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

0,3

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).

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)

Mathematica

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

Prog

(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

Crossrefs

Cf. A000045.

Sequence in context: A358703 A160698 A294114 * A045373 A075431 A184799

Adjacent sequences: A144704 A144705 A144706 * A144708 A144709 A144710

Keyword

easy,nonn

Author

Paul Barry, Sep 19 2008

Status

approved