A213585 Principal diagonal of the convolution array A213584.

1, 7, 22, 54, 116, 232, 443, 821, 1490, 2664, 4710, 8256, 14373, 24883, 42878, 73594, 125880, 214664, 365087, 619425, 1048666, 1771852, 2988362, 5031744, 8459401, 14201887, 23811238, 39873726, 66695420, 111440104, 186016835

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 + 3*x - 2*x^2 - 4*x^3 - 2*x^4)/(1 - 2*x + x^3)^2.

a(n) = Fibonacci(n+4) + n*Fibonacci(n+3) - (4*n + 3). - G. C. Greubel, Jul 08 2019

Mathematica

(* First program *)
b[n_]:= Fibonacci[n+1]; 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}]] (* A213584 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
d = Table[T[n, n], {n, 1, 40}] (* A213585 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213586 *)
(* Second program *)
Table[Fibonacci[n+4] + n*Fibonacci[n+3] -4*n-3, {n, 40}] (* G. C. Greubel, Jul 08 2019 *)

Prog

(PARI) vector(40, n, f=fibonacci; f(n+4) +n*f(n+3) -(4*n+3)) \\ G. C. Greubel, Jul 08 2019
(Magma) F:=Fibonacci; [F(n+4) +n*F(n+3) -(4*n+3): n in [1..40]]; // G. C. Greubel, Jul 08 2019
(Sage) f=fibonacci; [f(n+4) +n*f(n+3) -(4*n+3) for n in (1..40)] # G. C. Greubel, Jul 08 2019
(GAP) F:=Fibonacci;; List([1..40], n-> F(n+4) +n*F(n+3) -(4*n+3)) # G. C. Greubel, Jul 08 2019

Crossrefs

Cf. A000045, A213500, A213584.

Sequence in context: A101120 A151717 A188377 * A308579 A246831 A354430

Adjacent sequences: A213582 A213583 A213584 * A213586 A213587 A213588

Keyword

nonn,easy

Author

Clark Kimberling, Jun 18 2012

Status

approved