A093005 -id:A093005 - cp's OEIS frontend

A093006 Referring to the triangle in A093005, sequence contains the least term with maximal number of divisors.

1, 2, 6, 8, 12, 18, 24, 36, 40, 48, 60, 72, 84, 96, 120, 126, 144, 168, 180, 210, 216, 240, 264, 288, 312, 336, 360, 396, 420, 450, 480, 504, 540, 576, 600, 660, 672, 720, 756, 792, 840, 900, 924, 960, 1008, 1080, 1092, 1140, 1200, 1260, 1320, 1344, 1380, 1440

Offset: 1

Amarnath Murthy, Mar 29 2004

nonn

			a(4) = 8 has 4 divisors, though 10 also has 4 divisors.

More terms from Joshua Zucker, May 08 2006

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

Original entry on oeis.org

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

Amarnath Murthy, Aug 28 2002

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

			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

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).

Cf. A074147, A074149, A100795, A061925.
a(n) = A000982(n+1) - 1.
Antidiagonal sums of A237447 & A237448.

List([1..60],n->n+Int(n^2/2)); # Muniru A Asiru, Jan 04 2019

[(2*n^2+4*n+(-1)^n-1)/4: n in [1..60]]; // Vincenzo Librandi, Jun 16 2011

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

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 *)

a(n)=(2*n^2+4*n-1)\/4 \\ Charles R Greathouse IV, Apr 17 2012

def A074148(n): return n + n**2//2 # Chai Wah Wu, Jun 07 2022

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

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

A006578 Triangular numbers plus quarter squares: n*(n+1)/2 + floor(n^2/4) (i.e., A000217(n) + A002620(n)).

Original entry on oeis.org

0, 1, 4, 8, 14, 21, 30, 40, 52, 65, 80, 96, 114, 133, 154, 176, 200, 225, 252, 280, 310, 341, 374, 408, 444, 481, 520, 560, 602, 645, 690, 736, 784, 833, 884, 936, 990, 1045, 1102, 1160, 1220, 1281, 1344, 1408, 1474, 1541, 1610, 1680, 1752, 1825, 1900, 1976, 2054

Offset: 0

N. J. A. Sloane

Equals (1, 2, 3, 4, ...) convolved with (1, 2, 1, 2, ...). a(4) = 14 = (1, 2, 3, 4) dot (2, 1, 2, 1) = (2 + 2 + 6 + 4). - Gary W. Adamson, May 01 2009
We observe that is the transform of A032766 by the following transform T: T(u_0,u_1,u_2,u_3,...) = (u_0, u_0+u_1, u_0+u_1+u_2, u_0+u_1+u_2+u_3+u_4,...). In other words, v_p = Sum_{k=0..p} u_k and the g.f. phi_v of is given by phi_v = phi_u/(1-z). - Richard Choulet, Jan 28 2010
Equals row sums of a triangle with (1, 4, 7, 10, ...) in every column, shifted down twice for columns > 1. - Gary W. Adamson, Mar 03 2010
Number of pairs (x,y) with x in {0,...,n}, y odd in {0,...,2n}, and x < y. - Clark Kimberling, Jul 02 2012
Also A049451 and positives A000567 interleaved. - Omar E. Pol, Aug 03 2012
Similar to A001082. Members of this family are A093005, A210977, this sequence, A210978, A181995, A210981, A210982. - Omar E. Pol, Aug 09 2012

			G.f. = x + 4*x^2 + 8*x^3 + 14*x^4 + 21*x^5 + 30*x^6 + 40*x^7 + 52*x^8 + 65*x^9 + ...

Marc LeBrun, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Bruno Berselli, Hexagonal spiral containing the initial terms.
Marc Le Brun, Email to N. J. A. Sloane, Jul 1991
Emanuele Munarini, Topological indices for the antiregular graphs, Le Mathematiche (2021) Vol. 76, No. 1, see p. 283.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A001859, A077043, A002620, A002378.
Row sums of A104567.
Cf. A000034, A032766, A002717, A070893. - Richard Choulet, Jan 28 2010
Cf. A051125.

