A106523 Diagonal sums of number triangle A106522.

1, 1, 3, 4, 10, 14, 33, 49, 109, 170, 362, 586, 1207, 2011, 4037, 6878, 13536, 23464, 45475, 79891, 153011, 271612, 515460, 922372, 1738101, 3129565, 5865063, 10611336, 19802382, 35960970, 66888917, 121820229, 226016385, 412547222

Offset: 0

Author

Paul Barry, May 06 2005

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,3,1,0,-1).

Crossrefs

Cf. A106522, A000045, A000073.

Programs

Magma

I:=[1,1,3,4,10]; [n le 5 select I[n] else 3*Self(n-2) + Self(n-3) -Self(n-5): n in [1..41]]; // G. C. Greubel, Aug 10 2021

Mathematica

T[n_]:= T[n]= If[n<2, 0, If[n==2, 1, T[n-1] + T[n-2] + T[n-3]]]; (* A000073 *)
a[n_]:= (1/11)*((-1)^n*(Fibonacci[n+2] +2*Fibonacci[n]) +10*T[n+2] +5*T[n+1] + 3*T[n]);
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Aug 10 2021 *)

Sage

@CachedFunction
def T(n):
    if (n<2): return 0
    elif (n==2): return 2
    else: return T(n-1) + T(n-2) + T(n-3)
def a(n): return (1/11)*((-1)^n*(fibonacci(n+2) +2*fibonacci(n)) +10*T(n+2) +5*T(n+1) + 3*T(n))
[a(n) for n in (0..40)] # G. C. Greubel, Aug 10 2021

Formula

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

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

a(n) = Sum_{k=0..floor(n/2)} A106522(n-k, k)

a(n) = (1/11)*( 10*T(n+2) + 5*T(n+1) + 3*T(n) + (-1)^n*( F(n+1) + 3*F(n) ) ), where T(n) = A000073, and F(n) = A000045. - G. C. Greubel, Aug 10 2021