A061317 -id:A061317 - cp's OEIS frontend

A006331 a(n) = n(n+1)(2*n+1)/3.

0, 2, 10, 28, 60, 110, 182, 280, 408, 570, 770, 1012, 1300, 1638, 2030, 2480, 2992, 3570, 4218, 4940, 5740, 6622, 7590, 8648, 9800, 11050, 12402, 13860, 15428, 17110, 18910, 20832, 22880, 25058, 27370, 29820, 32412, 35150, 38038, 41080, 44280

Offset: 0

N. J. A. Sloane

Triangles in rhombic matchstick arrangement of side n.
Maximum accumulated number of electrons at energy level n. - Scott A. Brown, Feb 28 2000
Let M_n denote the n X n matrix M_n(i,j)=i^2+j^2; then the characteristic polynomial of M_n is x^n - a(n)x^(n-1) - .... - Michael Somos, Nov 14 2002
Convolution of odds (A005408) and evens (A005843). - Graeme McRae, Jun 06 2006
a(n) is the number of non-monotonic functions with domain {0,1,2} and codomain {0,1,...,n}. - Dennis P. Walsh, Apr 25 2011
For any odd number 2n+1, find Sum_{aJ. M. Bergot, Jul 16 2011
a(n) gives the number of (n+1) X (n+1) symmetric (0,1)-matrices containing three ones (see [Cameron]). - L. Edson Jeffery, Feb 18 2012
a(n) is the number of 4-tuples (w,x,y,z) with all terms in {0,...,n} and |w - x| < y. - Clark Kimberling, Jun 02 2012
Partial sums of A001105. - Omar E. Pol, Jan 12 2013
Total number of square diagonals (of any size) in an n X n square grid. - Wesley Ivan Hurt, Mar 24 2015
Number of diagonal attacks of two queens on (n+1) X (n+1) chessboard. - Antal Pinter, Sep 20 2015
a(n) is the minimum value obtainable by partitioning either the set {x in the natural numbers | 1 <= x <= 2n} or the set {x in the natural numbers | 0 <= x <= 2n+1} into pairs, taking the product of all such pairs, and taking the sum of all such products. - Thomas Anton, Oct 21 2020
a(n) is the irregularity of the n-th power of a path of length at least 3*n.  (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, Jun 16 2023
a(n) is the maximum possible total number of inversions in all rows and all columns of a Latin square of order n+1. - Ivaylo Kortezov, Jun 28 2025

			For n=2, a(2)=10 since there are 10 non-monotonic functions f from {0,1,2} to {0,1,2}, namely, functions f = <f(1),f(2),f(3)> given by <0,1,0>, <0,2,0>, <0,2,1>, <1,0,1>, <1,0,2>, <1,2,0>, <1,2,1>, <2,0,1>, <2,0,2>, and <2,1,2>. - _Dennis P. Walsh_, Apr 25 2011
Let n=4, 2*n+1 = 9. Since 9 = 1+8 = 3+6 = 5+4 = 7+2, a(4) = 1*8 + 3*6 + 5*4 + 7*2 = 60. - _Vladimir Shevelev_, May 11 2012

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
J. L. Bailey, Jr., A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., Vol. 2 (1931), pp. 355-359.
J. L. Bailey, A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., Vol. 2 (1931), pp. 355-359. [Annotated scanned copy]
Rowan Beckworth, Basic atomic information.
Allan Bickle and Zhongyuan Che, Irregularities of Maximal k-degenerate Graphs, Discrete Applied Math. 331 (2023) 70-87.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
P. Cameron, T. Prellberg and D. Stark, Asymptotics for incidence matrix classes, Electron. J. Combin. 13 (2006), #R85, p. 11.
Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, Semi-simplicial combinatorics of cyclinders and subdivisions, arXiv:2307.13749 [math.CO], 2023. See p. 25.
N. S. S. Gu, H. Prodinger and S. Wagner, Bijections for a class of labeled plane trees, Eur. J. Combinat., Vol. 31 (2010), pp. 720-732, doi|10.1016/j.ejc.2009.10.007, Theorem 2 at n=3.
JBMO 2025, 29th Junior Balkan Mathematical Olympiad, Problem 4, author: Boris Mihov
Germain Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, Vol. 6 (1965), circa p. 82.
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
Dennis Walsh, Notes on finite monotonic and non-monotonic functions.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

