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Convolution square of A040001.
2

%I #38 Mar 16 2023 12:13:57

%S 1,2,5,6,10,10,15,14,20,18,25,22,30,26,35,30,40,34,45,38,50,42,55,46,

%T 60,50,65,54,70,58,75,62,80,66,85,70,90,74,95,78,100,82,105,86,110,90,

%U 115,94,120,98,125,102,130,106,135,110,140,114,145,118,150,122,155,126

%N Convolution square of A040001.

%H G. C. Greubel, <a href="/A184418/b184418.txt">Table of n, a(n) for n = 0..10000</a>

%H Michael Somos, <a href="http://somos.crgr4.com/rfmc.txt">Rational Function Multiplicative Coefficients</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).

%F G.f.: (1 + x + x^2)^2 / (1 - x^2)^2 = 1 + x * (x + 2) * (2*x + 1) / (1 - x^2)^2. a(-n) = -a(n) except a(0) = 2.

%F Euler transform of length 3 sequence [2, 2, -2].

%F a(n) = 2 * b(n) where b() is multiplicative with b(2^e) = 5 * 2^(e-2) if e>0, b(p^e) = p^e if p>2.

%F a(2*n + 1) = 4*n + 2, a(2*n) = 5*n except a(0) = 2.

%F a(n) = (9+(-1)^n)*n/4 = (n/2)*A010710(n+1) for n>0. - _Bruno Berselli_, Mar 24 2011

%e G.f. = 1 + 2*x + 5*x^2 + 6*x^3 + 10*x^4 + 10*x^5 + 15*x^6 + 14*x^7 + 20*x^8 + ...

%t LinearRecurrence[{0,2,0,-1},{1,2,5,6,10},80] (* _Harvey P. Dale_, Jul 03 2017 *)

%o (PARI) {a(n) = (n==0) + n * ([5/2, 2] [n%2 + 1])};

%o (PARI) {a(n) = if( n==0, 1, sign(n) * polcoeff( (1 + x + x^2)^2 / (1 - x^2)^2 + x * O(x^abs(n)), abs(n)))};

%o (Magma) I:=[2,5,6,10]; [1] cat [n le 4 select I[n] else 2*Self(n-2) - Self(n-4): n in [1..30]]; // _G. C. Greubel_, Aug 14 2018

%K nonn

%O 0,2

%A _Michael Somos_, Feb 14 2011