A213580 - OEIS

%I #22 Nov 11 2024 12:59:57

%S 1,5,15,35,74,146,277,511,925,1651,2916,5108,8889,15385,26507,45491,

%T 77806,132678,225645,382835,648121,1095075,1846920,3109800,5228209,

%U 8777261,14716167,24643331,41220050,68873786,114964741,191719783

%N Principal diagonal of the convolution array A213579.

%H Clark Kimberling, <a href="/A213580/b213580.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,-2,4,0,-1).

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

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

%F a(n) = Fibonacci(n+3) + n*Fibonacci(n+2) - 2*(n+1). - _G. C. Greubel_, Jul 08 2019

%t (* First program *)

%t b[n_]:= Fibonacci[n]; c[n_]:= n;

%t T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]

%t TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]

%t Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213579 *)

%t r[n_]:= Table[T[n, k], {k, 40}]

%t d = Table[T[n, n], {n, 1, 40}] (* A213580 *)

%t s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]

%t s1 = Table[s[n], {n, 1, 50}] (* A053808 *)

%t (* Second program *)

%t Table[Fibonacci[n+3] + n*Fibonacci[n+2] -2*(n+1), {n, 40}] (* _G. C. Greubel_, Jul 08 2019 *)

%o (PARI) vector(40, n, f=fibonacci; f(n+3) +n*f(n+2) -2*(n+1)) \\ _G. C. Greubel_, Jul 08 2019

%o (Magma) F:=Fibonacci; [F(n+3) + n*F(n+2) -2*(n+1): n in [1..40]]; // _G. C. Greubel_, Jul 08 2019

%o (Sage) f=fibonacci; [f(n+3) +n*f(n+2) -2*(n+1) for n in (1..40)] # _G. C. Greubel_, Jul 08 2019

%o (GAP) F:=Fibonacci;; List([1..40], n-> F(n+3) +n*F(n+2) -2*(n+1)); # _G. C. Greubel_, Jul 08 2019

%Y Cf. A000045, A213500, A213579.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Jun 18 2012