A052941 -id:A052941 - cp's OEIS frontend

A356463 Sum of powers of roots of x^3 - 4*x^2 + x + 1.

3, 4, 14, 49, 178, 649, 2369, 8649, 31578, 115294, 420949, 1536924, 5611453, 20487939, 74803379, 273114124, 997165178, 3640743209, 13292693534, 48532865749, 177198026253, 646966545729, 2362135290914

Offset: 0

Greg Dresden and Ding Hao, Aug 08 2022

The three roots of x^3 - 4*x^2 + x + 1 are c1 = 1 + 2*cos(Pi/13) + 2*cos(5*Pi/13), c2 = 1 + 2*cos(3*Pi/13) + 2*cos(11*Pi/13), and c3 = 1 + 2*cos(7*Pi/13) + 2*cos(9*Pi/13), so our entries are a(n) = c1^n + c2^n + c3^n.

Index entries for linear recurrences with constant coefficients, signature (4,-1,-1).

Cf. A052941.

LinearRecurrence[{4, -1, -1}, {3, 4, 14},30]

a(n) = ([0,1,0;0,0,1;-1,-1,4]^n * [3;4;14])[1,1] \\ Jianing Song, Aug 09 2022

polsym(x^3 - 4*x^2 + x + 1, 33) \\ Joerg Arndt, Sep 13 2022

G.f.: (3 - 8*x + x^2)/(1 - 4*x + x^2 + x^3).
a(n) = 4*a(n-1) - a(n-2) - a(n-3).
a(n) = 3*b(n-1) + 5*b(n-2) + b(n-3) for b(n) = A052941(n).
a(n) = round(c1^n), c1 per the comment, n >= 3. - Bill McEachen, Sep 12 2022

A362379 Convolution triangle of A052547(n).

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 1, 4, 0, 1, 5, 2, 6, 0, 1, 5, 14, 3, 8, 0, 1, 14, 14, 27, 4, 10, 0, 1, 19, 49, 27, 44, 5, 12, 0, 1, 42, 68, 113, 44, 65, 6, 14, 0, 1, 66, 175, 159, 214, 65, 90, 7, 16, 0, 1, 131, 286, 465, 304, 360, 90, 119, 8, 18, 0, 1

Offset: 0

Philippe Deléham, Apr 20 2023

			Triangle begins, for n>=0, 0<=k<=n :
   1 ;
   0,  1 ;
   2,  0,   1 ;
   1,  4,   0,  1 ;
   5,  2,   6,  0,  1 ;
   5, 14,   3,  8,  0,  1 ;
  14, 14,  27,  4, 10,  0,  1 ;
  19, 49,  27, 44,  5, 12,  0, 1 ;
  42, 68, 113, 44, 65,  6, 14, 0, 1 ;
  ...

Cf. A052547, A077998 (row sums), A052964 (diagonal sums).

T(n,k) = T(n-1,k) + T(n-1,k-1) + 2*T(n-2,k) - T(n-2,k-1) - T(n-3,k) ; T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = T(2,1) = 0, T(2,0) = 2, T(n,k) = 0 if k<0 or if k>n .
Sum_{k = 0..n} T(n,k)*x^k = A052547(n), A077998(n), A052536(n), A052941(n) for x = 0, 1, 2, 3 respectively.
Sum_{k = 0..n} T(n,k)*2^(n-k) = A139818(n+1) = A001045(n+1)^2.

cp's OEIS Frontend

A356463 Sum of powers of roots of x^3 - 4*x^2 + x + 1.

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