A318774 - OEIS

%I #20 Sep 08 2022 08:46:22

%S 1,1,1,1,4,7,10,13,25,46,76,115,190,328,556,901,1471,2455,4123,6826,

%T 11239,18604,30973,51451,85168,140980,233899,388252,643756,1066696,

%U 1768393,2933149,4864417,8064505,13369684,22169131,36762382,60955897,101064949,167572342,277859488,460727179,763922026,1266639052

%N Coefficients in expansion of 1/(1 - x - 3*x^4).

%C The coefficients in the expansion of 1/(1 - x - 3*x^4) are given by the sequence generated by the row sums in triangle A318772.

%C Coefficients in expansion of 1/(1 - x - 3*x^4) are given by the sum of numbers along "third Layer" skew diagonals pointing top-right in triangle A013610 ((1+3x)^n) and by the sum of numbers along "third Layer" skew diagonals pointing top-left in triangle A027465 ((3+x)^n), see links.

%D Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

%H G. C. Greubel, <a href="/A318774/b318774.txt">Table of n, a(n) for n = 0..1000</a>

%H Zagros Lalo, <a href="/A318774/a318774_1.pdf">Third layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 3 x)^n</a>

%H Zagros Lalo, <a href="/A318774/a318774.pdf">Third layer skew diagonals in center-justified triangle of coefficients in expansion of (3 + x)^n</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,3).

%F a(n) = a(n-1) + 3*a(n-4) for n >= 0, a(n)=0 for n < 0, with a(0) = a(1) = a(2) = a(3) = 1.

%t CoefficientList[Series[1/(1-x-3x^4), {x, 0, 50}], x]

%t a[n_]:= a[n]= If[n<4, 1, a[n-1] + 3*a[n-4]]; Table[a[n], {n,0,50}]

%t LinearRecurrence[{1,0,0,3}, {1,1,1,1}, 51]

%o (Magma) [n le 4 select 1 else Self(n-1) +3*Self(n-4): n in [1..51]]; // _G. C. Greubel_, May 08 2021

%o (Sage)

%o def a(n): return 1 if (n<4) else a(n-1) + 3*a(n-4)

%o [a(n) for n in (0..50)] # _G. C. Greubel_, May 08 2021

%o (PARI) my(p=Mod('x,x^4-'x^3-3)); a(n) = vecsum(Vec(lift(p^n))); \\ _Kevin Ryde_, May 11 2021

%Y Cf. A013610, A027465, A318772, A318773.

%Y Essentially a duplicate of A143454.

%K nonn,easy

%O 0,5

%A _Zagros Lalo_, Sep 04 2018