|
%I #70 Aug 09 2022 12:03:58
%S 0,1,5,9,7,10,27,35,22,27,65,77,45,52,119,135,76,85,189,209,115,126,
%T 275,299,162,175,377,405,217,232,495,527,280,297,629,665,351,370,779,
%U 819,430,451,945,989,517,540,1127,1175,612,637,1325,1377,715,742,1539
%N Numerator of the Harary number for the star graph s_n.
%H G. C. Greubel, <a href="/A160050/b160050.txt">Table of n, a(n) for n = 1..5000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HararyIndex.html">Harary Index</a>.
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3,-6,10,-12,12,-10,6,-3,1).
%F Numerator of (1/4)*(n+2)*(n-1). - _Joerg Arndt_, Jan 04 2011
%F It appears that a(n + 1) = A060819(n-1) * A060819(n + 2). - _Paul Curtz_, Dec 23 2010 [Corrected by _Joerg Arndt_, Jan 04 2011]
%F G.f.: x^2*(-1-2*x-5*x^4+3*x^5-2*x^6+x^7) / ( (x-1)^3*(x^2+1)^3 ). - _R. J. Mathar_, Jan 04 2011
%F a(1+4*n) = (A000217(n+1)-1)/2, a(2+4*n) = (A000217(n+2)-1)/2, a(3+4*n) = A000217(n+3)-1, a(4+4*n) = A000217(n+4)-1. - _Paul Curtz_, Dec 23 2010.
%F a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12). This is not the shortest recurrence. -_Paul Curtz_, Mar 27 2011
%F a(1+3*n) = numerator of 9*n*(n+1)/4 = 9*A064038(1+n). - _Paul Curtz_, Apr 06 2011
%F a(n) = (n-1)*(n+2)*(3-i^((n-2)*(n-1)))/8, where i=sqrt(-1). - _Bruno Berselli_, Apr 07 2011, corrected by _Vaclav Kotesovec_, Aug 09 2022
%F Sum_{n>=2} 1/a(n) = 13/9 + Pi/6. - _Amiram Eldar_, Aug 09 2022
%e 0, 1, 5/2, 9/2, 7, 10, 27/2, 35/2, 22, 27, ...
%t f[n_] := n/GCD[n, 4]; Array[f[#] f[# + 3] &, 58]
%t Rest[CoefficientList[Series[x^2*(-1 - 2*x - 5*x^4 + 3*x^5 - 2*x^6 + x^7)/((x - 1)^3*(x^2 + 1)^3), {x, 0, 50}], x]] (* _G. C. Greubel_, Sep 21 2018 *)
%o (PARI) s=vector(40,n,1/4*(n+2)*(n-1)) /* fractions */
%o vector(#s,n,numerator(s[n])) /* this sequence */ \\ _Joerg Arndt_, Jan 04 2011
%o (PARI) x='x+O('x^50); concat([0], Vec(x^2*(-1 - 2*x - 5*x^4 + 3*x^5 - 2*x^6 + x^7)/((x - 1)^3*(x^2 + 1)^3))) \\ _G. C. Greubel_, Sep 21 2018
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x^2*(-1 - 2*x - 5*x^4 + 3*x^5 - 2*x^6 + x^7)/((x - 1)^3*(x^2 + 1)^3))); // _G. C. Greubel_, Sep 21 2018
%Y Cf. A130658 (denominators), A033954 (quadrisection), A001107 (quadrisection), A181890 (quadrisection).
%Y Cf. A000217, A060819, A064038.
%K nonn,easy,frac
%O 1,3
%A _Eric W. Weisstein_, Apr 30 2009
%E Edited by _N. J. A. Sloane_, Dec 23 2010
|
