A000982 -id:A000982 - cp's OEIS frontend

A001971 Nearest integer to n^2/8.

0, 0, 1, 1, 2, 3, 5, 6, 8, 10, 13, 15, 18, 21, 25, 28, 32, 36, 41, 45, 50, 55, 61, 66, 72, 78, 85, 91, 98, 105, 113, 120, 128, 136, 145, 153, 162, 171, 181, 190, 200, 210, 221, 231, 242, 253, 265, 276, 288, 300, 313, 325, 338, 351, 365, 378, 392, 406, 421, 435, 450

Offset: 0

N. J. A. Sloane

Restricted partitions.
a(0) = a(1) = 0; a(n) are the partitions of floor((3*n+3)/2) with 3 distinct numbers of the set {1, ..., n}; partitions of floor((3*n+3)/2)-C and ceiling((3*n+3)/2)+C have equal numbers. - Paul Weisenhorn, Jun 05 2009, corrected by M. F. Hasler, Jun 16 2022
Odd-indexed terms are the triangular numbers, even-indexed terms are the midpoint (rounded up where necessary) of the surrounding odd-indexed terms. - Carl R. White, Aug 12 2010
a(n+2) is the number of points one can surround with n stones in Go (including the points under the stones). - Thomas Dybdahl Ahle, May 11 2014
Corollary of above: a(n) is the number of points one can surround with n+2 stones in Go (excluding the points under the stones). - Juhani Heino, Aug 29 2015
From Washington Bomfim, Jan 13 2021: (Start)
For n >= 4, a(n) = A026810(n+2) - A026810(n-4).
Let \n,m\ be the number of partitions of n into m non-distinct parts.
For n >= 1, \n,4\ = round((n-2)^2/8).
For n >= 6, \n,4\ = A026810(n) - A026810(n-6).
(End)

A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
M. Jeger, Einfuehrung in die Kombinatorik, Klett, 1975, Bd.2, pages 110 ff. [Paul Weisenhorn, Jun 05 2009]
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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
G. Almkvist, Invariants, mostly old ones, Pacific J. Math. 86 (1980), no. 1, 1-13. MR0586866 (81j:14029)
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281. [Annotated scanned copy]
Shalosh B. Ekhad and Doron Zeilberger, In How many ways can I carry a total of n coins in my two pockets, and have the same amount in both pockets?, arXiv:1901.08172 [math.CO], 2019.
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.
D. Vainsencher and A. M. Bruckstein, On isoperimetrically optimal polyforms, Theoretical Computer Science 406.1-2, 2008, pp. 146-159.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

The 4th diagonal of A061857?
Kind of an inverse of A261491 (regarding Go).
Cf. A026810 (partitions with greatest part 4), A001400 (partitions in at most 4 parts), A000217 (a(2n+1): triangular numbers n(n+1)/2), A000982 (a(2n): round(n^2/2)).

a001971 = floor . (+ 0.5) . (/ 8) . fromIntegral . (^ 2)
-- Reinhard Zumkeller, May 08 2012

[Round(n^2/8): n in [0..60]]; // Vincenzo Librandi, Jun 23 2011

A001971:=-(1-z+z**2)/((z+1)*(z**2+1)*(z-1)**3); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation [Note that this "generating function" is Sum_{n >= 0} a(n+2)*z^n, not a(n)*z^n. - M. F. Hasler, Jun 16 2022]

LinearRecurrence[{2,-1,0,1,-2,1},{0,0,1,1,2,3},70] (* Harvey P. Dale, Jan 30 2014 *)

PARI
```
{a(n) = round(n^2 / 8)};
```

apply( {A001971(n)=n^2\/8}, [0..99]) \\ M. F. Hasler, Jun 16 2022

The listed terms through a(20)=50 satisfy a(n+2) = a(n-2) + n. - John W. Layman, Dec 16 1999
G.f.: x^2 * (1 - x + x^2) / (1 - 2*x + x^2 - x^4 + 2*x^5 - x^6) = x^2 * (1 - x^6) / ((1 - x) * (1 - x^2) * (1 - x^3) * (1 - x^4)). - Michael Somos, Feb 07 2004
a(n) = floor((n^2+4)/8). - Paul Weisenhorn, Jun 05 2009
a(2*n+1) = A000217(n), a(2*n) = floor((A000217(n-1)+A000217(n)+1)/2). - Carl R. White, Aug 12 2010
From Michael Somos, Aug 29 2015: (Start)
Euler transform of length 6 sequence [ 1, 1, 1, 1, 0, -1].
a(n) = a(-n) for all n in Z. (End)
a(2n) = A000982(n). - M. F. Hasler, Jun 16 2022
Sum_{n>=2} 1/a(n) = 2 + Pi^2/12 + tanh(Pi/2)*Pi/2. - Amiram Eldar, Jul 02 2023

Edited Feb 08 2004

A047838 a(n) = floor(n^2/2) - 1.

