A213504 Principal diagonal of the convolution array A213590.

1, 6, 35, 138, 488, 1564, 4733, 13734, 38711, 106846, 290496, 781264, 2084753, 5531846, 14619811, 38527834, 101328712, 266119228, 698218525, 1830665830, 4797572551, 12568780126, 32920653120, 86214096768, 225758326273

Offset

1,2

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (6,-10,-2,15,-2,-8,0,1).

Formula

a(n) = 6*a(n-1) - 10*a(n-2) - 2*a(n-3) + 15*a(n-4) - 2*a(n-5)- 8*a(n-6) + a(n-8).

G.f.: x*(1 + 9*x^2 - 10*x^3 + 7*x^4 - 2*x^5)/((1 - 3*x + x^2)*(1 - x - x^2)^3). [corrected by Georg Fischer, May 11 2019]

a(n) = Fibonacci(2*n+6) - Fibonacci(n+6) - 2*n*Fibonacci(n+3) - n^2*Fibonacci(n+1). - G. C. Greubel, Jul 06 2019

Mathematica

(* First program *)
b[n_]:= n^2; c[n_]:= Fibonacci[n];
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}]] (* A213590 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213504 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213557 *)
(* Second program *)
With[{F = Fibonacci}, Table[F[2*n+6] -F[n+6] -2*n*F[n+3] -n^2*F[n+1], {n, 40}]] (* G. C. Greubel, Jul 06 2019 *)

Prog

(PARI) vector(40, n, my(f=fibonacci); f(2*n+6) - f(n+6) - 2*n*f(n+3) - n^2*f(n+1)) \\ G. C. Greubel, Jul 06 2019
(Magma) F:=Fibonacci; [F(2*n+6) -F(n+6) -2*n*F(n+3) -n^2*F(n+1): n in [1..40]]; // G. C. Greubel, Jul 06 2019
(Sage) f=fibonacci; [f(2*n+6) -f(n+6) -2*n*f(n+3) -n^2*f(n+1) for n in (1..40)] # G. C. Greubel, Jul 06 2019
(GAP) F:=Fibonacci;; List([1..40], n-> F(2*n+6) -F(n+6) -2*n*F(n+3) -n^2*F(n+1)) # G. C. Greubel, Jul 06 2019

Crossrefs

Cf. A213590, A213500.

Sequence in context: A101077 A094952 A024526 * A089581 A132657 A161784

Adjacent sequences: A213501 A213502 A213503 * A213505 A213506 A213507

Keyword

nonn,easy

Author

Clark Kimberling, Jun 19 2012

Status

approved