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 A318774 Coefficients in expansion of 1/(1 - x - 3*x^4). 3
 1, 1, 1, 1, 4, 7, 10, 13, 25, 46, 76, 115, 190, 328, 556, 901, 1471, 2455, 4123, 6826, 11239, 18604, 30973, 51451, 85168, 140980, 233899, 388252, 643756, 1066696, 1768393, 2933149, 4864417, 8064505, 13369684, 22169131, 36762382, 60955897, 101064949, 167572342, 277859488, 460727179, 763922026, 1266639052 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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. 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. REFERENCES Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Zagros Lalo, Third layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 3 x)^n Zagros Lalo, Third layer skew diagonals in center-justified triangle of coefficients in expansion of (3 + x)^n Index entries for linear recurrences with constant coefficients, signature (1,0,0,3). FORMULA 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. MATHEMATICA CoefficientList[Series[1/(1-x-3x^4), {x, 0, 50}], x] a[n_]:= a[n]= If[n<4, 1, a[n-1] + 3*a[n-4]]; Table[a[n], {n, 0, 50}] LinearRecurrence[{1, 0, 0, 3}, {1, 1, 1, 1}, 51] PROG (Magma) [n le 4 select 1 else Self(n-1) +3*Self(n-4): n in [1..51]]; // G. C. Greubel, May 08 2021 (Sage) def a(n): return 1 if (n<4) else a(n-1) + 3*a(n-4) [a(n) for n in (0..50)] # G. C. Greubel, May 08 2021 (PARI) my(p=Mod('x, x^4-'x^3-3)); a(n) = vecsum(Vec(lift(p^n))); \\ Kevin Ryde, May 11 2021 CROSSREFS Cf. A013610, A027465, A318772, A318773. Essentially a duplicate of A143454. Sequence in context: A091290 A119256 A143454 * A341282 A065810 A123837 Adjacent sequences: A318771 A318772 A318773 * A318775 A318776 A318777 KEYWORD nonn,easy AUTHOR Zagros Lalo, Sep 04 2018 STATUS approved

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Last modified July 19 02:27 EDT 2024. Contains 374388 sequences. (Running on oeis4.)