Original entry on oeis.org

1, 3, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337, 363, 391, 419, 449, 479, 511, 543, 577, 611, 647, 683, 721, 759, 799, 839, 881, 923, 967, 1011, 1057, 1103, 1151, 1199, 1249, 1299, 1351, 1403

Offset: 2

Michael Somos, May 07 1999

Define the organization number of a permutation pi_1, pi_2, ..., pi_n to be the following. Start at 1, count the steps to reach 2, then the steps to reach 3, etc. Add them up. Then the maximal value of the organization number of any permutation of [1..n] for n = 0, 1, 2, 3, ... is given by 0, 1, 3, 7, 11, 17, 23, ... (this sequence). This was established by Graham Cormode (graham(AT)research.att.com), Aug 17 2006, see link below, answering a question raised by Tom Young (mcgreg265(AT)msn.com) and Barry Cipra, Aug 15 2006
From Dmitry Kamenetsky, Nov 29 2006: (Start)
This is the length of the longest non-self-intersecting spiral drawn on an n X n grid. E.g., for n=5 the spiral has length 17:
  1 0 1 1 1
  1 0 1 0 1
  1 0 1 0 1
  1 0 0 0 1
  1 1 1 1 1 (End)
It appears that a(n+1) is the maximum number of consecutive integers (beginning with 1) that can be placed, one after another, on an n-peg Towers of Hanoi, such that the sum of any two consecutive integers on any peg is a square. See the problem: http://online-judge.uva.es/p/v102/10276.html. - Ashutosh Mehra, Dec 06 2008
a(n) = number of (w,x,y) with all terms in {0,...,n} and w = |x+y-w|. - Clark Kimberling, Jun 11 2012
The same sequence also represents the solution to the "pigeons problem": maximal value of the sum of the lengths of n-1 line segments (connected at their end-points) required to pass through n trail dots, with unit distance between adjacent points, visiting all of them without overlaping two or more segments. In this case, a(0)=0, a(1)=1, a(2)=3, and so on. - Marco Ripà, Jan 28 2014
Also the longest path length in the n X n white bishop graph. - Eric W. Weisstein, Mar 27 2018
a(n) is the number of right triangles with sides n*(h-floor(h)), floor(h) and h, where h is the hypotenuse. - Andrzej Kukla, Apr 14 2021

			x^2 + 3*x^3 + 7*x^4 + 11*x^5 + 17*x^6 + 23*x^7 + 31*x^8 + 39*x^9 + 49*x^10 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000
Laurent Bulteau, Samuele Giraudo and Stéphane Vialette, Disorders and permutations , 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Article No. 18; pp. 18:1-18:14.
Graham Cormode, Notes on the organization number of a permutation.
Eric Weisstein's World of Mathematics, Longest Path Problem.
Eric Weisstein's World of Mathematics, White Bishop Graph.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Complement of A047839. First difference is A052928.
Cf. A000217, A007590, A080827, A134151, A135151, A135152, A188653.
Partial sums: A213759(n-1) for n > 1. - Guenther Schrack, May 12 2018

[Floor(n^2/2)-1 : n in [2..100]]; // Wesley Ivan Hurt, Aug 06 2015

seq(floor((n^2+4*n+2)/2), n=0..20) # Gary Detlefs, Feb 10 2010

Table[Floor[n^2/2] - 1, {n, 2, 60}] (* Robert G. Wilson v, Aug 31 2006 *)
LinearRecurrence[{2, 0, -2, 1}, {1, 3, 7, 11}, 60] (* Harvey P. Dale, Jan 16 2015 *)
Floor[Range[2, 20]^2/2] - 1 (* Eric W. Weisstein, Mar 27 2018 *)
Table[((-1)^n + 2 n^2 - 5)/4, {n, 2, 20}] (* Eric W. Weisstein, Mar 27 2018 *)
CoefficientList[Series[(-1 - x - x^2 + x^3)/((-1 + x)^3 (1 + x)), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 27 2018 *)

