A139570 -id:A139570 - cp's OEIS frontend

A046092 4 times triangular numbers: a(n) = 2n(n+1).

0, 4, 12, 24, 40, 60, 84, 112, 144, 180, 220, 264, 312, 364, 420, 480, 544, 612, 684, 760, 840, 924, 1012, 1104, 1200, 1300, 1404, 1512, 1624, 1740, 1860, 1984, 2112, 2244, 2380, 2520, 2664, 2812, 2964, 3120, 3280, 3444, 3612, 3784, 3960, 4140, 4324

Offset: 0

N. J. A. Sloane and Eric W. Weisstein

Consider all Pythagorean triples (X,Y,Z=Y+1) ordered by increasing Z; sequence gives Y values. X values are 1, 3, 5, 7, 9, ... (A005408), Z values are A001844.
In the triple (X, Y, Z) we have X^2=Y+Z. Actually, the triple is given by {x, (x^2 -+ 1)/2}, where x runs over the odd numbers (A005408) and x^2 over the odd squares (A016754). - Lekraj Beedassy, Jun 11 2004
a(n) is the number of edges in n X n square grid with all horizontal and vertical segments filled in. - Asher Auel, Jan 12 2000 [Corrected by Felix Huber, Apr 09 2024]
a(n) is the only number satisfying an inequality related to zeta(2) and zeta(3): Sum_{i>a(n)+1} 1/i^2 < Sum_{i>n} 1/i^3 < Sum_{i>a(n)} 1/i^2. - Benoit Cloitre, Nov 02 2001
Number of right triangles made from vertices of a regular n-gon when n is even. - Sen-Peng Eu, Apr 05 2001
Number of ways to change two non-identical letters in the word aabbccdd..., where there are n type of letters. - Zerinvary Lajos, Feb 15 2005
a(n) is the number of (n-1)-dimensional sides of an (n+1)-dimensional hypercube (e.g., squares have 4 corners, cubes have 12 edges, etc.). - Freek van Walderveen (freek_is(AT)vanwal.nl), Nov 11 2005
From Nikolaos Diamantis (nikos7am(AT)yahoo.com), May 23 2006: (Start)
Consider a triangle, a pentagon, a heptagon, ..., a k-gon where k is odd. We label a triangle with n=1, a pentagon with n=2, ..., a k-gon with n = floor(k/2). Imagine a player standing at each vertex of the k-gon.
Initially there are 2 frisbees, one held by each of two neighboring players. Every time they throw the frisbee to one of their two nearest neighbors with equal probability. Then a(n) gives the average number of steps needed so that the frisbees meet.
I verified this by simulating the processes with a computer program. For example, a(2) = 12 because in a pentagon that's the expected number of trials we need to perform. That is an exercise in Concrete Mathematics and it can be done using generating functions. (End)
A diagonal of A059056. - Zerinvary Lajos, Jun 18 2007
If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n-1) is equal to the number of 2-subsets of X containing none of X_i, (i=1,...,n). - Milan Janjic, Jul 16 2007
X values of solutions to the equation 2*X^3 + X^2 = Y^2. To find Y values: b(n) = 2n(n+1)(2n+1). - Mohamed Bouhamida, Nov 06 2007
Number of (n+1)-permutations of 3 objects u,v,w, with repetition allowed, containing n-1 u's. Example: a(1)=4 because we have vv, vw, wv and ww; a(2)=12 because we can place u in each of the previous four 2-permutations either in front, or in the middle, or at the end. - Zerinvary Lajos, Dec 27 2007
Sequence found by reading the line from 0, in the direction 0, 4, ... and the same line from 0, in the direction 0, 12, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, May 03 2008
a(n) is also the least weight of self-conjugate partitions having n different even parts. - Augustine O. Munagi, Dec 18 2008
From Peter Luschny, Jul 12 2009: (Start)
The general formula for alternating sums of powers of even integers is in terms of the Swiss-Knife polynomials P(n,x) A153641 (P(n,1)-(-1)^k P(n,2k+1))/2. Here n=2, thus
a(k) = |(P(2,1) - (-1)^k*P(2,2k+1))/2|. (End)
The sum of squares of n+1 consecutive numbers between a(n)-n and a(n) inclusive equals the sum of squares of n consecutive numbers following a(n). For example, for n = 2, a(2) = 12, and the corresponding equation is 10^2 + 11^2 + 12^2 = 13^2 + 14^2. - Tanya Khovanova, Jul 20 2009
Number of roots in the root system of type D_{n+1} (for n>2). - Tom Edgar, Nov 05 2013