[(6*n^2+4*n-1+(-1)^n)/8: n in [0..50] ]; // Vincenzo Librandi, Aug 20 2011

with (combinat): seq(count(Partition((3*n+1)), size=3), n=0..52); # Zerinvary Lajos, Mar 28 2008
# 2nd program
A006578 := proc(n)
    (6*n^2 + 4*n - 1 + (-1)^n)/8 ;
end proc: # R. J. Mathar, Apr 28 2017

Accumulate[LinearRecurrence[{1,1,-1}, {0,1,3}, 100]] (* Harvey P. Dale, Sep 29 2013 *)
a[ n_] := Quotient[n + 1, 2] (Quotient[n, 2] 3 + 1); (* Michael Somos, Jun 09 2014 *)
a[ n_] := Quotient[3 (n + 1)^2 + 1, 4] - (n + 1); (* Michael Somos, Jun 10 2015 *)
LinearRecurrence[{2, 0, -2, 1},{0, 1, 4, 8},53] (* Ray Chandler, Aug 03 2015 *)

{a(n) = (3*(n+1)^2 + 1)\4 - n - 1}; /* Michael Somos, Mar 10 2006 */

Expansion of x*(1+2*x) / ((1-x)^2*(1-x^2)). - Simon Plouffe in his 1992 dissertation
a(n) + A002620(n) = A002378(n) = n*(n+1).
Partial sums of A032766. - Paul Barry, May 30 2003
a(n) = a(n-1) + a(n-2) - a(n-3) + 3 = A002620(n) + A004526(n) = A001859(n) - A004526(n+1). - Henry Bottomley, Mar 08 2000
a(n) = (6*n^2 + 4*n - 1 + (-1)^n)/8. - Paul Barry, May 30 2003
a(n) = A001859(-1-n) for all n in Z. - Michael Somos, May 10 2006
a(n) = (A002378(n)/2 + A035608(n))/2. - Reinhard Zumkeller, Feb 07 2010
a(n) = (3*n^2 + 2*n - (n mod 2))/4. - Ctibor O. Zizka, Mar 11 2012
a(n) = Sum_{i=1..n} floor(3*i/2) = Sum_{i=0..n} (i + floor(i/2)). - Enrique Pérez Herrero, Apr 21 2012
a(n) = 3*n*(n+1)/2 - A001859(n). - Clark Kimberling, Jul 02 2012
a(n) = Sum_{i=1..n} (n - i + 1) * 2^( (i+1) mod 2 ). - Wesley Ivan Hurt, Mar 30 2014
a(n) = A002717(n) - A002717(n-1). - Michael Somos, Jun 09 2014
a(n) = Sum_{k=1..n} floor((n+k+1)/2). - Wesley Ivan Hurt, Mar 31 2017
a(n) = A002620(n+1)+2*A002620(n). - R. J. Mathar, Apr 28 2017
Sum_{n>=1} 1/a(n) = 3 - Pi/(4*sqrt(3)) - 3*log(3)/4. - Amiram Eldar, May 28 2022
E.g.f.: (x*(5 + 3*x)*cosh(x) - (1 - 5*x - 3*x^2)*sinh(x))/4. - Stefano Spezia, Aug 22 2023

Offset and description changed by N. J. A. Sloane, Nov 30 2006

A094727 Triangle read by rows: T(n,k) = n + k, 0 <= k < n, n >= 1.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 4, 5, 6, 7, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 11, 7, 8, 9, 10, 11, 12, 13, 8, 9, 10, 11, 12, 13, 14, 15, 9, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23

Offset: 1

Reinhard Zumkeller, May 24 2004

All numbers m occur ceiling(m/2) times, see A004526.

The LCM of the n-th row is A076100. - Michel Marcus, Mar 18 2018

			Triangle begins:
  1;
  2,  3;
  3,  4,  5;
  4,  5,  6,  7;
  5,  6,  7,  8,  9;
  6,  7,  8,  9, 10, 11;
  7,  8,  9, 10, 11, 12, 13;
  8,  9, 10, 11, 12, 13, 14, 15;
  9, 10, 11, 12, 13, 14, 15, 16, 17;
  ... - _Philippe Deléham_, Mar 30 2013

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
Bruno Berselli, Illustration of the initial terms.
László Németh, On the Binomial Interpolated Triangles, Journal of Integer Sequences, Vol. 20 (2017), Article 17.7.8.
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A002024, A002260, A003056, A004526, A004736, A005408, A005843.
Cf. A016777, A016813, A016861, A037213, A076100, A078112, A093005.
Cf. A094728, A214604, A214661, A319556.
Cf. A128076 (rows reversed).

