A213580 Principal diagonal of the convolution array A213579.

1, 5, 15, 35, 74, 146, 277, 511, 925, 1651, 2916, 5108, 8889, 15385, 26507, 45491, 77806, 132678, 225645, 382835, 648121, 1095075, 1846920, 3109800, 5228209, 8777261, 14716167, 24643331, 41220050, 68873786, 114964741, 191719783

Offset

1,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (4,-4,-2,4,0,-1).

Formula

a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) + a(n-5).

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

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

Mathematica

(* First program *)
b[n_]:= Fibonacci[n]; c[n_]:= 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}]] (* A213579 *)
r[n_]:= Table[T[n, k], {k, 40}]
d = Table[T[n, n], {n, 1, 40}] (* A213580 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A053808 *)
(* Second program *)
Table[Fibonacci[n+3] + n*Fibonacci[n+2] -2*(n+1), {n, 40}] (* G. C. Greubel, Jul 08 2019 *)

Prog

(PARI) vector(40, n, f=fibonacci; f(n+3) +n*f(n+2) -2*(n+1)) \\ G. C. Greubel, Jul 08 2019
(Magma) F:=Fibonacci; [F(n+3) + n*F(n+2) -2*(n+1): n in [1..40]]; // G. C. Greubel, Jul 08 2019
(Sage) f=fibonacci; [f(n+3) +n*f(n+2) -2*(n+1) for n in (1..40)] # G. C. Greubel, Jul 08 2019
(GAP) F:=Fibonacci;; List([1..40], n-> F(n+3) +n*F(n+2) -2*(n+1)); # G. C. Greubel, Jul 08 2019

Crossrefs

Cf. A000045, A213500, A213579.

Sequence in context: A270784 A368475 A261441 * A221140 A330911 A322048

Adjacent sequences: A213577 A213578 A213579 * A213581 A213582 A213583

Keyword

nonn,easy

Author

Clark Kimberling, Jun 18 2012

Status

approved