A242669 -id:A242669 - cp's OEIS frontend

A093353 a(n) = (n + (n mod 2))*(n + 1)/2.

0, 2, 3, 8, 10, 18, 21, 32, 36, 50, 55, 72, 78, 98, 105, 128, 136, 162, 171, 200, 210, 242, 253, 288, 300, 338, 351, 392, 406, 450, 465, 512, 528, 578, 595, 648, 666, 722, 741, 800, 820, 882, 903, 968, 990, 1058, 1081, 1152, 1176, 1250, 1275, 1352, 1378, 1458

Offset: 0

Reinhard Zumkeller, Apr 27 2004

Partial sums of A014682. - Paul Barry, Mar 31 2008
a(n) is the sum of all parts in the integer partitions of n+1 into two parts, see example. - Wesley Ivan Hurt, Jan 26 2013
Also the independence number of the n X n torus grid graph. - Eric W. Weisstein, Sep 06 2017
Also the number of circles we can draw on vertices of an (n+1)-sided regular polygon (using only a compass). - Matej Veselovac, Jan 21 2020

			a(1) = 2 since 2 = (1+1) and the sum of the first and second parts in the partition is 2; a(2) = 3 since 3 = (1+2) and the sum of these parts is 3; a(3) = 8 since 4 = (1+3) = (2+2) and the sum of all the parts is 8. - _Wesley Ivan Hurt_, Jan 26 2013

W. R. Hare, S. T. Hedetniemi, R. Laskar, and J. Pfaff, Complete coloring parameters of graphs, Proceedings of the sixteenth Southeastern international conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1985). Congr. Numer., Vol. 48 (1985), pp. 171-178. MR0830709 (87h:05088). See s_m on page 135. - N. J. A. Sloane, Apr 06 2012

G. C. Greubel, Table of n, a(n) for n = 0..5000
Math StackExchange, Number of circles on vertices of a regular polygon, 2020.
Eric Weisstein's World of Mathematics, Independence Number.
Eric Weisstein's World of Mathematics, Torus Grid Graph.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000217, A001105, A014105, A014682, A072691, A188859, A242669.

[(n+1)*(2*n+1-(-1)^n)/4: n in [0..60]]; // Vincenzo Librandi, Jan 23 2020

a:= n-> (n+1)*floor((n+1)/2); seq(a(n), n = 0..70);

(* Contributions from Harvey P. Dale, Nov 15 2013: Start *)
Table[(n+Mod[n,2])*(n+1)/2,{n,0,60}]
LinearRecurrence[{1,2,-2,-1,1},{0,2,3,8,10},60]
Join[{0},Module[{nn = 60, ab}, ab = Transpose[ Partition[ Accumulate[ Range[nn]], 2]]; Flatten[ Transpose[ {ab[[1]] + Range[nn/2], ab[[2]]}]]]]
(* End *)

a(n)=(n+1)\2*(n+1) \\ Charles R Greathouse IV, Jun 11 2015

[(n+1)*int((n+1)//2) for n in range(0,71)] # G. C. Greubel, Mar 14 2024

a(2*n) = a(2*n-1) + n = A014105(n).
a(2*n+1) = a(2*n) + 3*n + 2 = A001105(n+1).
G.f.: x*(2+x+x^2)/((1-x)^3*(1+x)^2).
a(n) = (n+1)*(2*n+1-(-1)^n)/4. - Paul Barry, Mar 31 2008
a(n) = (n+1)*floor((n+1)/2). - Wesley Ivan Hurt, Jan 26 2013
a(n) = Sum_{i=1..floor((n+1)/2)} i + Sum_{i=ceiling((n+1)/2)..n} i. - Wesley Ivan Hurt, Jun 08 2013
From Amiram Eldar, Mar 10 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/12 + 2*(1-log(2)) = A072691 + A188859.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - 2*(1-log(2)) = A072691 - A188859. (End)
E.g.f.: (x*(3 + x)*cosh(x) + (1 + x)^2*sinh(x))/2. - Stefano Spezia, Nov 13 2024

a(0)=0 prepended by Alois P. Heinz, Nov 13 2024

A010762 a(n) = floor(n/2) * floor(n/3).

Original entry on oeis.org

0, 0, 1, 2, 2, 6, 6, 8, 12, 15, 15, 24, 24, 28, 35, 40, 40, 54, 54, 60, 70, 77, 77, 96, 96, 104, 117, 126, 126, 150, 150, 160, 176, 187, 187, 216, 216, 228, 247, 260, 260, 294, 294, 308, 330, 345, 345, 384, 384, 400, 425, 442, 442, 486, 486, 504, 532, 551

Offset: 1

N. J. A. Sloane, Simon Plouffe

a(n) is also the number of 5 boxes polyomino (invert U patterns) packing into n X n square. The 6 boxes 2 X 3 (rectangular patterns) also gives the same sequence but difference in squares left. See illustration in links. - Kival Ngaokrajang, Nov 10 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Kival Ngaokrajang, Illustration of initial terms of invert u and 2 X 3 rectangular patterns.
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1,1,0,-1,-1,0,1).

Cf. A002264, A004526, A010761, A110533, A242669.