a094727 n k = n + k
a094727_row n = a094727_tabl !! (n-1)
a094727_tabl = iterate (\row@(h:_) -> (h + 1) : map (+ 2) row) [1]
-- Reinhard Zumkeller, Jul 22 2012

z:=12; &cat[ [m+n-1: m in [1..n] ]: n in [1..z] ];

Table[n + Range[0, n-1], {n, 12}]//Flatten (* Michael De Vlieger, Dec 16 2016 *)

from math import isqrt
def A094727(n): return ((a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(3-a)>>1)+n-1 # Chai Wah Wu, Jun 19 2025

flatten([[n+k for k in range(n)] for n in range(1,16)]) # G. C. Greubel, Mar 10 2024

T(n+1, k) = T(n, k) + 1 = T(n, k+1); T(n+1, k+1) = T(n, k) + 2.
T(n, n - A005843(k)) = A005843(n-k) for 0 <= k <= n/2.
T(n, n - A005408(k)) = A005408(n-k) for 0 <= k < n/2.
T(A005408(n), n) = A016777(n), n >= 0.
Sum_{k=1..n} T(n, k) = A000326(n) (row sums).
T(n, k) = A002024(n,k) + A002260(n,k) - 1. - Reinhard Zumkeller, Apr 27 2006
As a sequence rather than as a table: If m = floor((sqrt(8n-7)+1)/2), a(n) = n - m*(m-3)/2 - 1. - Carl R. White, Jul 30 2009
T(n, k) = n+k-1, n >= k >= 1. - Vincenzo Librandi, Nov 23 2009 [corrected by Klaus Brockhaus, Nov 23 2009]
T(n,k) = A037213((A214604(n,k) + A214661(n,k)) / 2). - Reinhard Zumkeller, Jul 25 2012
From Boris Putievskiy, Jan 16 2013: (Start)
a(n) = A002260(n) + A003056(n).
a(n) = i+t, where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). (End)
From G. C. Greubel, Mar 10 2024: (Start)
T(3*n-3, n) = A016813(n-1).
T(4*n-4, n) = A016861(n-1).
Sum_{k=0..n-1} (-1)^k*T(n, k) = A319556(n).
Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A093005(n).
Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = A078112(n-1).
Sum_{j=1..n} (Sum_{k=0..n-1} T(j, k)) = A002411(n) (sum of n rows). (End)

A123684 Alternate A016777(n) with A000027(n).

Original entry on oeis.org

1, 1, 4, 2, 7, 3, 10, 4, 13, 5, 16, 6, 19, 7, 22, 8, 25, 9, 28, 10, 31, 11, 34, 12, 37, 13, 40, 14, 43, 15, 46, 16, 49, 17, 52, 18, 55, 19, 58, 20, 61, 21, 64, 22, 67, 23, 70, 24, 73, 25, 76, 26, 79, 27, 82, 28, 85, 29, 88, 30, 91, 31, 94, 32, 97, 33, 100, 34, 103, 35, 106, 36

Offset: 1

Alford Arnold, Oct 11 2006

a(n) is a diagonal of Table A123685.
The arithmetic average of the first n terms gives the positive integers repeated (A008619). - Philippe Deléham, Nov 20 2013
Images under the modified '3x-1' map: a(n) = n/2 if n is even, (3n-1)/2 if n is odd. (In this sequence, the numbers at even indices n are n/2 [A000027], and the numbers at odd indices n are 3((n-1)/2) + 1 [A016777] = (3n-1)/2.) The latter correspondence interestingly mirrors an insight in David Bařina's 2020 paper (see below), namely that 3(n+1)/2 - 1 = (3n+1)/2. - Kevin Ge, Oct 30 2024

			The natural numbers begin 1, 2, 3, ... (A000027), the sequence 3*n + 1 begins 1, 4, 7, 10, ... (A016777), therefore A123684 begins 1, 1, 4, 2, 7, 3, 10, ...
1/1 = 1, (1+1)/2 = 1, (1+1+4)/3 = 2, (1+1+4+2)/4 = 2, ... - _Philippe Deléham_, Nov 20 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).
David Bařina, Convergence verification of the Collatz problem

Cf. A000027, A008619, A016777, A016825, A031878, A052928, A064455.
Cf. A071045, A092530, A093005, A105638, A123685, A137501, A225126.
Cf. A014682 (modified Collatz map)

import Data.List (transpose)
a123684 n = a123684_list !! (n-1)
a123684_list = concat $ transpose [a016777_list, a000027_list]
-- Reinhard Zumkeller, Apr 29 2013