PARI
```
a(n) = n^2\2 - 1
```

a(2)=1; for n > 2, a(n) = a(n-1) + n - 1 + (n-1 mod 2). - Benoit Cloitre, Jan 12 2003
a(n) = T(n-1) + floor(n/2) - 1 = T(n) - floor((n+3)/2), where T(n) is the n-th triangular number (A000217). - Robert G. Wilson v, Aug 31 2006
Equals (n-1)-th row sums of triangles A134151 and A135152. Also, = binomial transform of [1, 2, 2, -2, 4, -8, 16, -32, ...]. - Gary W. Adamson, Nov 21 2007
G.f.: x^2*(1+x+x^2-x^3)/((1-x)^3*(1+x)). - R. J. Mathar, Sep 09 2008
a(n) = floor((n^2 + 4*n + 2)/2). - Gary Detlefs, Feb 10 2010
a(n) = abs(A188653(n)). - Reinhard Zumkeller, Apr 13 2011
a(n) = (2*n^2 + (-1)^n - 5)/4. - Bruno Berselli, Sep 14 2011
a(n) = a(-n) = A007590(n) - 1.
a(n) = A080827(n) - 2. - Kevin Ryde, Aug 24 2013
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n > 4. - Wesley Ivan Hurt, Aug 06 2015
a(n) = A000217(n-1) + A004526(n-2), for n > 1. - J. Stauduhar, Oct 20 2017
From Guenther Schrack, May 12 2018: (Start)
Set a(0) = a(1) = -1, a(n) = a(n-2) + 2*n - 2 for n > 1.
a(n) = A000982(n-1) + n - 2 for n > 1.
a(n) = 2*A033683(n) - 3 for n > 1.
a(n) = A061925(n-1) + n - 3 for n > 1.
a(n) = A074148(n) - n - 1 for n > 1.
a(n) = A105343(n-1) + n - 4 for n > 1.
a(n) = A116940(n-1) - n for n > 1.
a(n) = A179207(n) - n + 1 for n > 1.
a(n) = A183575(n-2) + 1 for n > 2.
a(n) = A265284(n-1) - 2*n + 1 for n > 1.
a(n) = 2*A290743(n) - 5 for n > 1. (End)
E.g.f.: 1 + x + ((x^2 + x - 2)*cosh(x) + (x^2 + x - 3)*sinh(x))/2. - Stefano Spezia, May 06 2021
Sum_{n>=2} 1/a(n) = 3/2 + tan(sqrt(3)*Pi/2)*Pi/(2*sqrt(3)) - cot(Pi/sqrt(2))*Pi/(2*sqrt(2)). - Amiram Eldar, Sep 15 2022

Edited by Charles R Greathouse IV, Apr 23 2010

A002731 Numbers k such that (k^2 + 1)/2 is prime.

Original entry on oeis.org

3, 5, 9, 11, 15, 19, 25, 29, 35, 39, 45, 49, 51, 59, 61, 65, 69, 71, 79, 85, 95, 101, 121, 131, 139, 141, 145, 159, 165, 169, 171, 175, 181, 195, 199, 201, 205, 209, 219, 221, 231, 245, 261, 271, 275, 279, 289, 299, 309, 315, 321, 325, 329, 335, 345, 349, 371, 375, 379, 391, 399, 405

Offset: 1

N. J. A. Sloane

From Wolfdieter Lang, Feb 24 2012: (Start)
a(n) = sqrt(8*A129307(n)+1) = sqrt(2*A027862(n)-1), n >= 1.
a(n) is the nontrivial solution of the congruence a(n)^2 == 1 (Modd A027862(n)). The trivial one is +1. For Modd n see a comment on A203571. E.g., a(3)^2 = 81 == 1 (Modd 41), see a comment on A027862.
(End)

L. Euler, De numeris primis valde magnis (E283), reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 3, p. 24.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
L. Euler, De numeris primis valde magnis (E283), The Euler Archive.

Cf. A027861. A027862 gives primes, A091277 gives prime indices.

