A084850 -id:A084850 - cp's OEIS frontend

A014848 a(n) = n^2 - floor( n/2 ).

0, 1, 3, 8, 14, 23, 33, 46, 60, 77, 95, 116, 138, 163, 189, 218, 248, 281, 315, 352, 390, 431, 473, 518, 564, 613, 663, 716, 770, 827, 885, 946, 1008, 1073, 1139, 1208, 1278, 1351, 1425, 1502, 1580, 1661, 1743, 1828, 1914, 2003, 2093, 2186, 2280, 2377, 2475

Offset: 0

N. J. A. Sloane

Quasipolynomial of order 2. - Charles R Greathouse IV, Jan 19 2012

The binomial transform is 0, 1, 5, 20,... which is A084850 with offset 1. - R. J. Mathar, Nov 26 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A033951, A033991, A042963 (first differences), A084850.

[n^2-Floor(n/2) : n in [0..50]]; // Wesley Ivan Hurt, Nov 14 2014

A014848:=n->n^2 - floor(n/2); seq(A014848(n), n=0..50); # Wesley Ivan Hurt, Oct 11 2013

Table[n^2-Floor[n/2],{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Apr 12 2011 *)
LinearRecurrence[{2,0,-2,1},{0,1,3,8},60] (* Harvey P. Dale, Jun 13 2022 *)

PARI
```
a(n)=n^2-n\2
```

def A014848(n): return n**2-(n>>1) # Chai Wah Wu, Jan 18 2023

[n^2 - (n//2) for n in range(71)] # G. C. Greubel, Mar 14 2024

a(2*n) = A033991(n).
a(2*n+1) = A033951(n).
G.f.: x*(1+x+2*x^2)/((1-x)^2*(1-x^2)).
a(n) = (2*n*(2*n-1) + 1 - (-1)^n)/4. - Bruno Berselli, Feb 17 2011
a(n) = round(n/(exp(1/n) - 1)), n > 0. - Richard R. Forberg, Nov 14 2014
E.g.f.: (1/4)*((1 + 2*x + 4*x^2)*exp(x) - exp(-x)). - G. C. Greubel, Mar 14 2024

A134083 A007318 * A134082.

Original entry on oeis.org

1, 3, 1, 5, 6, 1, 7, 15, 9, 1, 9, 28, 30, 12, 1, 11, 45, 70, 50, 15, 1, 13, 66, 135, 140, 75, 18, 1, 15, 91, 231, 315, 245, 105, 21, 1, 17, 120, 364, 616, 630, 392, 140, 24, 1, 19, 153, 540, 1092, 1386, 1134, 588, 180, 27, 1

Offset: 0

Gary W. Adamson, Oct 07 2007

Row sums = A001787: (1, 4, 12, 32, 80, 192, ...).

A134083 * [1,2,3,...] = A084850: (1, 5, 20, 68, 208, 592, ...).

			First few rows of the triangle:
   1;
   3,  1;
   5,  6,   1;
   7, 15,   9,   1;
   9, 28,  30,  12,   1;
  11, 45,  70,  50,  15,   1;
  13, 66, 135, 140,  75,  18,  1;
  15, 91, 231, 315, 245, 105, 21, 1;
  ...

Cf. A134082, A001787, A084850.

Binomial transform of A134082
From formalism in A132382, e.g.f. = I_o[2*(u*x)^(1/2)] exp(x)(1+2x) where I_o is the zeroth modified Bessel function of the first kind, i.e., I_o[2*(u*x)^(1/2)] = Sum_{j>=0} u^j/j! * x^j/j!. - Tom Copeland, Dec 07 2007
Row polynomial e.g.f.: exp(x*y) * exp(x) * (1+2x). - Tom Copeland, Dec 03 2013

cp's OEIS Frontend

A014848 a(n) = n^2 - floor( n/2 ).

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