&cat[ [ 3*n-2, n ]: n in [1..36] ]; // Klaus Brockhaus, May 12 2007

/* From the fourteenth formula: */ [&+[1+k*(-1)^k: k in [0..n]]: n in [0..80]]; // Bruno Berselli, Jul 16 2013

A123684:=n->n-1/4-(1/2*n-1/4)*(-1)^n: seq(A123684(n), n=1..70); # Wesley Ivan Hurt, Jul 26 2014

CoefficientList[Series[(1 +x +2*x^2)/((1-x)^2*(1+x)^2), {x,0,70}], x] (* Wesley Ivan Hurt, Jul 26 2014 *)
LinearRecurrence[{0,2,0,-1},{1,1,4,2},80] (* Harvey P. Dale, Apr 14 2025 *)

print(vector(72, n, if(n%2==0, n/2, (3*n-1)/2))) \\ Klaus Brockhaus, May 12 2007

print(vector(72, n, n-1/4-(1/2*n-1/4)*(-1)^n)); \\ Klaus Brockhaus, May 12 2007

[(n + (2*n-1)*(n%2))//2 for n in range(1,71)] # G. C. Greubel, Mar 15 2024

From Klaus Brockhaus, May 12 2007: (Start)
G.f.: x*(1+x+2*x^2)/((1-x)^2*(1+x)^2).
a(n) = (1/4)*(4*n - 1 - (2*n - 1)*(-1)^n).
a(2n-1) = A016777(n-1) = 3(n-1) + 1.
a(2n) = A000027(n) = n.
a(n) = A071045(n-1) + 1.
a(n) = A093005(n) - A093005(n-1) for n > 1.
a(n) = A105638(n+2) - A105638(n+1) for n > 1.
a(n) = A092530(n) - A092530(n-1) - 1.
a(n) = A031878(n+1) - A031878(n) - 1. (End)
a(2*n+1) + a(2*n+2) = A016825(n). - Paul Curtz, Mar 09 2011
a(n)= 2*a(n-2) - a(n-4). - Paul Curtz, Mar 09 2011
From Jaroslav Krizek, Mar 22 2011 (Start):
a(n) = n + a(n-1) for odd n; a(n) = n - A064455(n-1) for even n.
a(n) = A064455(n) - A137501(n).
Abs(a(n) - A064455(n)) = A052928(n). (End)
a(n) = A225126(n) for n > 1. - Reinhard Zumkeller, Apr 29 2013
a(n) = Sum_{k=1..n} (1 + (k-1)*(-1)^(k-1)). - Bruno Berselli, Jul 16 2013
a(n) = n + floor(n/2) for odd n; a(n) = n/2 for even n. - Reinhard Muehlfeld, Jul 25 2014

More terms from Klaus Brockhaus, May 12 2007

A227906 Coins left after packing heart patterns (fixed orientation) into n X n coins.

Original entry on oeis.org

2, 4, 4, 9, 6, 13, 8, 17, 10, 21, 12, 25, 14, 29, 16, 33, 18, 37, 20, 41, 22, 45, 24, 49, 26, 53, 28, 57, 30, 61, 32, 65, 34, 69, 36, 73, 38, 77, 40, 81, 42, 85, 44, 89, 46, 93, 48, 97, 50, 101, 52, 105, 54, 109, 56, 113, 58, 117, 60, 121

Offset: 2

Kival Ngaokrajang, Oct 19 2013

On the Japanese TV show "Tsuki no Koibito", a girl told her boyfriend that she saw a heart in 4 coins. Actually there are a total of 6 distinct patterns appearing in 2 X 2 coins in which each pattern consists of a part of the perimeter of each coin and forms a continuous area.

a(n) is the number of coins left after packing fixed orientation heart patterns (type 4c2s1: 4-curve cover 2 coins and symmetry) into n X n coins. The total number of hearts is A093005 and the number of voids left is A093353. See illustration in links.

Kival Ngaokrajang, Illustration of initial terms (U)
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008795, A230370 (3-curves), A074148, A229093, A229154 (4-curves), A001399, A230267, A230276 (5-curves), A229593, A228949, A229598 (6-curves).