Cf. A000982, A010051, A116945, A129307, A188546, A188547, A187431.

a002731 n = a002731_list !! (n-1)
a002731_list = filter ((== 1) . a010051 . a000982) [1, 3 ..]
-- Reinhard Zumkeller, Jul 13 2014

[n: n in [3..410] | IsPrime((n^2+1) div 2) ]; // Vincenzo Librandi, Sep 25 2012

Select[Range[400], PrimeQ[(#^2 + 1)/2] &] (* Alonso del Arte, Feb 24 2012 *)

forstep(n=1,10^3,2, if(isprime((n^2+1)/2),print1(n,", ")));
/* Joerg Arndt, Sep 02 2012 */

a(n) = 2*A027861(n) + 1.

A061925 a(n) = ceiling(n^2/2) + 1.

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 19, 26, 33, 42, 51, 62, 73, 86, 99, 114, 129, 146, 163, 182, 201, 222, 243, 266, 289, 314, 339, 366, 393, 422, 451, 482, 513, 546, 579, 614, 649, 686, 723, 762, 801, 842, 883, 926, 969, 1014, 1059, 1106, 1153, 1202, 1251, 1302, 1353, 1406

Offset: 0

Henry Bottomley, May 17 2001

a(n+1) gives index of the first occurrence of n in A100795. - Amarnath Murthy, Dec 05 2004
First term in each group in A074148. - Amarnath Murthy, Aug 28 2002
From Christian Barrientos, Jan 01 2021: (Start)
For n >= 3, a(n) is the number of square polyominoes with at least 2n - 2 cells whose bounding box has size 2 X n.
For n = 3, there are 6 square polyominoes with a bounding box of size 2 X 3:
      _       _       _       _       _      
   |||_|   |||_|   |||_|   |||_|   |||_|   |||_
   |||_|   |||     || ||       ||     ||       |||
(End)
Except for a(2), a(n) agrees with the lower matching number of the (n+1) X (n+1) bishop graph up to at least n = 13. - Eric W. Weisstein, Dec 23 2024

Harry J. Smith, Table of n, a(n) for n = 0..1000
Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Cyclic permutations avoiding patterns in both one-line and cycle forms, arXiv:2312.05145 [math.CO], 2023. See p. 2.
Eric Weisstein's World of Mathematics, Bishop Graph.
Eric Weisstein's World of Mathematics, Lower Matching Number.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A100795, A074147-A074149.

seq(floor((n^2+3)/2),n=0..25); # Gary Detlefs, Feb 13 2010

Table[Ceiling[n^2/2]+1,{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011 *)
LinearRecurrence[{2,0,-2,1},{1,2,3,6},60] (* Harvey P. Dale, Jan 03 2024 *)

a(n) = { ceil(n^2/2) + 1 } \\ Harry J. Smith, Jul 29 2009

a(n) = a(n-1) + 2*floor((n-1)/2) + 1 = A061926(3, k) = 2*A002620(n+1) - (n-1) = A000982(n) + 1.
a(2*n) = a(2*n-1) + 2*n - 1 = 2*n^2 + 1 = A058331(n).
a(2*n+1) = a(2*n) + 2*n + 1 = 2*(n^2 + n + 1) = A051890(n+1).
a(n) = floor((n^2+3)/2). - Gary Detlefs, Feb 13 2010
From R. J. Mathar, Feb 19 2010: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1-x^2+2*x^3)/((1+x) * (1-x)^3). (End)
a(n) = (2*n^2 - (-1)^n + 5)/4. - Bruno Berselli, Sep 29 2011
a(n) = A007590(n+1) - n + 1. - Wesley Ivan Hurt, Jul 15 2013
a(n) + a(n+1) = A027688(n). a(n+1) - a(n) = A109613(n). - R. J. Mathar, Jul 20 2013
E.g.f.: ((2 + x + x^2)*cosh(x) + (3 + x + x^2)*sinh(x))/2. - Stefano Spezia, May 07 2021

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 09 2007

A163102 a(n) = n^2*(n+1)^2/2.

Original entry on oeis.org

0, 2, 18, 72, 200, 450, 882, 1568, 2592, 4050, 6050, 8712, 12168, 16562, 22050, 28800, 36992, 46818, 58482, 72200, 88200, 106722, 128018, 152352, 180000, 211250, 246402, 285768, 329672, 378450, 432450, 492032, 557568, 629442, 708050, 793800, 887112, 988418

