A135276 a(0)=0, a(1)=1; for n>1, a(n) = a(n-1) + n^0 if n odd, a(n) = a(n-1) + n^1 if n is even.

0, 1, 3, 4, 8, 9, 15, 16, 24, 25, 35, 36, 48, 49, 63, 64, 80, 81, 99, 100, 120, 121, 143, 144, 168, 169, 195, 196, 224, 225, 255, 256, 288, 289, 323, 324, 360, 361, 399, 400, 440, 441, 483, 484, 528, 529, 575, 576, 624, 625, 675, 676, 728, 729, 783, 784, 840, 841, 899, 900, 960, 961

Offset: 0

Author

Artur Jasinski, May 12 2008, corrected May 17 2008

Keywords

• nonn
• easy

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

r=0, A000027, this seq, A135301, A135332, A140142, A140143;

r=1, A140144, A000217, A140113, A140145, A140146, A140147;

r=2, A140148, A136047, A000330, A140149, A140150, A140151;

r=3, A140152, A140153, A140154, A000537, A140155, A140156;

r=4, A140157, A140158, A140159, A140160, A000538, A140161;

r=5, A140162, A140163, A135095, A135099, A135214, A000539.

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).

Crossrefs

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A004526, A136047, A140113.

Cf. A070909. - Gary W. Adamson, May 16 2010

Programs

Magma

[(2*n^2+6*n+1+(2*n-1)*(-1)^n)/8 : n in [0..100]]; // Wesley Ivan Hurt, Mar 22 2016

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 *)

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

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

a(n) = A004526((n+1)^2) - A004526(n+1)^2. - Bruno Berselli, Oct 21 2016

Sum_{n>=1} 1/a(n) = 3/4 + Pi^2/6. - Amiram Eldar, Sep 08 2022

Extensions

Offset corrected by R. J. Mathar, Feb 22 2009

Edited by Michel Marcus, Apr 07 2023