A131780 Row sums of triangle A131779.

1, 4, 5, 10, 15, 26, 41, 68, 109, 178, 287, 466, 753, 1220, 1973, 3194, 5167, 8362, 13529, 21892, 35421, 57314, 92735, 150050, 242785, 392836, 635621, 1028458, 1664079, 2692538, 4356617, 7049156, 11405773, 18454930, 29860703, 48315634, 78176337, 126491972

Offset: 1

Author

Gary W. Adamson, Jul 14 2007

Keywords

• nonn
• easy

Comments

a(n)/a(n-1) tends to phi; e.g., a(10)/a(9) = 178/109 = 1.633...

Examples

			a(4) = 10 = sum of row 4 terms of triangle A131779: (3 + 1 + 5 + 1).
		

Links

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

Crossrefs

Row sums of A131779.

Programs

Mathematica

LinearRecurrence[{1,2,-1,-1},{1,4,5,10},40] (* or *) CoefficientList[ Series[(1+3*x-x^2-2*x^3)/(1-x-2*x^2+x^3+x^4),{x,0,40}],x] (* Harvey P. Dale, Aug 27 2021 *)

PARI

a(n) = 2*fibonacci(n+1) - (1 - (-1)^n)/2; \\ Andrew Howroyd, Sep 01 2018

PARI

Vec((1 + 3*x - x^2 - 2*x^3)/((1 - x)*(1 + x)*(1 - x - x^2)) + O(x^40)) \\ Andrew Howroyd, Sep 01 2018

Formula

From Andrew Howroyd, Sep 01 2018: (Start)

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

a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n > 4.

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

E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5 - sinh(x) - 2. - Stefano Spezia, Jul 11 2025

Extensions

Terms a(11) and beyond from Andrew Howroyd, Sep 01 2018