With[{nn=60},Join[{2,4},Riffle[Range[4,nn,2],Range[9,2nn+1,4]]]] (* Harvey P. Dale, Feb 11 2015 *)

Vec(-x^2*(x^5-x^3-4*x-2)/((x-1)^2*(x+1)^2) + O(x^100)) \\ Colin Barker, Oct 30 2013

From Colin Barker, Oct 30 2013: (Start)
a(n) = (-1 + (-1)^n - (-3 + (-1)^n)*n)/2 for n>3.
a(n) = n for n>3 and even.
a(n) = 2*n-1 for n > 3 and odd.
a(n) = 2*a(n-2) - a(n-4) for n>7.
G.f.: -x^2*(x^5-x^3-4*x-2) / ((x-1)^2*(x+1)^2).(End)

A183207 Termwise products of the natural numbers and odd integers repeated.

Original entry on oeis.org

1, 2, 9, 12, 25, 30, 49, 56, 81, 90, 121, 132, 169, 182, 225, 240, 289, 306, 361, 380, 441, 462, 529, 552, 625, 650, 729, 756, 841, 870, 961, 992, 1089, 1122, 1225, 1260, 1369, 1406, 1521, 1560, 1681, 1722, 1849, 1892, 2025

Offset: 1

Gary W. Adamson, Feb 11 2011

There is a chessboard of n^2 squares. A pawn is standing on the lower left corner of the chessboard O (0,0) and its primary goal is to reach the upper right corner of the chessboard N (n,n). The only moves allowed are diagonal shortcuts through squares. Once a square is crossed it is destroyed so that it is impossible to cross again. The secondary goal of the pawn on its way to N is to destroy as many squares as possible. a(n) is the maximum possible number of destroyed squares, provided the pawn has reached its primary goal. - Ivan N. Ianakiev, Feb 23 2014

The sequence gives the number of curved edges created from circle intersections when a circle of radius r is drawn around each of n equally spaced points on the circumference of a circle of radius r. The number of regions in these constructions is A093005(n) and the number of vertices is A370980(n). See the attached images. - Scott R. Shannon, Jul 07 2024

			a(4) = 4*3 = 12.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
Scott R. Shannon, Image for n = 3. In this and other images the center of each circle of shown as a white dot.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 8.

Cf. A093005, A370980.

I:=[1,2,9,12,25]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Feb 23 2014

a[n_] := n (2 Floor[(n - 1)/2] + 1); Array[a, 45] (* Robert G. Wilson v, Feb 11 2011 *)
CoefficientList[Series[(-1 - x - 5 x^2 - x^3)/((1 + x)^2 (x - 1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Feb 23 2014 *)

PARI
```
a(n) = n^2-n*(1+(-1)^n)/2;
```

Termwise products of (1, 2, 3, 4, 5, 6, 7, 8...) and (1, 1, 3, 3, 5, 5, 7, 7,...).
From R. J. Mathar, Feb 12 2011: (Start)
G.f.: x*( -1-x-5*x^2-x^3 ) / ( (1+x)^2*(x-1)^3 ).
a(n) = n^2-n*(1+(-1)^n)/2. (End)
Sum_{n>=1} 1/a(n) = Pi^2/8 + log(2). - Amiram Eldar, Mar 15 2024
a(n) = A093005(n) + A370980(n) - 1, by Euler's formula. - Scott R. Shannon, Jul 07 2024

A210977 A005475 and positive terms of A000566 interleaved.

Original entry on oeis.org

0, 1, 3, 7, 11, 18, 24, 34, 42, 55, 65, 81, 93, 112, 126, 148, 164, 189, 207, 235, 255, 286, 308, 342, 366, 403, 429, 469, 497, 540, 570, 616, 648, 697, 731, 783, 819, 874, 912, 970, 1010, 1071, 1113, 1177, 1221, 1288, 1334, 1404, 1452, 1525, 1575, 1651, 1703, 1782, 1836, 1918, 1974, 2059

Offset: 0

Omar E. Pol, Aug 03 2012

Vertex number of a square spiral similar to A085787.

Partial sums of the sequence formed by A005843 and A016777 interleaved.

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

Members of this family are A093005, this sequence, A006578, A210978, A181995, A210981, A210982.

Cf. A000566, A005843, A016777, A085787,

G.f.: -x*(1+2*x+2*x^2) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Aug 07 2012

a(n) = (10*n^2+6*n-1-(2*n-1)*(-1)^n)/16. - Luce ETIENNE, Oct 04 2014

A210982 Zero together with A126264 and positive terms of A051624 interleaved.

