A138289 Row sums of A138060.

1, 3, 6, 10, 11, 14, 20, 30, 41, 55, 75, 105, 146, 201, 276, 381, 527, 728, 1004, 1385, 1912, 2640, 3644, 5029, 6941, 9581, 13225, 18254, 25195, 34776, 48001, 66255, 91450, 126226, 174227, 240482, 331932, 458158, 632385, 872867, 1204799, 1662957, 2295342, 3168209, 4373008, 6035965, 8331307, 11499516

Offset: 1

Author

Roger L. Bagula, May 02 2008

Keywords

• nonn

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1).

Programs

Maple

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:
A138289 := proc(n) add(k,k=A138060(n)) ; end proc: # R. J. Mathar, Jul 08 2011

Mathematica

LinearRecurrence[{1, 0, 0, 1}, {1, 3, 6, 10}, 35] (* Jean-François Alcover, Jul 01 2023 *)

Formula

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

From Charlie Neder, Jun 22 2018: (Start)

The conjectured recurrence is true. Proof:

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).

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