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A181435 First column in matrix inverse of a mixed convolution of A052906. 4
1, -4, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, 1, -1, 0, 0, 1, 0, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, 1, 1, 0, -1, -1, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, 1, 1, -1, 0, -1, 1, 0, 0, 1, -1, -1, 0, 1, -1, -1, 0, -1, 1, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

It appears that except for the second term, the sequence is identical to the Möbius function.

LINKS

R. J. Mathar, Table of n, a(n) for n = 1..1091

FORMULA

From Mats Granvik, Sep 16 2017: (Start)

a(n) as the matrix inverse of a mixed convolution: Let c = 3 and let the sequence b be defined by the recurrence: b(1) = 1, b(2) = c, b(3) = c^2; for n >= 4, b(n) = c*b(n-1) + b(n-2), so b(n) = A052906(n-1), and let the lower triangular matrix A be: If n >= k then A(n,k) = b(n - k + 1) else A(n,k) = 0, and let B be the lower triangular matrix A051731. Then the matrix inverse (A.B)^-1 will have a(n) as its first column.

The matrix product T = A.B can be defined as follows: Let c = 3 and the sequence b be defined by the recurrence b(0) = 1, b(1) = 1; for b >= 2, b(n) = c*b(n - 1) + b(n - 2); and let T be the lower triangular matrix defined by the recurrence: T(n, 1) = If n >= 1 then T(n, 1) = b(n) else T(n, 1) = 0; for k >= 2, T(n, k) = If n >= k then (Sum_{i=1..k-1} T(n - i, k - 1) - T(n - i, k)) else 0. (Then the matrix inverse of T will have a(n) as its first column.)

(End)

MAPLE

b := proc(n)

    option remember;

    local c;

    c := 3;

    if n <= 3 then

        op(n, [1, c, c^2]) ;

    else

        c*procname(n-1)+procname(n-2) ;

    end if;

end proc:

A := proc(n, k)

    if n >= k then

        b(n-k+1) ;

    else

        0 ;

    end if;

end proc:

B := proc(n, k)

    if modp(n, k) = 0 then

        1;

    else

        0;

    end if;

end proc:

AB := proc(n, k)

    option remember;

    add( A(n, j)*B(j, k), j=1..n) ;

end proc:

ABinv := proc(n, k)

    option remember;

    if k > n then

        0;

    elif k = n then

        1;

    else

        -add( AB(n, j)*procname(j, k), j=k..n-1) ;

    end if;

end proc:

A181435 := proc(n)

    ABinv(n, 1) ;

end proc:

for n from 1 do

    printf("%d %d\n", n, ABinv(n, 1)) ;

end do: # R. J. Mathar, Oct 06 2017

MATHEMATICA

Clear[t, n, k, nn, b, A, c]; nn = 77; c = 3; b[0] = 1; b[1] = 1; b[n_] := b[n] = c*b[n - 1] + b[n - 2]; t[n_, 1] = If[n >= 1, b[n], 0]; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}] - Sum[t[n - i, k], {i, 1, k - 1}], 0]; MatrixForm[A = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}]]; Inverse[A][[All, 1]] (* Mats Granvik, Sep 16 2017 *)

CROSSREFS

Cf. A003688, A006190, A013946, A008683, A178536, A181434.

Sequence in context: A094924 A056968 A213541 * A206774 A307850 A290455

Adjacent sequences:  A181432 A181433 A181434 * A181436 A181437 A181438

KEYWORD

sign

AUTHOR

Mats Granvik, Oct 20 2010

STATUS

approved

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Last modified July 19 09:33 EDT 2019. Contains 325155 sequences. (Running on oeis4.)