A144707 Diagonal sums of the triangle A132047.

1, 1, 2, 7, 11, 22, 35, 61, 98, 163, 263, 430, 695, 1129, 1826, 2959, 4787, 7750, 12539, 20293, 32834, 53131, 85967, 139102, 225071, 364177, 589250, 953431, 1542683, 2496118, 4038803, 6534925, 10573730, 17108659, 27682391, 44791054, 72473447, 117264505

Offset: 0

Author

Paul Barry, Sep 19 2008

Keywords

• easy
• nonn

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Crossrefs

Cf. A000045.

Programs

Mathematica

Table[3*Fibonacci[n+1] -3 -(-1)^n +2*Boole[n==0], {n,0,40}] (* G. C. Greubel, Jun 16 2022 *)

PARI

Vec((1-x^2+4*x^3+2*x^4) / ((1-x^2)*(1-x-x^2)) + O(x^50)) \\ Colin Barker, Jul 12 2017

SageMath

[3*fibonacci(n+1) -2 -2*((n+1)%2) +2*bool(n==0) for n in (0..40)] # G. C. Greubel, Jun 16 2022

Formula

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

a(n) = 3*Fibonacci(n+1) - 3 - (-1)^n + 2*0^n.

a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>4. - Philippe Deléham, Dec 16 2008

From Colin Barker, Jul 12 2017: (Start)

a(n) = (3*2^(-n-1)*((1 + sqrt(5))^(n+1) - (1-sqrt(5))^(n+1))) / sqrt(5) - 4 for n>0 and even.

a(n) = (3*2^(-n-1)*((1+sqrt(5))^(n+1) - (1-sqrt(5))^(n+1)))/sqrt(5) - 2 for n odd.

(End)