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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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:

r=0, s=0: A000027;

r=0, s=1: this entry;

r=0, s=2: A135301;

r=0, s=3: A135332;

r=0, s=4: A140142;

r=0, s=5: A140143;

r=1, s=0: A140144;

r=1, s=1: A000217;

r=1, s=2: A140113;

r=1, s=3: A140145;

r=1, s=4: A140146;

r=1, s=5: A140147;

r=2, s=0: A140148;

r=2, s=1: A136047;

r=2, s=2: A000330;

r=2, s=3: A140149;

r=2, s=4: A140150;

r=2, s=5: A140151;

r=3, s=0: A140152;

r=3, s=1: A140153;

r=3, s=2: A140154;

r=3, s=3: A000537;

r=3, s=4: A140155;

r=3, s=5: A140156;

r=4, s=0: A140157;

r=4, s=1: A140158;

r=4, s=2: A140159;

r=4, s=3: A140160;

r=4, s=4: A000538;

r=4, s=5: A140161;

r=5, s=0: A140162;

r=5, s=1: A140163;

r=5, s=2: A135095;

r=5, s=3: A135099;

r=5, s=4: A135214;

r=5, s=5: 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

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

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

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

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

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

Sequence in context: A177986 A186775 A285440 * A058074 A319875 A123722

Adjacent sequences: A135273 A135274 A135275 * A135277 A135278 A135279

KEYWORD

nonn,easy

AUTHOR

Artur Jasinski, May 12 2008, corrected May 17 2008

EXTENSIONS

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

STATUS

approved

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Last modified December 9 13:46 EST 2022. Contains 358700 sequences. (Running on oeis4.)