A row of A132339.
Cf. A002378, A048147, A098077, A163102.
Cf. A005900, A084980, A077415, A112524, A188475, A100178, A035597, A096000.
Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).

a006331 n = sum $ zipWith (*) [2*n-1, 2*n-3 .. 1] [2, 4 ..]
-- Reinhard Zumkeller, Feb 11 2012

[n*(n+1)*(2*n+1)/3: n in [0..40]]; // Vincenzo Librandi, Aug 15 2011

A006331 := proc(n)
    n*(n+1)*(2*n+1)/3 ;
end proc:
seq(A006331(n),n=0..80) ; # R. J. Mathar, Sep 27 2013

Table[n(n+1)(2n+1)/3,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,2,10,28},50] (* Harvey P. Dale, Apr 12 2013 *)

PARI
```
a(n)=if(n<0,0,n*(n+1)*(2*n+1)/3)
```

G.f.: 2*x*(1 + x)/(1 - x)^4. - Simon Plouffe (in his 1992 dissertation)
a(n) = 2*binomial(n+1,3) + 2*binomial(n+2,3).
a(n) = 2*A000330(n) = A002492(n)/2.
a(n) = Sum_{i=0..n} T(i,n-i), array T as in A048147. - N. J. A. Sloane, Dec 11 1999
From the formula for the sum of squares of positive integers 1^2 + 2^2 + 3^2 + ... + n^2 = n*(n+1)(2*n+1)/6, if we multiply both sides by 2 we get Sum_{k=0..n} 2*k^2 = n*(n+1)*(2*n+1)/3, which is an alternative formula for this sequence. - Mike Warburton, Sep 08 2007
10*a(n) = A016755(n) - A001845(n); since A016755 are odd cubes and A001845 centered octahedral numbers, 10*a(n) are the "odd cubes without their octahedral contents." - Damien Pras, Mar 19 2011
a(n) = sum(a*b), where the summing is over all unordered partitions 2*n+1=a+b. - Vladimir Shevelev, May 11 2012
a(n) = binomial(2*n+2, 3)/2. - Ronan Flatley, Dec 13 2012
a(n) = A000292(n) + A002411(n). - Omar E. Pol, Jan 11 2013
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3, with a(0)=0, a(1)=2, a(2)=10, a(3)=28. - Harvey P. Dale, Apr 12 2013
a(n) = A208532(n+1,2). - Philippe Deléham, Dec 05 2013
Sum_{n>0} 1/a(n) = 9 - 12*log(2). - Enrique Pérez Herrero, Dec 03 2014
a(n) = A000292(n-1) + (n+1)*A000217(n). - J. M. Bergot, Sep 02 2015
a(n) = 2*(A000332(n+3) - A000332(n+1)). - Antal Pinter, Sep 20 2015
From Bruno Berselli, May 17 2018: (Start)
a(n) = n*A002378(n) - Sum_{k=0..n-1} A002378(k) for n>0, a(0)=0. Also:
A163102(n) = n*a(n) - Sum_{k=0..n-1} a(k) for n>0, A163102(0)=0. (End)
a(n) = A005900(n) - A000290(n) = A096000(n) - A000578(n+1) = A000578(n+1) - A084980(n+1) = A000578(n+1) - A077415(n)-1 = A112524(n) + 1 = A188475(n) - 1 = A061317(n) - A100178(n) = A035597(n+1) - A006331(n+1). - Bruce J. Nicholson, Jun 24 2018
E.g.f.: (1/3)*exp(x)*x*(6 + 9*x + 2*x^2). - Stefano Spezia, Jan 05 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi - 9. - Amiram Eldar, Jan 04 2022

A007588 Stella octangula numbers: a(n) = n(2n^2 - 1).

