|
%I #53 Feb 13 2024 02:19:10
%S 0,5,7,22,26,51,57,92,100,145,155,210,222,287,301,376,392,477,495,590,
%T 610,715,737,852,876,1001,1027,1162,1190,1335,1365,1520,1552,1717,
%U 1751,1926,1962,2147,2185,2380,2420,2625,2667,2882,2926,3151,3197,3432,3480
%N Numbers of the form m*(6*m-1) and m*(6*m+1), where m is an integer.
%C PartitionQ[ p ] is odd and contains an extra even partition; series term z^p in Product_{n>=1}(1-z^n) has coefficient (+1). - _Wouter Meeussen_
%C Numbers k such that the number of partitions of k into distinct parts with an even number of parts exceed by 1 the number of partitions of k into distinct parts with an odd number of parts. [See, e.g., the Freitag-Busam reference given under A036499, p. 410. - _Wolfdieter Lang_, Jan 18 2016]
%C In formal power series, A010815 = Product_{k>0}(1-x^k), ranks of coefficients 1 (A001318 = ranks of nonzero (1 or -1) in A010815 = ranks of odds terms in A000009).
%H Vincenzo Librandi, <a href="/A036498/b036498.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F a(n) = n(n+1)/6 for n=0 or 5 (mod 6).
%F a(n) = 1/8*(-1+(-1)^n+2*n)*(-3+(-1)^n+6*n) (see MATHEMATICA code).
%F G.f.: x^2*(5+2*x+5*x^2)/((1+x)^2*(1-x)^3). - _Colin Barker_, Apr 02 2012
%F a(1)=0, a(2)=5, a(3)=7, a(4)=22, a(5)=26, a(n)=a(n-1)+2*a(n-2)- 2*a(n-3)- a(n-4)+a(n-5). - _Harvey P. Dale_, Aug 13 2012
%F Bisections: a(2*k+1) = A001318(4*k) = k*(1+6*k) = A049453(k), k >= 0; a(2*k) = A001318(4*k-1) = k*(-1+6*k) = A049452(k), k >= 1. - _Wolfdieter Lang_, Jan 18 2016
%F From _Amiram Eldar_, Feb 13 2024: (Start)
%F Sum_{n>=2} 1/a(n) = 6 - sqrt(3)*Pi.
%F Sum_{n>=2} (-1)^n/a(n) = 4*log(2) + 3*log(3) - 6. (End)
%p p1 := n->n*(6*n-1): p2 := n->n*(6*n+1): S:={}: for n from 0 to 100 do S := S union {p1(n), p2(n)} od: S
%t Table[ 1/8*(-1 + (-1)^k + 2*k)*(-3 + (-1)^k + 6*k), {k, 64} ]
%t CoefficientList[Series[x*(5+2*x+5*x^2)/((1+x)^2*(1-x)^3),{x,0,50}],x] (* _Vincenzo Librandi_, Apr 24 2012 *)
%t Rest[Flatten[{#(6#-1),#(6#+1)}&/@Range[0,30]]] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{0,5,7,22,26},60] (* _Harvey P. Dale_, Aug 13 2012 *)
%o (PARI) \ps 5000; for(n=1,5000,if(polcoeff(eta(x),n,x)==1,print1(n,",")))
%o (PARI) concat(0, Vec(x^2*(5+2*x+5*x^2)/((1+x)^2*(1-x)^3) + O(x^100))) \\ _Altug Alkan_, Jan 19 2016
%o (Magma) [1/8*(-1+(-1)^n+2*n)*(-3+(-1)^n+6*n): n in [1..50]]; // _Vincenzo Librandi_, Apr 24 2012
%o (Magma) /* By definition: */ A036498:=func<n | n*(6*n+1)>; [0] cat [A036498(n*m): m in [-1,1], n in [1..25]]; // _Bruno Berselli_, Nov 13 2012
%Y Union of A049452 and A049453.
%Y Cf. A000009, A001318, A010815, A036499.
%K nonn,easy
%O 1,2
%A _Wouter Meeussen_
%E Better description from Claude Lenormand (claude.lenormand(AT)free.fr), Feb 12 2001
%E Additional comments and more terms from _James A. Sellers_, Feb 14 2001
|
