A301653 Expansion of x*(1 + 2*x)/((1 - x)*(1 + x)*(1 - x - x^2)).

0, 1, 3, 5, 10, 16, 28, 45, 75, 121, 198, 320, 520, 841, 1363, 2205, 3570, 5776, 9348, 15125, 24475, 39601, 64078, 103680, 167760, 271441, 439203, 710645, 1149850, 1860496, 3010348, 4870845, 7881195, 12752041, 20633238, 33385280, 54018520, 87403801, 141422323, 228826125, 370248450

Offset: 0

Author

Ilya Gutkovskiy, Mar 25 2018

Keywords

• nonn
• easy

Comments

Apparently (for n > 0), numbers that have a unique partition into a sum of distinct Lucas numbers (A000204).

Links

• Eric Weisstein's World of Mathematics, Lucas Number
• Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Crossrefs

Cf. A000071, A000204, A001622, A003263, A014217, A054770, A294203.

Programs

Mathematica

CoefficientList[Series[x (1 + 2 x)/((1 - x) (1 + x) (1 - x - x^2)) , {x, 0, 40}], x]
LinearRecurrence[{1, 2, -1, -1}, {0, 1, 3, 5}, 41]
Table[LucasL[n + 1] - (3 - (-1)^n)/2, {n, 0, 40}]
Table[Floor[GoldenRatio^(n + 1)] - 1, {n, 0, 40}]

PARI

a(n) = fibonacci(n) + fibonacci(n+2) + ((-1)^n - 3)/2; \\ Altug Alkan, Mar 25 2018

Formula

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

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

a(n) = Lucas(n+1) - (3 - (-1)^n)/2.

a(n) = floor(phi^(n+1)) - 1, where phi = (1 + sqrt(5))/2 is the golden ratio (A001622).

a(n) = Sum_{k>=0} A051601(n-k,k) (conjectured). - Greg Dresden, May 18 2023