Offset: 0

Omar E. Pol, Jul 24 2009

Row sums of triangle A163282.
Also, the number of nonattacking placements of 2 rooks on an (n+1) X (n+1) board. - Thomas Zaslavsky, Jun 26 2010
If P_{k}(n) is the n-th k-gonal number, then a(n) = P_{s}(n+1)*P_{t}(n+1) - P_{s+1}(n+1)*P_{t-1}(n+1) for s=t+1. - Bruno Berselli, Sep 05 2014
Subsequence of A000982, see formula. - David James Sycamore, Jul 31 2018
Number of edges in the (n+1) X (n+1) rook complement graph. - Freddy Barrera, May 02 2019
Number of paths from (0,0) to (n+2,n+2) consisting of exactly three forward horizontal steps and three upward vertical steps. - Greg Dresden and Snezhana Tuneska, Aug 24 2023

Seth Chaiken, Christopher R. H. Hanusa, and Thomas Zaslavsky, A q-queens problem, in preparation. - Thomas Zaslavsky, Jun 26 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Seth Chaiken, Christopher R. H. Hanusa, and Thomas Zaslavsky, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853 [math.CO], 2016-2020.
A. Umar, Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations, J. Int. Seq., Vol. 14 (2011), Article 11.7.5.
Eric Weisstein's World of Mathematics, Rook Complement Graph.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Column k=2 of A144084.
Cf. A000217, A000537, A000982, A035287.
Cf. A006002, A099903, A163274, A163275, A163276, A163277, A163282.
Cf. A060300, A254371.

List([0..40],n->(n*(n+1))^2/2); # Muniru A Asiru, Aug 02 2018

[n^2*(n+1)^2/2: n in [0..40]]; // Vincenzo Librandi, Mar 26 2012

CoefficientList[Series[2*x*(1+4*x+x^2)/(1-x)^5,{x,0,40}],x] (* Vincenzo Librandi, Mar 26 2012 *)

a(n)=n^2*(n+1)^2/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 2*A000537(n) = A035287(n+1)/2. - Omar E. Pol, Nov 29 2011
G.f.: 2*x*(1+4*x+x^2)/(1-x)^5. - R. J. Mathar, Nov 30 2011
Let t(n) = A000217(n). Then a(n) = (t(n-1)*(t(n)+t(n+1)) + t(n)*(t(n-1)+t(n+1)) + t(n+1)*(t(n-1)+t(n)))/3. - J. M. Bergot, Jun 21 2012
a(n) = A000982(n*(n+1)). - David James Sycamore, Jul 31 2018
From Amiram Eldar, Nov 02 2021: (Start)
Sum_{n>=1} 1/a(n) = 2*Pi^2/3 - 6.
Sum_{n>=1} (-1)^(n+1)/a(n) = 6 - 8*log(2). (End)
Another identity: ..., a(4) = 200 = 1*(2+4+6+8) + 3*(4+6+8) + 5*(6+8) + 7*(8), a(5) = 450 = 1*(2+4+6+8+10) + 3*(4+6+8+10) + 5*(6+8+10) + 7*(8+10) + 9*(10) = 30+84+120+126+90, and so on. - J. M. Bergot, Aug 25 2022
From Elmo R. Oliveira, Aug 14 2025: (Start)
E.g.f.: x*(2 + x)*(2 + 6*x + x^2)*exp(x)/2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A254371(n)/4 = A060300(n)/8. (End)

A188181 T(n,k) is the number of strictly increasing arrangements of n numbers in -(n+k-2)..(n+k-2) with sum zero.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 5, 1, 4, 8, 12, 12, 1, 5, 13, 24, 32, 32, 1, 6, 18, 43, 73, 94, 94, 1, 7, 25, 69, 141, 227, 289, 289, 1, 8, 32, 104, 252, 480, 734, 910, 910, 1, 9, 41, 150, 414, 920, 1656, 2430, 2934, 2934, 1, 10, 50, 207, 649, 1636, 3370, 5744, 8150, 9686, 9686, 1, 11

