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

0,3

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).

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A106522, A000045, A000073.

Sequence in context: A173285 A025084 A134512 * A007068 A121720 A056515

Adjacent sequences: A106520 A106521 A106522 * A106524 A106525 A106526

Keyword

easy,nonn

Author

Paul Barry, May 06 2005

Status

approved