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
Also the length of a longest induced (or snake) path in the (n+1) X (n+1) king graph. - _David Radcliffe_, Jun 24 2026
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Nikolai Beluhov, Snake paths in king and knight graphs, arXiv:2301.01152 [math.CO], 2023.
David Bevan, Cheyne Homberger, and Bridget Eileen Tenner, Prolific permutations and permuted packings: downsets containing many large patterns, arXiv preprint arXiv:1608.06931 [math.CO], 2016-2017.
Peter M. Chema, Illustration of initial terms, n > 1.
Alan Edelman and Michael 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.
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)
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..60); # _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 *)
(* Alternative: *)
Table[n + Floor[n^2/2], {n, 60}] (* _Eric W. Weisstein_, Mar 27 2018 *)
(* Alternative: *)
Table[((-1)^n + 2 n (n + 2) - 1)/4, {n, 60}] (* _Eric W. Weisstein_, Mar 27 2018 *)
(* Alternative: *)
LinearRecurrence[{2, 0, -2, 1}, {1, 4, 7, 12}, 60] (* _Eric W. Weisstein_, Mar 27 2018 *)
(* Alternative: *)
CoefficientList[Series[(-1 - 2 x + x^2)/((-1 + x)^3 (1 + x)), {x, 0, 60}], 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
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