A074148 a(n) = n + floor(n^2/2).

1, 4, 7, 12, 17, 24, 31, 40, 49, 60, 71, 84, 97, 112, 127, 144, 161, 180, 199, 220, 241, 264, 287, 312, 337, 364, 391, 420, 449, 480, 511, 544, 577, 612, 647, 684, 721, 760, 799, 840, 881, 924, 967, 1012, 1057, 1104, 1151, 1200, 1249, 1300, 1351, 1404, 1457

Offset

1,2

Comments

Last term in each group in A074147.

Index of the last occurrence of n in A100795.

Equals row sums of an infinite lower triangular matrix with alternate columns of (1, 3, 5, 7, ...) and (1, 1, 1, ...). - Gary W. Adamson, May 16 2010

a(n) = A214075(n+2,2). - Reinhard Zumkeller, Jul 03 2012

The heart pattern appears in (n+1) X (n+1) coins. Abnormal orientation heart is A065423. Normal heart is A093005 (A074148 - A065423). Void is A007590. See illustration in links. - Kival Ngaokrajang, Sep 11 2013

a(n+1) is the smallest size of an n-prolific permutation; a permutation of s letters is n-prolific if each (s - n)-subset of the letters in its one-line notation forms a unique pattern. - David Bevan, Nov 30 2016

For n > 2, a(n-1) is the smallest size of a nontrivial permuted packing of diamond tiles with diagonal length n; a permuted packing is a translational packing for which the set of translations is the plot of a permutation. - David Bevan, Nov 30 2016

Also the length of a longest path in the (n+1) X (n+1) bishop and black bishop graphs. - Eric W. Weisstein, Mar 27 2018

Row sums of A143182 triangle - Nikita Sadkov, Oct 10 2018

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

D. Bevan, C. Homberger, and B. E. Tenner, Prolific permutations and permuted packings: downsets containing many large patterns, arXiv preprint arXiv:1608.06931 [math.CO], 2016.

Peter M. Chema, Illustration of initial terms, n > 1.

A. Edelman and M. La Croix, The Singular Values of the GUE (Less is More), arXiv preprint arXiv:1410.7065 [math.PR], 2014-2015. See Section 7.

Kival Ngaokrajang, Illustration of initial terms.

Eric Weisstein's World of Mathematics, Bishop Graph.

Eric Weisstein's World of Mathematics, Black Bishop Graph.

Eric Weisstein's World of Mathematics, Longest Path Problem.

Wikipedia, Cartan decomposition.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Formula

a(n) = (2*n^2 + 4*n + (-1)^n - 1)/4. - Vladeta Jovovic, Apr 06 2003

a(n) = A109225(n,2) for n > 1. - Reinhard Zumkeller, Jun 23 2005

a(n) = +2*a(n-1) - 2*a(n-3) + 1*a(n-4). - Joerg Arndt, Apr 02 2011

a(n) = a(n-2) + 2*n, a(0) = 0, a(1) = 1. - Paul Barry, Jul 17 2004

From R. J. Mathar, Aug 30 2008: (Start)

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

a(n) + a(n+1) = A028387(n).

a(n+1) - a(n) = A109613(n+1). (End)

a(n) = floor(n^4/(2n^2 + 1)) with offset 2..a(2) = 1. - Gary Detlefs, Feb 11 2010

a(n) = n + floor(n^2/2). - Wesley Ivan Hurt, Jun 14 2013

From Franck Maminirina Ramaharo, Jan 04 2019: (Start)

a(n) = n*(n + 1)/2 + floor(n/2) = A000217(n) + A004526(n).

E.g.f.: (exp(-x) - (1 - 6*x - 2*x^2)*exp(x))/4. (End)

Sum_{n>=1} 1/a(n) = 1 - cot(Pi/sqrt(2))*Pi/(2*sqrt(2)). - Amiram Eldar, Sep 16 2022

Example

Equals row sums of the generating triangle:
   1;
   3,  1;
   5,  1,  1;
   7,  1,  3,  1;
   9,  1,  5,  1,  1;
  11,  1,  7,  1,  3,  1;
  13,  1,  9,  1,  5,  1,  1;
  15,  1, 11,  1,  7,  1,  3,  1;
  ...
Example: a(5) = 17 = (9 + 1 + 5 + 1 + 1). - Gary W. Adamson, May 16 2010
The smallest 1-prolific permutations are 3142 and its symmetries; a(2) = 4. The smallest 2-prolific permutations are 3614725 and its symmetries; a(3) = 7. - David Bevan, Nov 30 2016

Maple

seq(floor(n^4/(2*n^2+1)), n=2..25); # Gary Detlefs, Feb 11 2010

Mathematica

f[x_, y_] := Floor[Abs[y/x - x/y]]; Table[Floor[f[1, n^2 + 2 n + 1]/2], {n, 60}] (* Robert G. Wilson v, Aug 11 2010 *)
Table[n + Floor[n^2/2], {n, 20}] (* Eric W. Weisstein, Mar 27 2018 *)
Table[((-1)^n + 2 n (n + 2) - 1)/4, {n, 10}] (* Eric W. Weisstein, Mar 27 2018 *)
LinearRecurrence[{2, 0, -2, 1}, {1, 4, 7, 12}, 20] (* Eric W. Weisstein, Mar 27 2018 *)
CoefficientList[Series[(-1 - 2 x + x^2)/((-1 + x)^3 (1 + x)), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 27 2018 *)

Prog

(Magma) [(2*n^2+4*n+(-1)^n-1)/4: n in [1..60]]; // Vincenzo Librandi, Jun 16 2011
(PARI) a(n)=(2*n^2+4*n-1)\/4 \\ Charles R Greathouse IV, Apr 17 2012
(GAP) List([1..60], n->n+Int(n^2/2)); # Muniru A Asiru, Jan 04 2019
(Python)
def A074148(n): return n + n**2//2 # Chai Wah Wu, Jun 07 2022

Crossrefs

Cf. A074147, A074149, A100795, A061925.

a(n) = A000982(n+1) - 1.

Antidiagonal sums of A237447 & A237448.

Sequence in context: A310791 A246399 A276222 * A336660 A310792 A178907

Adjacent sequences: A074145 A074146 A074147 * A074149 A074150 A074151

Keyword

nonn,easy

Author

Amarnath Murthy, Aug 28 2002

Extensions

More terms from Vladeta Jovovic, Apr 06 2003

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 31 2007

Further edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar

Description simplified by Eric W. Weisstein, Mar 27 2018

Status

approved