Original entry on oeis.org

0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, 2651, 3444, 4381, 5474, 6735, 8176, 9809, 11646, 13699, 15980, 18501, 21274, 24311, 27624, 31225, 35126, 39339, 43876, 48749, 53970, 59551, 65504, 71841, 78574, 85715, 93276, 101269, 109706, 118599, 127960

Offset: 0

N. J. A. Sloane

Also as a(n)=(1/6)*(12*n^3-6*n), n>0: structured hexagonal anti-diamond numbers (vertex structure 13) (Cf. A005915 = alternate vertex; A100188 = structured anti-diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004
The only known square stella octangula number for n>1 is a(169) = 169*(2*169^2 - 1) = 9653449 = 3107^2. - Alexander Adamchuk, Jun 02 2008
Ljunggren proved that 9653449 = (13*239)^2 is the only square stella octangula number for n>1. See A229384 and the Wikipedia link. - Jonathan Sondow, Sep 30 2013
4*A007588 = A144138(ChebyshevU[3,n]). - Vladimir Joseph Stephan Orlovsky, Jun 30 2011
If A016813 is regarded as a regular triangle (with leading terms listed in A001844), a(n) provides the row sums of this triangle: 1, 5+9=14, 13+17+21=51 and so on. - J. M. Bergot, Jul 05 2013
Shares its digital root, A267017, with n*(n^2 + 1)/2 ("sum of the next n natural numbers" see A006003). - Peter M. Chema, Aug 28 2016

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 51.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.
W. Ljunggren, Zur Theorie der Gleichung x^2 + 1 = Dy^4, Avh. Norske Vid. Akad. Oslo. I. 1942 (5): 27.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alexander Adamchuk and Vincenzo Librandi, Table of n, a(n) for n = 0..10000 [Alexander Adamchuk computed terms 0 - 169, Jun 02, 2008; Vincenzo Librandi computed the first 10000 terms, Aug 18 2011]
A. Bremner, R. Høibakk and D. Lukkassen, Crossed ladders and Euler’s quartic, Annales Mathematicae et Informaticae, 36 (2009) pp. 29-41. See p. 33.
John Elias, Nesting Cubes of the Surface Points of a Hexagonal Prism
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Amelia Carolina Sparavigna, Generalized Sum of Stella Octangula Numbers, Politecnico di Torino (Italy, 2021).
Amelia Carolina Sparavigna, Cardano Formula and Some Figurate Numbers, Politecnico di Torino (Italy, 2021).
Eric Weisstein's World of Mathematics, Stella Octangula Number
Wikipedia, Stella octangula number
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Backwards differences give star numbers A003154: A003154(n)=a(n)-a(n-1).
1/12*t*(n^3-n)+ n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
Cf. A001653 = Numbers n such that 2*n^2 - 1 is a square.
a(169) = (A229384(3)*A229384(4))^2.
Cf. A267017, A006003, A135503.
Cf. A002378, A005843, A061317, A062392.

[n*(2*n^2 - 1): n in [0..40]]; // Vincenzo Librandi, Aug 18 2011

A007588:=n->n*(2*n^2 - 1); seq(A007588(n), n=0..40); # Wesley Ivan Hurt, Mar 10 2014

Table[ n(2n^2-1), {n,0,169} ] (* Alexander Adamchuk, Jun 02 2008 *)
LinearRecurrence[{4,-6,4,-1},{0,1,14,51},50] (* Harvey P. Dale, Sep 16 2011 *)

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

def A007588(n): return n*(2*n**2-1) # Chai Wah Wu, Feb 18 2022

G.f.: x*(1+10*x+x^2)/(1-x)^4.
a(n) = n*A056220(n).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3. - Harvey P. Dale, Sep 16 2011
From Ilya Gutkovskiy, Jul 02 2016: (Start)
E.g.f.: x*(1 + 6*x + 2*x^2)*exp(x).
Dirichlet g.f.: 2*zeta(s-3) - zeta(s-1). (End)
a(n) = A004188(n) + A135503(n). - Miquel Cerda, Dec 25 2016
a(n) = A061317(n) - A005843(n) = A062392(n) - A062392(n-1). - J.S. Seneschal, Jul 01 2025

In the formula given in the 1995 Encyclopedia of Integer Sequences, the second 2 should be an exponent.

A098547 a(n) = n^3 + n^2 + 1.

Original entry on oeis.org

1, 3, 13, 37, 81, 151, 253, 393, 577, 811, 1101, 1453, 1873, 2367, 2941, 3601, 4353, 5203, 6157, 7221, 8401, 9703, 11133, 12697, 14401, 16251, 18253, 20413, 22737, 25231, 27901, 30753, 33793, 37027, 40461, 44101, 47953, 52023, 56317, 60841, 65601, 70603, 75853

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Oct 26 2004

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A001093, A011379, A027444, A033431, A033562, A034262, A053698, A061317, A066023, A071568.