Offset: 1

R. H. Hardin, Mar 23 2011

			Table starts
....1....1.....1.....1......1......1......1.......1.......1.......1.......1
....1....2.....3.....4......5......6......7.......8.......9......10......11
....2....5.....8....13.....18.....25.....32......41......50......61......72
....5...12....24....43.....69....104....150.....207.....277.....362.....462
...12...32....73...141....252....414....649.....967....1394....1944....2649
...32...94...227...480....920...1636...2739....4370....6698....9926...14293
...94..289...734..1656...3370...6375..11322...19138...30982...48417...73316
..289..910..2430..5744..12346..24591..46029...81805..139143..227930..361384
..910.2934..8150.20094..45207..94257.184717..343363..610358.1043534.1724882
.2934.9686.27718.70922.165821.360002.734517.1421530.2628824.4672836.8022362
Some solutions for n=7 and k=5:
.-7...-9...-8..-10...-6...-6...-9...-8...-8...-7...-9..-10...-9...-8...-9...-7
.-5...-7...-6...-7...-5...-5...-4...-6...-7...-3...-8...-5...-3...-7...-5...-4
.-3...-1...-5...-4...-4...-3...-1...-4...-5...-2...-6...-3...-1...-3...-4...-3
.-1....0...-1....3...-3...-2....0...-1....0...-1....4....0....1....1...-1...-1
..3....3....2....5...-1....1....1....4....3....0....5....3....3....2....4....0
..4....4....8....6....9....7....4....5....7....4....6....5....4....5....6....7
..9...10...10....7...10....8....9...10...10....9....8...10....5...10....9....8

R. H. Hardin, Table of n, a(n) for n = 1..550

Column 1 is A076822.
Column 2 is A002838.
Cf. A000982.

T(3,n) = A000982(n+1).

A080827 Rounded up staircase on natural numbers.

Original entry on oeis.org

1, 3, 5, 9, 13, 19, 25, 33, 41, 51, 61, 73, 85, 99, 113, 129, 145, 163, 181, 201, 221, 243, 265, 289, 313, 339, 365, 393, 421, 451, 481, 513, 545, 579, 613, 649, 685, 723, 761, 801, 841, 883, 925, 969, 1013, 1059, 1105, 1153, 1201, 1251, 1301, 1353, 1405, 1459

Offset: 1

Paul Barry, Feb 28 2003

Represents the 'rounded up' staircase diagonal on A000027, arranged as a square array. A000982 is the 'rounded down' staircase.
Partial sums of A131055. - Paul Barry, Jun 14 2008
The same sequence arises in the triangular array of integers >= 1 according to a simple "zig-zag" rule for selection of terms. a(n-1) lies in the (n-1)-th row of the array and the second row of that subarray (with apex a(n-1)) contains just two numbers, one odd one even. The one with the same (odd) parity as a(n-1) is a(n). - David James Sycamore, Jul 29 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
J. C. F. de Winter, Using the Student's t-test with extremely small sample sizes, Practical Assessment, Research & Evaluation, 18(10), 2013.
Girtrude Hamm, Classification of lattice triangles by their two smallest widths, arXiv:2304.03007 [math.CO], 2023.
David James Sycamore, Triangular array.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Apart from leading term identical to A099392.

Cf. A000027, A000982, A007590, A131055.

List([1..10],n->Int(n^2/2)+1); # Muniru A Asiru, Aug 02 2018

[n*(n+1)/2-Floor((n-1)/2) : n in [1..60]]; // Vincenzo Librandi, Aug 05 2013

A080827:=n->(n^2+2-(1-(-1)^n)/2)/2: seq(A080827(n), n=1..100); # Wesley Ivan Hurt, Sep 08 2015

CoefficientList[Series[(1 + x - x^2 + x^3) / ((1 + x) (1 - x)^3), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 05 2013 *)
Ceiling[(Range[100]^2 + 1)/2] (* Paolo Xausa, Jul 19 2025 *)

a(n) = ceiling((n^2+1)/2).
a(1) = 1, a(2n) = a(2n-1) + 2n, a(2n+1) = a(2n) + 2n. - Amarnath Murthy, May 07 2003
From Paul Barry, Apr 12 2008: (Start)
G.f.: x*(1+x-x^2+x^3)/((1+x)(1-x)^3).
a(n) = n*(n+1)/2-floor((n-1)/2). [corrected by R. J. Mathar, Jul 14 2013] (End)
From Wesley Ivan Hurt, Sep 08 2015: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n > 4.
a(n) = (n^2 + 2 - (1 - (-1)^n)/2)/2.
a(n) = floor(n^2/2) + 1 = A007590(n-1) + 1. (End)
Sum_{n>=1} 1/a(n) = tanh(Pi/2)*Pi/2 + coth(Pi/sqrt(2))*Pi/(2*sqrt(2)) - 1/2. - Amiram Eldar, Sep 15 2022
E.g.f.: ((2 + x + x^2)*cosh(x) + (1 + x + x^2)*sinh(x) - 2)/2. - Stefano Spezia, Jan 27 2024

A132188 Number of 3-term geometric progressions with no term exceeding n.