Original entry on oeis.org

0, 1, 8, 12, 26, 33, 54, 64, 92, 105, 140, 156, 198, 217, 266, 288, 344, 369, 432, 460, 530, 561, 638, 672, 756, 793, 884, 924, 1022, 1065, 1170, 1216, 1328, 1377, 1496, 1548, 1674, 1729, 1862, 1920, 2060, 2121, 2268, 2332, 2486, 2553, 2714, 2784, 2952, 3025, 3200, 3276, 3458, 3537, 3726

Offset: 0

Omar E. Pol, Aug 03 2012

Vertex number of a square spiral similar to A195162.

This is the case k=5 of the formula b(n,k) = ( 2*(k+5)*n^2+2*(k+3)*n-(k+1)+(2*(k-1)*n+k+1)*(-1)^n )/16. Sequences of the same family: A093025 (k=-1, with an initial 0), A210977 (k=0), A006578 (k=1), A210978 (k=2), A181995 (k=3, with one 0 only), A210981 (k=4). - Luce ETIENNE, Oct 30 2014

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

Members of this family are A093005, A210977, A006578, A210978, A181995, A210981, this sequence.

Cf. A001622, A051624, A126264, A195162.

[(10*n^2+8*n-3+(4*n+3)*(-1)^n )/8: n in [0..60]]; // Vincenzo Librandi, Oct 31 2014

Table[(10*n^2 + 8*n - 3 + (4*n + 3)*(-1)^n)/8, {n, 0, 50}] (* G. C. Greubel, Aug 23 2017 *)

my(x='x+O('x^50)); Vec(x*(1+7*x+2*x^2)/((1+x)^2*(1-x)^3)) \\ G. C. Greubel, Aug 23 2017

G.f.: x*(1+7*x+2*x^2) / ( (1+x)^2*(1-x)^3 ). - R. J. Mathar, Aug 07 2012
a(n) = (10*n^2 +8*n -3 +(4*n+3)*(-1)^n)/8. - Luce ETIENNE, Oct 14 2014
E.g.f.: (1/8)*((10*x^3 + 18*x -3)*exp(x) - (4*x - 3)*exp(-x)). - G. C. Greubel, Aug 23 2017
Sum_{n>=1} 1/a(n) = 5/9 + (sqrt(1-2/sqrt(5))/6 + sqrt(1+2/sqrt(5))/8)*Pi + 7*log(phi)*sqrt(5)/24 - 5*log(5)/48, where phi is the golden ratio (A001622). - Amiram Eldar, Aug 21 2022

A278688 Triangle read by rows T(n, k) = number of non-equivalent ways to place k non-attacking ferses on an n X n board.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 6, 7, 6, 2, 1, 1, 3, 17, 45, 92, 99, 76, 27, 7, 1, 6, 43, 225, 832, 2102, 3773, 4860, 4643, 3356, 1868, 795, 248, 56, 8, 1, 1, 6, 84, 709, 4500, 19987, 66201, 164423, 314224, 465230, 540247, 492206, 352300, 195717, 83247, 26083, 5754, 780, 55

Offset: 1

Heinrich Ludwig, Nov 27 2016

The triangle T(n, k) is irregularly shaped: 0 <= k <= A093005(n), which means that A093005(n) is the maximal number of non-attacking ferses that can be placed on an n X n board. First row corresponds to n = 1. First column corresponds to k = 0.
Two placements that differ by rotation or reflection are counted only once.
A fers is a fairy chess piece attacking one step ne-nw-sw-se.

			Triangle begins:
1, 1;
1, 1,  1;
1, 3,  6,  7,  6,  2,  1;
1, 3, 17, 45, 92, 99, 76, 27, 7;

Heinrich Ludwig, Table of n, a(n) for n = 1..130
Wikipedia, Fairy chess piece

Cf. A008805, A232567, A278682, A278683, A278684, A278685, A278686, (columns 2 through 8 of this sequence, respectively), A278687, A093005 (row length - 1).

cp's OEIS Frontend

A093006 Referring to the triangle in A093005, sequence contains the least term with maximal number of divisors.

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

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A006578 Triangular numbers plus quarter squares: n*(n+1)/2 + floor(n^2/4) (i.e., A000217(n) + A002620(n)).

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A094727 Triangle read by rows: T(n,k) = n + k, 0 <= k < n, n >= 1.

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A123684 Alternate A016777(n) with A000027(n).

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A227906 Coins left after packing heart patterns (fixed orientation) into n X n coins.

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