Cf. A000217, A081423.

[(n^3+n^2+1): n in [1..60]]; // Vincenzo Librandi, Apr 06 2011

with(combinat): seq(fibonacci(3,n)+n^3, n=0..40); # Zerinvary Lajos, May 25 2008

Table[n^3+n^2+1,{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Mar 09 2011 *)

Vec(-(x^3-7*x^2+x-1)/(x-1)^4 + O(x^100)) \\ Colin Barker, Aug 29 2014

From Colin Barker, Aug 29 2014: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: (1 - x + 7*x^2 - x^3)/(1-x)^4. (End)
a(n) = A081423(n) + A000217(n-1). - Bruce J. Nicholson, Jan 06 2019
E.g.f.: exp(x)*(1 + 2*x + 4*x^2 + x^3). - Elmo R. Oliveira, Apr 20 2025

A099721 a(n) = n^2(2n+1).

Original entry on oeis.org

0, 3, 20, 63, 144, 275, 468, 735, 1088, 1539, 2100, 2783, 3600, 4563, 5684, 6975, 8448, 10115, 11988, 14079, 16400, 18963, 21780, 24863, 28224, 31875, 35828, 40095, 44688, 49619, 54900, 60543, 66560, 72963, 79764, 86975, 94608, 102675, 111188, 120159, 129600

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Nov 07 2004

For a right triangle with sides of lengths 8*n^3 + 12*n^2 + 8*n + 2, 4*n^4 + 8*n^3 + 4*n^2, and 4*n^4 + 8*n^3 + 12*n^2 + 8*n + 2, dividing the area by the perimeter gives a(n). - J. M. Bergot, Jul 30 2013
This sequence is the difference between the centered icosahedral (or cuboctahedral) numbers (A005902(n)) and the centered octagonal pyramidal numbers (A000447(n+1)). - Peter M. Chema, Jan 09 2016
a(n) is the sum of the integers in the closed interval (n-1)*n to n*(n+1). - J. M. Bergot, Apr 19 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Row n=3 of A229079.
Cf. A000578, A001093, A011379, A015237, A027444, A033431, A033562, A034262, A053698, A061317, A066023, A071568, A098547.
Cf. A000290, A000447, A002378, A005408, A005902, A024196.

[n^2*(2*n+1): n in [0..50]]; // Vincenzo Librandi, May 01 2011

A099721 := proc(n) n^2*(2*n+1) ; end proc:
seq(A099721(n),n=0..10) ;

a[n_]:=2*n^3+n^2; (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
LinearRecurrence[{4,-6,4,-1},{0,3,20,63},40] (* Harvey P. Dale, Aug 19 2022 *)

a(n) = ceil(sum(i=n^2-(n-1), n^2+(n-1), if(!issquare(4*i+1), (2*i+1+sqrt(4*i+1))/2, 0))); \\ Michel Marcus, Nov 14 2014, after Richard R. Forberg

G.f.: x*(3 + 8*x + x^2)/(x-1)^4.
a(n) = A024196(n) - A024196(n-1). - Philippe Deléham, May 07 2012
a(n) = ceiling(Sum_{i=n^2-(n-1)..n^2+(n-1)} s(i)), for n > 0 and integer i, where s(i) are the real solutions to x = i + sqrt(x), and the summation range excludes the integer solutions which occur where i is an oblong number (A002378). The fractional portion of the summation converges to 2/3 for large n. If s(i) is replaced with i, then the summation equals n^2*(2*n-1) = A015237. - Richard R. Forberg, Oct 15 2014
a(n) = A005902(n) - A000447(n+1). - Peter M. Chema, Jan 09 2016
From Amiram Eldar, May 17 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/6 + 4*log(2) - 4.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - Pi - 2*log(2) + 4. (End)
From Elmo R. Oliveira, Aug 08 2025: (Start)
E.g.f.: x*(1 + 2*x)*(3 + x)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = A000290(n)*A005408(n). (End)

A110450 a(n) = n(n+1)(n^2+n+1)/2.