Original entry on oeis.org

1, 2, 3, 6, 7, 8, 9, 12, 17, 18, 19, 22, 23, 24, 25, 32, 33, 38, 39, 42, 43, 44, 45, 48, 57, 58, 63, 66, 67, 68, 69, 76, 77, 78, 79, 90, 91, 92, 93, 96, 97, 98, 99, 102, 107, 108, 109, 116, 129, 138, 139, 142, 143, 148, 149, 152, 153, 154, 155, 158

Offset: 1

Gerry Myerson, Nov 21 2007

nonn

a(n) = number of pairs (i,j) in [1..n] X [1..n] with integral geometric mean sqrt(i*j). Cf. A000982, A362931. - N. J. A. Sloane, Aug 28 2023
Also the number of 2 X 2 symmetric singular matrices with entries from {1, ..., n} - cf. A064368.
Rephrased: Number of ordered triples (w,x,y) with all terms in {1,...,n} and w^2=x*y. See A211422. - Clark Kimberling, Apr 14 2012

			a(4) counts these six (w,x,y) - triples: (1,1,1), (2,1,4), (2,4,1), (2,2,2), (3,3,3), (4,4,4). - _Clark Kimberling_, Apr 14 2012

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Gerry Myerson, Trifectas in Geometric Progression, Australian Mathematical Society Gazette 35 (3) (2008) pp. 189--194 (pages 47--52 in PDF).

Cf. A057918, A064368, A120486, A132189, A132345, A211422, A338894.

Cf. also A000982, A362931.

a132188 0 = 0
a132188 n = a132345 n + (a120486 $ fromInteger n)
-- Reinhard Zumkeller, Apr 21 2012

a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+
      1+2*add(`if`(issqr(i*n), 1, 0), i=1..n-1))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Aug 28 2023

t[n_] := t[n] = Flatten[Table[w^2 - x*y, {w, 1, n}, {x, 1, n}, {y, 1, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 80}]  (* Clark Kimberling, Apr 14 2012 *)

from sympy.ntheory.primetest import is_square
def A132188(n): return n+(sum(1 for x in range(1,n+1) for y in range(1,x) if is_square(x*y))<<1) # Chai Wah Wu, Aug 28 2023

a(n) = Sum [sqrt(n/k)]^2, where the sum is over all squarefree k not exceeding n.
If we call A120486, this sequence and A132189 F(n), P(n) and S(n), respectively, then P(n) = 2 F(n) - n = S(n) + n. The Finch-Sebah paper cited at A000188 proves that F(n) is asymptotic to (3 / pi^2) n log n. In the reference, we prove that F(n) = (3 / pi^2) n log n + O(n), from which it follows that P(n) = (6 / pi^2) n log n + O(n) and similarly for S(n).
a(n) = Sum_{1 <=x,y <=n} A010052(x*y). - Clark Kimberling, Apr 14 2012
a(n) = n+2*Sum_{1<=xA010052(x*y). - Chai Wah Wu, Aug 28 2023

A137928 The even principal diagonal of a 2n X 2n square spiral.

Original entry on oeis.org

2, 4, 10, 16, 26, 36, 50, 64, 82, 100, 122, 144, 170, 196, 226, 256, 290, 324, 362, 400, 442, 484, 530, 576, 626, 676, 730, 784, 842, 900, 962, 1024, 1090, 1156, 1226, 1296, 1370, 1444, 1522, 1600, 1682, 1764, 1850, 1936, 2026, 2116, 2210, 2304, 2402, 2500, 2602, 2704, 2810

Offset: 1

William A. Tedeschi, Feb 29 2008

This is concerned with 2n X 2n square spirals of the form illustrated in the Example section.

			Example with n = 2:
.
   7---8---9--10
   |           |
   6   1---2  11
   |       |   |
   5---4---3  12
               |
  16--15--14--13
.
a(1) = 2(1) + 4*floor((1-1)/4) = 2;
a(2) = 2(2) + 4*floor((2-1)/4) = 4.

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

Cf. A000982, A002061 (odd diagonal), A002620, A080335, A171218.

