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A227168 a(n) = gcd(2*n, n*(n+1)/2)^2.

%I A227168 #66 Aug 21 2022 04:18:53
%S A227168 1,1,36,4,25,9,196,16,81,25,484,36,169,49,900,64,289,81,1444,100,441,
%T A227168 121,2116,144,625,169,2916,196,841,225,3844,256,1089,289,4900,324,
%U A227168 1369,361,6084,400
%N A227168 a(n) = gcd(2*n, n*(n+1)/2)^2.
%C A227168 a(n) is defined as A062828(n)^2 for n >= 1. If we extend the sequence to n=0 and negative n by use of the recurrence that relates a(n) to a(n+12), a(n+8) and a(n+4), we obtain a(0)=0, a(-1)=4 and a(-n) = A176743(n-2)^2 for n >= 2.
%C A227168 Define c(n) = a(n+2) - a(n-2) for c >= 0. Because a(n) is a shuffle of three interleaved 2nd-order polynomials, c(n) is a shuffle of three interleaved 1st-order polynomials: c(n) = 4* A062828(n)*(periodically repeated 1, 8, 1, 1).
%C A227168 The sequence a(n) is case p=0 of the family A062828(n)*A062828(n+p):
%C A227168 0, 1, 1, 36,  4, 25,  9, 196, ... = a(n).
%C A227168 0, 1, 6, 12, 10, 15, 42,  56, ... = A130658(n)*A000217(n) = A177002(n-1)*A064038(n+1).
%C A227168 0, 6, 2, 30,  6, 70, 12, 126, ... = 2*A198148(n)
%C A227168 0, 2, 5, 18, 28, 20, 27,  70, ... = A177002(n+2)*A160050(n+1) = A014695(n+2)*A000096(n).
%H A227168 G. C. Greubel, <a href="/A227168/b227168.txt">Table of n, a(n) for n = 1..5000</a>
%H A227168 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,3,0,0,0,-3,0,0,0,1).
%F A227168 a(n) = A062828(n)^2.
%F A227168 a(4n) = (4*n+1)^2; a(2n+1) = (n+1)^2; a(4n+2) = 4*(4*n+3)^2.
%F A227168 a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
%F A227168 a(n) * (period 4: repeat 4, 1, 1, 4) = A061038(n).
%F A227168 A005565(n-3) = a(n+1) * A061037(n). - Corrected by _R. J. Mathar_, Jul 25 2013
%F A227168 a(n) = A130658(n-1)^2 * A181318(n). - Corrected by _R. J. Mathar_, Aug 01 2013
%F A227168 G.f.: -x*(1 + x + 36*x^2 + 4*x^3 + 22*x^4 + 6*x^5 + 88*x^6 + 4*x^7 + 9*x^8 + x^9 + 4*x^10) / ( (x-1)^3*(1+x)^3*(x^2+1)^3 ). - _R. J. Mathar_, Jul 20 2013
%F A227168 Sum_{n>=1} 1/a(n) = 47*Pi^2/192 + 3*G/8, where G is Catalan's constant (A006752). - _Amiram Eldar_, Aug 21 2022
%p A227168 A227168 := proc(n)
%p A227168     A062828(n)^2 ;
%p A227168 end proc: # _R. J. Mathar_, Jul 25 2013
%t A227168 a[n_] := GCD[2*n, n*(n + 1)/2]^2; Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Jul 03 2013 *)
%o A227168 (PARI) a(n)=if(n%2, n*if(n%4>2, 2, 1), n/2)^2 \\ _Charles R Greathouse IV_, Jul 07 2013
%o A227168 (Magma) [GCD(2*n, n*(n+1)/2)^2: n in [1..50]]; // _G. C. Greubel_, Sep 20 2018
%Y A227168 Cf. A062828, A016814, A017138, A005565, A006752, A061037, A061038.
%K A227168 nonn,easy,less
%O A227168 1,3
%A A227168 _Paul Curtz_, Jul 03 2013