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%I #62 Aug 12 2022 09:22:34
%S 0,0,1,1,5,5,35,14,42,30,165,55,143,91,455,140,340,204,969,285,665,
%T 385,1771,506,1150,650,2925,819,1827,1015,4495,1240,2728,1496,6545,
%U 1785,3885,2109,9139,2470,5330,2870,12341,3311,7095,3795,16215,4324
%N a(4*n) = n*(16*n^2-1)/3, a(2*n+1) = n*(n+1)*(2*n+1)/6, a(4*n+2) = (4*n+1)*(4*n+2)*(4*n+3)/6.
%C a(n+2) is divisible by A060819(floor(n/3)).
%C a(n) is divisible by A176672(floor(n/3)).
%C Denominator of a(n)/n is of period 24: 1,1,3,4,1,6,1,4,3,1,1,12,1,2,3,4,1,3,1,4,3,2,1,12 (two successive palindromes).
%C This is the fifth column of the triangle A107711, hence the formula involving gcd(n+2,4) given below follows. - _Wolfdieter Lang_, Feb 24 2014
%H Vincenzo Librandi, <a href="/A208950/b208950.txt">Table of n, a(n) for n = 0..5000</a>
%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,4,0,0,0,-6,0,0,0,4,0,0,0,-1).
%F a(n) = 4*a(n-4) - 6*a(n-8) + 4*a(n-12) - a(n-16).
%F a(n+1) = A002415(n+1)/A145979(n-1).
%F a(n) = A051724(n-1) * A051724(n) * A051724(n+1).
%F a(n) = A060819(n-1) * A060819(n) * A060819(n+1) / 3.
%F a(n) * a(n+4) = A061037(n+1) * A061037(n+2) * A061037(n+3) / 9.
%F a(n) = A138190(n)/A000034(n) for n > 0.
%F a(n) = A000292(n-1)/A176895(n+2) for n > 0.
%F a(n)/a(n+4) = n*(n^2-1)/((n+3)*(n+4)*(n+5)).
%F a(n)/a(n+12) = (n-1)*n*(n+1)/((n+11)*(n+12)*(n+13)).
%F G.f.: (x^2 + x^3 + 5*x^4 + 5*x^5 + 31*x^6 + 10*x^7 + 22*x^8 + 10*x^9 + 31*x^10 + 5*x^11 + 5*x^12 + x^13 + x^14) / ((1-x)^4*(1+x)^4*(1 + 4*x^2 + 6*x^4 + 4*x^6 + x^8)). - _R. J. Mathar_, Mar 10 2012
%F From _Wolfdieter Lang_, Feb 24 2014: (Start)
%F G.f.: (1 + x^12 + x*(1+x^10) + 5*x^2*(1+x^8) + 5*x^3*(1+x^7) + 31*x^4*(1+x^4) + 10*x^5*(1+x^2) + 22*x^6)/(1-x^4)^4. This is the preceding g.f. rewritten.
%F a(n) = binomial(n+1,3)*gcd(n+2,4)/4, n >= 0. From the g.f., see a comment above on A107711. (End)
%F a(n) = (n*(n-1)*((n+1)*(4+2*(-1)^n + (1+(-1)^n)*(-1)^((2*n+3+(-1)^n)/4))))/48. - _Luce ETIENNE_, Jan 01 2015
%F Sum_{n>=2} 1/a(n) = 12 - 27*log(2)/2. - _Amiram Eldar_, Aug 12 2022
%t CoefficientList[Series[(x^2 + x^3 + 5 x^4 + 5 x^5 + 31 x^6 + 10 x^7 + 22 x^8 + 10 x^9 + 31 x^10 + 5 x^11 + 5 x^12 + x^13 + x^14)/((1 - x)^4 (1 + x)^4 (1 + 4 x^2 + 6 x^4 + 4 x^6 + x^8)), {x, 0, 47}], x] (* _Bruno Berselli_, Mar 11 2012 *)
%o (Maxima) A208950(n) := block(
%o [a,npr] ,
%o if equal(mod(n,4), 0) then (
%o a : n/12*(n^2-1)
%o ) else if equal(mod(n,2),0) then (
%o a : (n-1)*n*(n+1)/6
%o ) else (
%o npr : (n-1)/2,
%o a : npr*(npr+1)*n/6
%o ) ,
%o return(a)
%o )$ /* _R. J. Mathar_, Mar 10 2012 */
%o (PARI) vector(50, n, n--; binomial(n+1,3)*gcd(n+2,4)/4) \\ _G. C. Greubel_, Sep 20 2018
%o (Magma) [Binomial(n+1,3)*GCD(n+2,4)/4: n in [0..50]]; // _G. C. Greubel_, Sep 20 2018
%Y Cf. A107711 (fifth column).
%Y Cf. A000034, A000292, A002415, A051724, A060819, A061037, A107711, A138190, A145979, A176672, A176895.
%K nonn,easy
%O 0,5
%A _Paul Curtz_, Mar 03 2012
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