OFFSET
0,3
COMMENTS
Index to family of sequences of the form a(n) = a(n-1) + n^r if n odd, a(n) = a(n-1)+ n^s if n is even, for n > 1 and a(1)=1:
s=0, s=1, s=2, s=3, s=4, s=5
Equals triangle A070909 * [1,2,3,...]. - Gary W. Adamson, May 16 2010
Right edge of the triangle in A199332: a(n) = A199332(n,n), for n > 0. - Reinhard Zumkeller, Nov 23 2011
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Girtrude Hamm, Classification of lattice triangles by their two smallest widths, arXiv:2304.03007 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
FORMULA
a(n) = (n/2 + 1)^2 - 1 if n is even, ((n+1)/2)^2 if n is odd. - M. F. Hasler, May 17 2008
From R. J. Mathar, Feb 22 2009: (Start)
G.f.: x*(1+2*x-x^2)/((1+x)^2*(1-x)^3).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). (End)
a(n) = (2*n^2 + 6*n + 1 + (2*n-1)*(-1)^n)/8. - Luce ETIENNE, Jul 08 2014
a(n) = (floor(n/2)+1)^2 + (n mod 2) - 1. - Wesley Ivan Hurt, Mar 22 2016
Sum_{n>=1} 1/a(n) = 3/4 + Pi^2/6. - Amiram Eldar, Sep 08 2022
MAPLE
A135276:=n->( 2*n^2 + 6*n + 1 + (2*n-1)*(-1)^n )/8: seq(A135276(n), n=0..100); # Wesley Ivan Hurt, Mar 22 2016
MATHEMATICA
a = {}; r = 0; s = 1; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 0, 100}]; a (* Artur Jasinski *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 3, 4, 8}, 50] (* G. C. Greubel, Oct 08 2016 *)
PROG
(PARI) A135276(n)=if(n%2, ((n+1)/2)^2, (n/2+1)^2-1) \\ M. F. Hasler, May 17 2008
(PARI) my(x='x+O('x^200)); concat(0, Vec(x*(1+2*x-x^2)/((1+x)^2*(1-x)^3))) \\ Altug Alkan, Mar 23 2016
(Magma) [(2*n^2+6*n+1+(2*n-1)*(-1)^n)/8 : n in [0..100]]; // Wesley Ivan Hurt, Mar 22 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, May 12 2008, corrected May 17 2008
EXTENSIONS
Offset corrected by R. J. Mathar, Feb 22 2009
Edited by Michel Marcus, Apr 07 2023
STATUS
approved