A137928:=n->2*ceil(n^2/2): seq(A137928(n), n=1..100); # Wesley Ivan Hurt, Jul 25 2017

LinearRecurrence[{2,0,-2,1},{2,4,10,16},60] (* Harvey P. Dale, Aug 28 2017 *)

a(n)=2*n+(n-1)^2\4*4 \\ Charles R Greathouse IV, May 21 2015

a = lambda n: 2*n + 4*floor((n-1)**2/4)

a(n) = 2*n + 4*floor((n-1)^2/4) = 2*n + 4*A002620(n-1).
a(n) = A171218(n) - A171218(n-1). - Reinhard Zumkeller, Dec 05 2009
From R. J. Mathar, Jun 27 2011: (Start)
G.f.: 2*x*(1 + x^2) / ( (1 + x)*(1 - x)^3 ).
a(n) = 2*A000982(n). (End)
a(n+1) = (3 + 4*n + 2*n^2 + (-1)^n)/2 = A080335(n) + (-1)^n. - Philippe Deléham, Feb 17 2012
a(n) = 2 * ceiling(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
a(n) = n^2 + (n mod 2). - Bruno Berselli, Oct 03 2017
Sum_{n>=1} 1/a(n) = Pi*tanh(Pi/2)/4 + Pi^2/24. - Amiram Eldar, Jul 07 2022

A200154 T(n,k) = number of 0..k arrays x(0..n-1) of n elements with zero (n-1)-st difference.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 5, 4, 1, 5, 8, 9, 2, 1, 6, 13, 22, 15, 8, 1, 7, 18, 41, 40, 39, 2, 1, 8, 25, 66, 103, 112, 45, 16, 1, 9, 32, 107, 202, 275, 182, 129, 6, 1, 10, 41, 158, 381, 730, 685, 688, 149, 32, 1, 11, 50, 219, 636, 1589, 2036, 2525, 844, 243, 2, 1, 12, 61, 304, 1033, 3000, 5153, 7488, 5221, 2090, 369, 64, 1

Offset: 1

R. H. Hardin, Nov 13 2011

Table starts
 1    1     1     1      1      1       1       1        1        1
 3    4     5     6      7      8       9      10       11       12
 5    8    13    18     25     32      41      50       61       72
 9   22    41    66    107    158     219     304      403      516
15   40   103   202    381    636    1033    1550     2287     3212
39  112   275   730   1589   3000    5181    8350    13871    21588
45  182   685  2036   5153  11370   23035   43284    76523   129052
129  688  2525  7488  18809  52166  121921  253768   484977   867086
149  844  5221 19262  68813 194818  514113 1171190  2531421  5019770
243 2090 13897 62772 256859 841122 2347671 6169890 14503751 31169760
T(n,k) is the number of integer lattice points in k*C(n) where C(n) is a certain polytope with vertices having rational entries (the intersection of [0,1]^n with a hyperplane).  Thus row n is an Ehrhart quasi-polynomial of degree n-1. - Robert Israel, Dec 12 2019

			Some solutions for n=7, k=6:
  5  6  5  3  6  0  0  5  4  1  2  2  0  2  1  2
  3  1  5  1  6  5  4  0  2  5  2  0  2  0  4  0
  3  3  6  5  6  1  6  2  0  1  1  4  3  4  6  2
  3  2  3  6  5  1  3  6  0  2  1  6  3  3  6  3
  2  0  2  5  5  3  2  6  1  6  2  5  3  1  5  2
  1  1  6  5  6  2  6  1  2  6  3  3  4  3  4  1
  4  1  1  3  1  2  0  1  5  0  3  1  6  1  2  4

R. H. Hardin, Table of n, a(n) for n = 1..321

Row 3 is A000982(n+1).

Cf. A187202 (for 3rd PARI function).

pad(d, n) = while(#d != n, d = concat([0], d)); d;
mydigits(i,n) = if (n<2, vector(i), digits(i,n));
bedt(n) = {for(i=2, #n=n, n=vecextract(n, "^1")-vecextract(n, "^-1")); n[1];}
T(n, k) = {k++; my(nbok = 0); for (i=0, k^n-1, d = pad(mydigits(i,k), n); if (bedt(d) == 0, nbok++);); nbok;} \\ Michel Marcus, Apr 08 2017

cp's OEIS Frontend

A001971 Nearest integer to n^2/8.

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A047838 a(n) = floor(n^2/2) - 1.

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A002731 Numbers k such that (k^2 + 1)/2 is prime.

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A061925 a(n) = ceiling(n^2/2) + 1.

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A163102 a(n) = n^2*(n+1)^2/2.

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A188181 T(n,k) is the number of strictly increasing arrangements of n numbers in -(n+k-2)..(n+k-2) with sum zero.

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