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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.)