cp's OEIS Frontend

A219190 Numbers of the form k*(5*k+1), where k = 0,-1,1,-2,2,-3,3,...

%I A219190 #65 Mar 16 2022 03:20:03
%S A219190 0,4,6,18,22,42,48,76,84,120,130,174,186,238,252,312,328,396,414,490,
%T A219190 510,594,616,708,732,832,858,966,994,1110,1140,1264,1296,1428,1462,
%U A219190 1602,1638,1786,1824,1980,2020,2184,2226,2398,2442,2622,2668,2856,2904,3100
%N A219190 Numbers of the form k*(5*k+1), where k = 0,-1,1,-2,2,-3,3,...
%C A219190 Equivalently, numbers m such that 20*m+1 is a square.
%C A219190 Also, integer values of h*(h+1)/5.
%C A219190 More generally, for the numbers of the form n*(k*n+1) with n in A001057, we have:
%C A219190 . generating function (offset 1): x^2*(k-1+2*x+(k-1)*x^2)/((1+x)^2*(1-x)^3);
%C A219190 . n-th term: b(n) = (2*k*n*(n-1)+(k-2)*(-1)^n*(2*n-1)+k-2)/8;
%C A219190 . first differences: (n-1)*((-1)^n*(k-2)+k)/2;
%C A219190 . b(2n+1)-b(2n) = 2*n (independent from k);
%C A219190 . (4*k)*b(n)+1 = (2*k*n+(k-2)*(-1)^n-k)^2/4.
%H A219190 Bruno Berselli, <a href="/A219190/b219190.txt">Table of n, a(n) for n = 1..1000</a>
%H A219190 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A219190 G.f.: 2*x^2*(2 + x + 2*x^2)/((1 + x)^2*(1 - x)^3).
%F A219190 a(n) = a(-n+1) = (10*n*(n-1) + 3*(-1)^n*(2*n - 1) + 3)/8.
%F A219190 a(n) = 2*A057569(n) = A008851(n+1)*A047208(n)/5.
%F A219190 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - _Harvey P. Dale_, Jan 21 2015
%F A219190 Sum_{n>=2} 1/a(n) = 5 - sqrt(1+2/sqrt(5))*Pi. - _Amiram Eldar_, Mar 15 2022
%F A219190 a(n) = A132356(n-1)/2, n >= 1. - _Bernard Schott_, Mar 15 2022
%t A219190 Rest[Flatten[{# (5 # - 1), # (5 # + 1)} & /@ Range[0, 25]]]
%t A219190 CoefficientList[Series[2 x (2 + x + 2 x^2) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%t A219190 LinearRecurrence[{1,2,-2,-1,1},{0,4,6,18,22},50] (* _Harvey P. Dale_, Jan 21 2015 *)
%o A219190 (Magma) k:=5; f:=func<n | n*(k*n+1)>; [0] cat [f(n*m): m in [-1,1], n in [1..25]];
%o A219190 (Magma) I:=[0,4,6,18,22]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // _Vincenzo Librandi_, Aug 18 2013
%Y A219190 Subsequence of A011858.
%Y A219190 Cf. A090771: square roots of 20*a(n)+1 (see the first comment).
%Y A219190 Cf. numbers of the form n*(k*n+1) with n in A001057: k=0, A001057; k=1, A110660; k=2, A000217; k=3, A152749; k=4, A074378; k=5, this sequence; k=6, A036498; k=7, A219191; k=8, A154260.
%Y A219190 Cf. similar sequences listed in A219257.
%K A219190 nonn,easy
%O A219190 1,2
%A A219190 _Bruno Berselli_, Nov 14 2012