[Floor(n/2)*Floor(n/3) : n in [1..50]]; // Wesley Ivan Hurt, Jun 22 2014

[ seq(floor(n/2)*floor(n/3), n=1..64) ];

Table[Floor[n/2]*Floor[n/3], {n, 1, 70}] (* Clark Kimberling, May 18 2012 *)
CoefficientList[Series[- x^2 (x^7 + x^6 + x^5 + 2 x^4 + 3 x^3 + x^2 + 2 x+1)/((x - 1)^3 (x + 1)^2 (x^2 - x + 1) (x^2 + x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 15 2013 *)
LinearRecurrence[{0,1,1,0,-1,1,0,-1,-1,0,1},{0,0,1,2,2,6,6,8,12,15,15},60] (* Harvey P. Dale, Jan 09 2016 *)

a(n) = (n\2) * (n\3) \\ Charles R Greathouse IV, Oct 07 2015; corrected by Michel Marcus, Jun 01 2025

a(n) = A004526(n) * A002264(n). - Reinhard Zumkeller, Jul 25 2005
a(n) = a(n-2) + a(n-3) - a(n-5) + a(n-6) - a(n-8) - a(n-9) + a(n-11). - Clark Kimberling, May 18 2012
G.f.: -x^3*(x^7+x^6+x^5+2*x^4+3*x^3+x^2+2*x+1) / ((x-1)^3*(x+1)^2*(x^2-x+1)*(x^2+x+1)^2). - Colin Barker, Apr 05 2013
Sum_{n>=3} (-1)^(n+1)/a(n) = sqrt(3)*Pi/4 + 9*log(3)/4 - 2*log(2) - 3/2. - Amiram Eldar, Mar 30 2023
E.g.f.: (3*(x - 1)*x*cosh(x) - sqrt(3)*exp(-x/2)*(1 + exp(x) + 4*x)*sin(sqrt(3)*x/2)/2 + 3*cos(sqrt(3)*x/2)*sinh(x/2) + 3*(1 + x^2)*sinh(x))/18. - Stefano Spezia, Jun 01 2025

A233035 a(n) = n * floor(n/4).

Original entry on oeis.org

0, 0, 0, 4, 5, 6, 7, 16, 18, 20, 22, 36, 39, 42, 45, 64, 68, 72, 76, 100, 105, 110, 115, 144, 150, 156, 162, 196, 203, 210, 217, 256, 264, 272, 280, 324, 333, 342, 351, 400, 410, 420, 430, 484, 495, 506, 517, 576, 588, 600, 612, 676, 689, 702, 715, 784, 798, 812, 826, 900, 915, 930, 945, 1024, 1040

Offset: 1

Kival Ngaokrajang, Dec 03 2013

The maximum number of I patterns tetrominos that can be packed into an n X n array of squares with rotation is prohibited.
u(n) = n*(n mod 4), where u(n) is total number of squares left after packing I patterns into n X n squares.
a(n) = A132028(n) for 4 <= n <= 31.

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

Cf. A132028, A233036, A242669.

Table[n*Floor[n/4],{n,80}] (* or *) LinearRecurrence[{1,0,0,2,-2,0,0,-1,1},{0,0,0,4,5,6,7,16,18},80] (* Harvey P. Dale, Aug 22 2020 *)

a(n) = n * floor(n/4); \\ Joerg Arndt, Dec 08 2013

a(n) = (n^2 - n*(n mod 4))/4.

G.f.: (x^7 + x^6 + x^5 + x^4 + 4*x^3)/((1-x)*(1-x^4)^2). - Ralf Stephan, Dec 08 2013

A213033 n[n/2][n/3], where [] = floor.

Original entry on oeis.org

0, 0, 0, 3, 8, 10, 36, 42, 64, 108, 150, 165, 288, 312, 392, 525, 640, 680, 972, 1026, 1200, 1470, 1694, 1771, 2304, 2400, 2704, 3159, 3528, 3654, 4500, 4650, 5120, 5808, 6358, 6545, 7776, 7992, 8664, 9633, 10400, 10660, 12348, 12642, 13552

Offset: 0

Clark Kimberling, Jun 05 2012

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

Cf. A242669.

a[n_] := n*Floor[n/2]*Floor[n/3]
Table[a[n], {n, 0, 90}]    (* A213033 *)
LinearRecurrence[{0, 1, 1, 0, -1, 2, 0, -2, -2, 0, 2, -1, 0, 1, 1, 0, -1}, {0, 0, 0, 3, 8, 10, 36, 42, 64, 108, 150, 165, 288, 312, 392, 525, 640}, 60]

a(n) = a(n-2)+a(n-3)-a(n-5)+2*a(n-6)-2*a(n-8)+2*a(n-11)-a(n-12)+a(n-14)+a(n-15)-a(n-17).

G.f.: (3*x^3 + 8*x^4 + 7*x^5 + 25*x^6 + 24*x^7 + 21*x^8 + 32*x^9 + 38*x^10 + 15*x^11 + 22*x^12 + 13*x^13 + 5*x^14 + 2*x^15 + x^16)/(1 - x^2 - x^3 + x^5 - 2*x^6 + 2*x^8 + 2*x^9 - 2*x^11 + x^12 - x^14 - x^15 + x^17).

cp's OEIS Frontend

A093353 a(n) = (n + (n mod 2))*(n + 1)/2.

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A010762 a(n) = floor(n/2) * floor(n/3).

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A233035 a(n) = n * floor(n/4).

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A213033 n*[n/2]*[n/3], where [] = floor.

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A213033 n[n/2][n/3], where [] = floor.