A187275 a(n) = (n/4)*5^(n/2)*((1+sqrt(5))^2+(-1)^n*(1-sqrt(5))^2).
0, 5, 30, 75, 300, 625, 2250, 4375, 15000, 28125, 93750, 171875, 562500, 1015625, 3281250, 5859375, 18750000, 33203125, 105468750, 185546875, 585937500, 1025390625, 3222656250, 5615234375, 17578125000, 30517578125, 95214843750, 164794921875, 512695312500, 885009765625, 2746582031250
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- R. Kemp, On the number of words in the language {w in Sigma* | w = w^R }^2, Discrete Math., 40 (1982), 225-234. See Lemma 1.
- Index entries for linear recurrences with constant coefficients, signature (0,10,0,-25).
Programs
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Magma
/* By definition: */ Z
:=PolynomialRing(Integers()); N :=NumberField(x^2-5); [Integers()!((n/4)*r^n*((1+r)^2+(-1)^n*(1-r)^2)): n in [0..30]]; // Bruno Berselli, Mar 29 2016 -
Magma
I:=[0,5,30,75]; [n le 4 select I[n] else 10*Self(n-2)-25*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Mar 29 2016
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Maple
See A187272.
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Mathematica
LinearRecurrence[{0, 10, 0, -25}, {0, 5, 30, 75}, 30] (* Vincenzo Librandi, Mar 29 2016 *) -
Python
def A187275(n): return n*5**(1+(n>>1)) if n&1 else 3*n*5**(n>>1) # Chai Wah Wu, Feb 19 2024
Formula
a(n) = 10*a(n-2) - 25*a(n-4). - Colin Barker, Jul 25 2013
G.f.: 5*x*(x+1)*(5*x+1) / (5*x^2-1)^2. - Colin Barker, Jul 25 2013
a(2*n) = 6*n*5^n, a(2*n+1) = (2*n+1)*5^(n+1). - Andrew Howroyd, Mar 28 2016