -2/(n+3) -1/(n+4) +1/(n+5) +2/(n+6))

...

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.

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.

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.

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

FORMULA

T(n,k) = A023900(GCD(n,k)), for n>=k.

MATHEMATICA

Clear[nn, t, n, k]; nn = 12; t[n_, 1] = 1; t[1, k_] = 1;

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}]]

CROSSREFS

Cf. A191898, A014963, A008683, A023900.

Sequence in context: A025429 A325561 A076250 * A136713 A224782 A160322

Adjacent sequences:  A231422 A231423 A231424 * A231426 A231427 A231428

KEYWORD

sign,tabl

AUTHOR

Mats Granvik, Nov 19 2013

STATUS

approved

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Last modified May 23 12:38 EDT 2019. Contains 323514 sequences. (Running on oeis4.)