Original entry on oeis.org

0, 3, 21, 78, 210, 465, 903, 1596, 2628, 4095, 6105, 8778, 12246, 16653, 22155, 28920, 37128, 46971, 58653, 72390, 88410, 106953, 128271, 152628, 180300, 211575, 246753, 286146, 330078, 378885, 432915, 492528, 558096, 630003, 708645, 794430

Offset: 0

Reinhard Zumkeller, Jul 21 2005

This sequence is related to A085461 by 3*A085461(n) = n*a(n) - Sum_{i=0..n-1} a(i) for n>0. - Bruno Berselli, Dec 27 2010

Subsequence of the triangular numbers A000217, see formulas below. - David James Sycamore, Jul 31 2018

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A002378, A061317, A085461, A110449, A014105.

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

[n*(n+1)*(n^2+n+1)/2: n in [0..40]]; // Vincenzo Librandi, Dec 26 2010

A110450:=n->n*(n+1)*(n^2+n+1)/2; seq(A110450(k), k=0..50); # Wesley Ivan Hurt, Sep 27 2013

Table[n (n + 1) (n^2 + n + 1)/2, {n, 0, 100}] (* Wesley Ivan Hurt, Sep 27 2013 *)
CoefficientList[Series[-3 x (x^2 + 2 x + 1)/(x - 1)^5, {x, 0, 36}], x] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 3, 21, 78, 210}, 36] (* Robert G. Wilson v, Jul 31 2018 *)

a(n)=n*(n+1)*(n^2+n+1)/2 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = Sum_{k=0..n} A110449(n,k), sums of rows in triangle A110449.
From Bruno Berselli, Dec 27 2010: (Start)
G.f.: 3*x*(1 + x)^2/(1 - x)^5.
a(n) = A014105(A000217(n)). (End)
a(n) = Sum_{i=1..n*(n+1)} i. - Wesley Ivan Hurt, Sep 27 2013
a(n) = Sum_{i=0..n} i*(2*i^2+1), and these are the partial sums of A061317. - Bruno Berselli, Feb 09 2017
a(n) = t(n,t(n,A000217(n))), where t(n,k) = n*(n+1)/2 + k*n and k=0. - Bruno Berselli, Feb 28 2017
E.g.f.: (x/2)*(6 + 15*x + 8*x^2 + x^3)*exp(x). - G. C. Greubel, Aug 24 2017
a(n) = A000217(n*(n+1)). - David James Sycamore, Jul 31 2018
a(n) = A000217(2*A000217(n)) = A000217(A002378(n)). - Alois P. Heinz, Jul 31 2018
From R. J. Mathar, Mar 23 2021: (Start)
a(n) = A002378(n)+A062392(n).
a(n) = 3*A006325(n+1). (End)
Sum_{n>=1} 1/a(n) = 4 - 2*Pi*tanh(sqrt(3)*Pi/2)/sqrt(3). - Amiram Eldar, May 10 2025

A185870 (Even,odd)-polka dot array in the natural number array A000027, by antidiagonals.

Original entry on oeis.org

3, 8, 10, 17, 19, 21, 30, 32, 34, 36, 47, 49, 51, 53, 55, 68, 70, 72, 74, 76, 78, 93, 95, 97, 99, 101, 103, 105, 122, 124, 126, 128, 130, 132, 134, 136, 155, 157, 159, 161, 163, 165, 167, 169, 171, 192, 194, 196, 198, 200, 202, 204, 206, 208, 210, 233, 235, 237, 239, 241, 243, 245, 247, 249, 251, 253, 278, 280, 282, 284, 286, 288, 290, 292, 294, 296, 298, 300, 327, 329, 331, 333, 335, 337, 339, 341, 343, 345, 347, 349, 351, 380, 382, 384, 386, 388, 390, 392, 394, 396, 398, 400, 402, 404, 406

Offset: 1

Clark Kimberling, Feb 05 2011

This is the third of four polka dot arrays in the array A000027.  See A185868.
row 1: A033816
col 1: A014105
col 2: -A168244
antidiagonal sums: A061317
antidiagonal sums: 3*(octahedral numbers) = 3*A005900.

