cp's OEIS Frontend

A084263 a(n) = (-1)^n/2+(n^2+n+1)/2.

%I A084263 #31 Jun 10 2023 13:34:53
%S A084263 1,1,4,6,11,15,22,28,37,45,56,66,79,91,106,120,137,153,172,190,211,
%T A084263 231,254,276,301,325,352,378,407,435,466,496,529,561,596,630,667,703,
%U A084263 742,780,821,861,904,946,991,1035,1082,1128,1177,1225,1276,1326,1379,1431
%N A084263 a(n) = (-1)^n/2+(n^2+n+1)/2.
%C A084263 Old name was "Modified triangular numbers".
%C A084263 Starting with offset 1 = row sums of an infinite lower triangular matrix with alternate columns of (1, 3, 5, 7, ...) and (1, 0, 0, 0, ...) (see example). - _Gary W. Adamson_, May 14 2010
%H A084263 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A084263 E.g.f.: cosh(x)+exp(x)*(x+x^2/2).
%F A084263 a(n) = Sum_{k=0..n} k+(-1)^k.
%F A084263 a(n) = A000217(n)+A059841(n). Partial sums are A084570. Binomial transform is A084264.
%F A084263 G.f.: (1-x+2*x^2)/((1-x)^3*(1+x)). - _R. J. Mathar_, Apr 02 2008
%F A084263 a(0) = 1, a(n) = n^2 - a(n-1) + 1 for n >= 1. - _Richard R. Forberg_, Jun 05 2013
%F A084263 a(n) = 1 + floor(n/2) + floor(n^2/2). - _Wesley Ivan Hurt_, Jun 15 2013
%F A084263 a(n) + a(n+1) = A002522(n+1). - _R. J. Mathar_, May 21 2018
%F A084263 a(n) = 2*a(n-1)-2*a(n-3)+a(n-4). - _Wesley Ivan Hurt_, Dec 23 2021
%e A084263 From _Gary W. Adamson_, May 14 2010: (Start)
%e A084263 First few rows of the triangle with row sums = A084263 =
%e A084263 1;
%e A084263 3, 1;
%e A084263 5, 0, 1;
%e A084263 7, 0, 3, 1;
%e A084263 9, 0, 5, 0, 1;
%e A084263 11, 0, 7, 0, 3, 1;
%e A084263 ...
%e A084263 Example: a(4) = 11 = (7 + 0 + 3 + 1). (End)
%t A084263 Table[(-1)^n/2 + (n^2 + n + 1)/2, {n, 0, 100}] (* _Wesley Ivan Hurt_, Dec 23 2021 *)
%t A084263 LinearRecurrence[{2,0,-2,1},{1,1,4,6},60] (* _Harvey P. Dale_, Jun 10 2023 *)
%o A084263 (PARI) a(n)=n*(n+1)/2+(1-n)%2 \\ _Charles R Greathouse IV_, Jun 04 2013
%o A084263 (Magma) [(-1)^n/2+(n^2+n+1)/2 : n in [0..80]]; // _Wesley Ivan Hurt_, Dec 23 2021
%Y A084263 Partial sums of A004442.
%Y A084263 Cf. A000217, A002522, A059841, A084264, A084570.
%K A084263 easy,nonn
%O A084263 0,3
%A A084263 _Paul Barry_, May 31 2003
%E A084263 Name changed by _Wesley Ivan Hurt_, Dec 23 2021