A213589 Antidiagonal sums of the convolution array A213587.

1, 6, 20, 55, 135, 308, 668, 1395, 2830, 5610, 10914, 20904, 39515, 73860, 136720, 250937, 457137, 827260, 1488190, 2662905, 4741946, 8407236, 14846100, 26120400, 45801925, 80064018, 139553708, 242597035, 420678315, 727792580

Offset

1,2

Links

Clark Kimberling, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Formula

a(n) = 3*a(n-1) - 5*a(n-3) + 3*a(n-5) + a(n-6).

G.f.: x*(1 + 3*x + 2*x^2)/(1 - x - x^2)^3.

a(n) = (n+1)/2*Sum_{j=0..(n+1)/2}((n-j)*binomial(n-j+1,j)). - Vladimir Kruchinin, Apr 09 2016

a(n) = (n+1)*(n*Lucas(n+3) - 2*Fibonacci(n))/10 = (n+1)*((n+2) *Fibonacci(n+3) + 2*(n-2)*Fibonacci(n+2))/10. - G. C. Greubel, Jul 08 2019

Mathematica

(* First program *)
b[n_]:= Fibonacci[n+1]; c[n_]:= Fibonacci[n+1];
T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213587 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213588 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213589 *)
(* Second program *)
Table[(n+1)*(n*LucasL[n+3] -2*Fibonacci[n])/10, {n, 35}]] (* G. C. Greubel, Jul 08 2019 *)

Prog

(Maxima)
a(n):=(n+1)/2*sum((n-j)*binomial(n-j+1, j), j, 0, (n+1)/2); /* Vladimir Kruchinin, Apr 09 2016 */
(PARI) vector(35, n, f=fibonacci; (n+1)*((n+2)*f(n+3)+ 2*(n-2)*f(n+2) )/10) \\ G. C. Greubel, Jul 08 2019
(Magma) F:=Fibonacci; [(n+1)*((n+2)*F(n+3) + 2*(n-2)*F(n+2))/10: n in [1..35]]; // G. C. Greubel, Jul 08 2019
(Sage) f=fibonacci; [(n+1)*((n+2)*f(n+3)+ 2*(n-2)*f(n+2) )/10 for n in (1..35)] # G. C. Greubel, Jul 08 2019
(GAP) F:=Fibonacci;; List([1..35], n-> (n+1)*((n+2)*F(n+3) + 2*(n-2)*F(n+2))/10) # G. C. Greubel, Jul 08 2019

Crossrefs

Cf. A213587, A213500.

Sequence in context: A027993 A028492 A059822 * A152959 A328681 A109903

Adjacent sequences: A213586 A213587 A213588 * A213590 A213591 A213592

Keyword

nonn,easy

Author

Clark Kimberling, Jun 19 2012

Status

approved