A081490 - OEIS

%I #24 Oct 18 2022 15:01:48

%S 1,2,4,9,19,36,62,99,149,214,296,397,519,664,834,1031,1257,1514,1804,

%T 2129,2491,2892,3334,3819,4349,4926,5552,6229,6959,7744,8586,9487,

%U 10449,11474,12564,13721,14947,16244,17614,19059,20581,22182,23864,25629

%N Leading term of n-th row of A081491.

%C First differences are given by A002522 = n^2 + 1. Second differences are odd numbers given by A005408.

%C a(1)=1, a(2)=2, (a(n+1) -a(n)) - (a(n) -a(n-1)) = 2*(n-1)-1. - _Ben Paul Thurston_, Aug 22 2009

%H G. C. Greubel, <a href="/A081490/b081490.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(1) = 1, a(n) = A081489(n-1) + 1.

%F From _R. J. Mathar_, Feb 06 2010: (Start)

%F G..f: x*(1-2*x+2*x^2+x^3)/(x-1)^4.

%F a(n) = n*(2*n^2 -9*n +19)/6 -1. (End)

%F a(n) = (n-2)^2 + a(n-1)+1, n>1. - _Gary Detlefs_, Jun 29 2010

%F a(1)=1, a(2)=2, a(3)=4, a(4)=9, a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - _Harvey P. Dale_, Apr 30 2011

%p with (combinat):a:=n->sum(fibonacci(3,i), i=0..n):seq(a(n)+1, n=-1..42); # _Zerinvary Lajos_, Apr 25 2008

%t Rest[CoefficientList[Series[x (1-2x+2x^2+x^3)/(x-1)^4,{x,0,50}],x]] (* or *) LinearRecurrence[{4,-6,4,-1}, {1,2,4,9}, 50] (* _Harvey P. Dale_, Apr 30 2011 *)

%o (PARI) vector(50, n, (2*n^3-9*n^2+19*n-6)/6) \\ _G. C. Greubel_, Aug 13 2019

%o (Magma) [(2*n^3-9*n^2+19*n-6)/6: n in [1..50]]; // _G. C. Greubel_, Aug 13 2019

%o (Sage) [(2*n^3-9*n^2+19*n-6)/6 for n in (1..50)] # _G. C. Greubel_, Aug 13 2019

%o (GAP) List([1..50], n-> (2*n^3-9*n^2+19*n-6)/6); # _G. C. Greubel_, Aug 13 2019

%Y Cf. A002522, A005408, A081489, A081491, A081492.

%K nonn,easy

%O 1,2

%A _Amarnath Murthy_, Mar 25 2003

%E More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 29 2003