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URL: https://oeis.org/A389120

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A389120
a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4) with a(0)=0, a(1)=1, a(2)=2, a(3)=7.
2
0, 1, 2, 7, 20, 57, 166, 479, 1384, 4001, 11562, 33415, 96572, 279097, 806606, 2331135, 6737104, 19470593, 56271058, 162626375, 469998948, 1358322169, 3925623926, 11345263711, 32788420664, 94760294433, 273862333690, 791476833863, 2287410503308, 6610739047289, 19105390435230
OFFSET
0,3
COMMENTS
The triples (X(n) = a(n) * phi^(3/2) - A387018(n) * phi^(1/2), Y(n) = A138573(n) * phi - A138574(n), 1) where phi = (1+sqrt(5))/2 form right triangles with hypotenuse 1. X(n)^2 + Y(n)^2 = 1 for all n > 0
Alternatively the sequences are related with this integer matrix
+- -+
| 1, -1, 1, -1 |
| -1, 0, -1, 0 |
M= | 1, 0, 1, -1 |
| 0, 1, -1, 0 |
+- -+
+- -+ +- -+
|-A387018(n+1) | |-A387018(n) |
| a(n+1) | = M. | a(n) |
|-A138574(n+1) | |-A138574(n) |
| A138573(n+1) | | A138573(n) |
+- -+ +- -+
FORMULA
G.f.: x*(1 + x^2) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4).
EXAMPLE
When n = 1: (Kepler triangle)
X(1) = 1 * phi^(3/2) - 1 * phi^(1/2) = 0.78615137775742328606955858584295892953,
Y(1) = 1 * phi - 1 = 0.61803398874989484820458683436563811772.
When n = 2:
X(2) = 2 * phi^(3/2) - 4 * phi^(1/2) = -0.97173654351329135636572775178906512438,
Y(2) = 2 * phi - 3 = 0.23606797749978969640917366873127623544.
When n = 3:
X(3) = 7 * phi^(3/2) - 11 * phi^(1/2) = 0.41498104624568714547722025395074653983,
Y(3) = 5 * phi - 9 = -0.90983005625052575897706582817180941141.
MATHEMATICA
LinearRecurrence[{2, 2, 2, -1}, {0, 1, 2, 7}, 30] (* Paolo Xausa, Jan 03 2026 *)
PROG
(PARI) my(x='x+O('x^50)); concat([0], Vec( x*(1 + x^2) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) ))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved