A138289 - OEIS

%I #31 Jul 01 2023 09:21:41

%S 1,3,6,10,11,14,20,30,41,55,75,105,146,201,276,381,527,728,1004,1385,

%T 1912,2640,3644,5029,6941,9581,13225,18254,25195,34776,48001,66255,

%U 91450,126226,174227,240482,331932,458158,632385,872867,1204799,1662957,2295342,3168209,4373008,6035965,8331307,11499516

%N Row sums of A138060.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1).

%F Conjecture: a(n) = a(n-1) + a(n-4) with g.f. x*(-1 - 2*x - 3*x^2 - 4*x^3) / (-1 + x + x^4). - _R. J. Mathar_, Jul 08 2011

%F From _Charlie Neder_, Jun 22 2018: (Start)

%F The conjectured recurrence is true. Proof:

%F Denote by W(n) the word formed from the n-th row of A138060 and by M^k() the morphism sending W(n) to W(n+1) applied k times. We have W(5) = W(4) + W(1), so for any k >= 0, W(k+5) = M^k(W(5)) = M^k(W(4) + W(1)) = M^k(W(4)) + M^k(W(1)) = W(k+4) + W(k+1).

%F Setting n=k+5 completes the proof. (End)

%p A138060 := proc() option remember; if n = 1 then return [1] ; else  L := procname(n-1) ; a := [] ; for i from 1 to nops(L) do if op(i,L) = 1 then a := [op(a),1,2] ; elif op(i,L) <=3 then a := [op(a),op(i,L)+1] ; else a := [op(a),1] ; end if; end do: end if; a ;end proc:

%p A138289 := proc(n) add(k,k=A138060(n)) ; end proc: # _R. J. Mathar_, Jul 08 2011

%t LinearRecurrence[{1, 0, 0, 1}, {1, 3, 6, 10}, 35] (* _Jean-François Alcover_, Jul 01 2023 *)

%K nonn

%O 1,2

%A _Roger L. Bagula_, May 02 2008