A184005 a(n) = n - 1 + ceiling(3*n^2/4); complement of A184004.

1, 4, 9, 15, 23, 32, 43, 55, 69, 84, 101, 119, 139, 160, 183, 207, 233, 260, 289, 319, 351, 384, 419, 455, 493, 532, 573, 615, 659, 704, 751, 799, 849, 900, 953, 1007, 1063, 1120, 1179, 1239, 1301, 1364, 1429, 1495, 1563, 1632, 1703, 1775, 1849, 1924, 2001, 2079, 2159, 2240, 2323, 2407, 2493, 2580, 2669, 2759

Offset: 1

Author

Clark Kimberling, Jan 08 2011

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A184004.

Programs

Magma

[(6*n^2 + 8*n - (-1)^n - 7)/8: n in [1..80]]; // Vincenzo Librandi, Feb 09 2011

Mathematica

a=4/3; b=0;
Table[n+Floor[(a*n+b)^(1/2)],{n,80}]
Table[n-1+Ceiling[(n*n-b)/a],{n,60}]
Table[n - 1 + Ceiling[3 n^2/4], {n, 60}] (* or *) CoefficientList[ Series[x (1 + 2 x + x^2 - x^3)/((1 + x) (1 - x)^3), {x, 0, 60}], x] (* or *) Table[Round[(6 n^2 + 8 n - 7)/8], {n, 60}] (* Michael De Vlieger, Mar 23 2016 *)
LinearRecurrence[{2,0,-2,1},{1,4,9,15},60] (* Harvey P. Dale, Sep 16 2016 *)

PARI

my(x='x+O('x^200)); Vec(x*(1+2*x+x^2-x^3)/((1+x)*(1-x)^3)) \\ Altug Alkan, Mar 23 2016

Python

def A184005(n): return n+((m:=3*n**2)>>2)-(not m&3) # Chai Wah Wu, Oct 01 2024

Formula

a(n) = 2*a(n-1) - 2*a(n-3) + 1*a(n-4).

From Bruno Berselli, Jan 25 2011: (Start)

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

a(n) = (6*n^2 + 8*n - (-1)^n - 7)/8. (End)

a(n) = round((6*n^2 + 8*n - 7)/8). - Bruno Berselli, Jan 25 2011

From Paul Curtz, Feb 09 2011: (Start)

a(n) - a(n-1) = A007494(n).

a(n) - a(n-2) = 3*n - 1 = A016789(n-1).

a(n) - a(n-4) = 6*n - 8 = A016957(n-2).

a(n) - a(n-8) = 12*n - 40 = A017617(n-4).

a(n) - a(n-16) = 24*n - 176 = 8*A016789(n-8).

a(n) - a(n-32) = 48*n - 736 = 16*A016789(n-16). (End)

a(n) = n^2 - floor((n-2)^2/4). - Bruno Berselli, Jan 17 2017

a(n) = A002061(n+2) - A002620(n+4). - Anton Zakharov, May 17 2017

E.g.f.: (1/8)*(8 + (6*x^2 + 14*x -7)*exp(x) - exp(-x)). - G. C. Greubel, Jul 22 2017