			Northwest corner:
  3....8....17...30...47
  10...19...32...49...70
  21...34...51...72...97
  36...53...74...99...128

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000027 (as an array), A185868, A185869, A185871.

f[n_,k_]:=2n+(2n+2k-3)(n+k-1);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

from math import comb, isqrt
def A185870(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    x = n-comb(a,2)
    y = a-x+1
    return y*((y+(c:=x<<1)<<1)-5)+x*(c-3)+3 # Chai Wah Wu, Jun 18 2025

T(n,k) = 2*n + (n+k-1)*(2*n+2*k-3), k>=1, n>=1.

A235355 0 followed by the sum of (1),(2), (3,4),(5,6), (7,8,9),(10,11,12) from the natural numbers.

Original entry on oeis.org

0, 1, 2, 7, 11, 24, 33, 58, 74, 115, 140, 201, 237, 322, 371, 484, 548, 693, 774, 955, 1055, 1276, 1397, 1662, 1806, 2119, 2288, 2653, 2849, 3270, 3495, 3976, 4232, 4777, 5066, 5679, 6003, 6688, 7049, 7810, 8210, 9051, 9492, 10417, 10901, 11914, 12443, 13548

Offset: 0

Paul Curtz, Jan 07 2014

Difference table for 0 followed by a(n):
  0,  0,  1,   2,   7,  11,  24,  33,...
  0,  1,  1,   5,   4,  13,   9,  25,... =A147685(n)
  1,  0,  4,  -1,   9,  -4,  16,  -9,... =interleave A000290(n+1),-A000290(n)
  -1, 4, -5,  10, -13,  20, -25,  34,...
  5, -9, 15, -23,  33, -45,  59, -75,... =(-1)^n*A027688(n+2).
a(-n) = -a(n-1).
From the second row, signature (0,3,0,-3,0,1).
Consider a(n+2k+1)+a(2k-n):
  1,     2,   6,   9,  17,  22,  34,...
  9,    12,  24,  33,  57,  72, 108,...
  35,   40,  60,  75, 115, 140, 200,...
  91,   98, 126, 147, 203, 238, 322,...
  189, 198, 234, 261, 333, 378, 486,... .
The first column is A005898(n).
The rows are successively divisible by 2*k+1. Hence
  1,   2,  6,  9, 17, 22, 34,...
  3,   4,  8, 11, 19, 24, 36,...
  7,   8, 12, 15, 23, 28, 40,...
  13, 14, 18, 21, 29, 34, 46,...
  21, 22, 26, 29, 37, 42, 54,...
The first column is A002061(n+1).
The main diagonal is A212965(n).
The first difference of every row is A022998(n+1).
Compare to the (2k+1)-sections of A061037 in A165943.

			a(1)=1, a(2)=2, a(3)=3+4=7, a(4)=5+6=11, a(5)=7+8+9=24, a(6)=10+11+12=33.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A075356, A234305.

LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,1,2,7,11,24,33},50] (* Harvey P. Dale, Nov 22 2014 *)

Vec(x*(x^2+1)*(x^2+x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ Colin Barker, Jan 20 2014

a(n) = 4*a(n-2) -6*a(n-4) +4*a(n-6) -a(n-8), n>7.
a(2n) = 0 followed by A085786(n). a(2n+1) =  A081436(n).
a(2n) + a(2n+1) = A005898(n).
a(2n-1) + a(2n) = A061317(n).
a(n) = (-1)*((-1+(-1)^n-2*n)*(2+n+n^2))/16. a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7). G.f.: x*(x^2+1)*(x^2+x+1) / ((x-1)^4*(x+1)^3). - Colin Barker, Jan 20 2014

More terms from Colin Barker, Jan 20 2014

cp's OEIS Frontend

A006331 a(n) = n*(n+1)*(2*n+1)/3.

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A007588 Stella octangula numbers: a(n) = n*(2*n^2 - 1).

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A098547 a(n) = n^3 + n^2 + 1.

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A099721 a(n) = n^2*(2*n+1).

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A110450 a(n) = n*(n+1)*(n^2+n+1)/2.

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A185870 (Even,odd)-polka dot array in the natural number array A000027, by antidiagonals.

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A006331 a(n) = n(n+1)(2*n+1)/3.

A007588 Stella octangula numbers: a(n) = n(2n^2 - 1).

A099721 a(n) = n^2(2n+1).

A110450 a(n) = n(n+1)(n^2+n+1)/2.

A385897 a(n) = 1 - 5(n + 1)^2 + 5(n + 1)^4.