Draw n ellipses in the plane (n>0), any 2 meeting in 4 points; sequence gives number of intersections of these ellipses (cf. A051890, A001844); a(n) = A051890(n+1) - 2 = A001844(n) - 1. - Jaroslav Krizek, Dec 27 2013
a(n) appears also as the second member of the quartet [p0(n), a(n), p2(n), p3(n)] of the square of [n, n+1, n+2, n+3] in the Clifford algebra Cl_2 for n >= 0. p0(n) = -A147973(n+3), p2(n) = A054000(n+1) and p3(n) = A139570(n). See a comment on A147973, also with a reference. - Wolfdieter Lang, Oct 15 2014
a(n) appears also as the third and fourth member of the quartet [p0(n), p0(n), a(n), a(n)] of the square of [n, n, n+1, n+1] in the Clifford algebra Cl_2 for n >= 0. p0(n) = A001105(n). - Wolfdieter Lang, Oct 16 2014
Consider two equal rectangles composed of unit squares. Then surround the 1st rectangle with 1-unit-wide layers to build larger rectangles, and surround the 2nd rectangle just to hide the previous layers. If r(n) and h(n) are the number of unit squares needed for n layers in the 1st case and the 2nd case, then for all rectangles, we have a(n) = r(n) - h(n) for n>=1. - Michel Marcus, Sep 28 2015
When greater than 4, a(n) is the perimeter of a Pythagorean triangle with an even short leg 2*n. - Agola Kisira Odero, Apr 26 2016
Also the number of minimum connected dominating sets in the (n+1)-cocktail party graph. - Eric W. Weisstein, Jun 29 2017
a(n+1) is the harmonic mean of A000384(n+2) and A014105(n+1). - Bob Andriesse, Apr 27 2019
Consider a circular cake from which wedges of equal center angle c are cut out in clockwise succession and turned around so that the bottom comes to the top. This goes on until the cake shows its initial surface again. An interesting case occurs if 360°/c is not an integer. Then, with n = floor(360°/c), the number of wedges which have to be cut out and turned equals a(n). (For the number of cutting line segments see A005408.) - According to Peter Winkler's book "Mathematical Mind-Benders", which presents the problem and its solution (see Winkler, pp. 111, 115) the problem seems to be of French origin but little is known about its history. - Manfred Boergens, Apr 05 2022
a(n-3) is the maximum irregularity over all maximal 2-degenerate graphs with n vertices.  The extremal graphs are 2-stars (K_2 joined to n-2 independent vertices).  (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, May 29 2023
Number of ways of placing a domino on a (n+1)X(n+1) board of squares. - R. J. Mathar, Apr 24 2024
The sequence terms are the exponents in the expansion of (1/(1 + x)) * Sum_{n >= 0} x^n * Product_{k = 1..n} (1 - x^(2*k-1))/(1 + x^(2*k+1)) = 1 - x^4 + x^12 - x^24 + x^40 - x^60 + - ... (Andrews and Berndt, Entry 9.3.3, p. 229). Cf. A153140. - Peter Bala, Feb 15 2025
Number of edges in an (n+1)-dimensional orthoplex. 2D orthoplexes (diamonds) have 4 edges, 3D orthoplexes (octahedrons) have 12 edges, 4D orthoplexes (16-cell) have 24 edges, and so on. - Aaron Franke, Mar 23 2025

			a(7)=112 because 112 = 2*7*(7+1).
The first few triples are (1,0,1), (3,4,5), (5,12,13), (7,24,25), ...
The first such partitions, corresponding to a(n)=1,2,3,4, are 2+2, 4+4+2+2, 6+6+4+4+2+2, 8+8+6+6+4+4+2+2. - _Augustine O. Munagi_, Dec 18 2008

George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook, Part I, Springer, 2005.
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.
Albert H. Beiler, Recreations in the Theory of Numbers. New York: Dover, p. 125, 1964.
Ronald L. Graham, D. E. Knuth and Oren Patashnik, Concrete Mathematics, Reading, Massachusetts: Addison-Wesley, 1994.
Peter Winkler, Mathematical Mind-Benders, Wellesley, Massachusetts: A K Peters, 2007.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
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.
H. J. Brothers, Pascal's Prism: Supplementary Material.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, preprint, 2016.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, Vol. 13, No. 4 (2017), Article #47.
Z. Janelidze, F. van Niekerk, and J. Viljoen, What is the maximal connected partial symmetry index of a connected graph of a given size?, arXiv:2502.00704 [math.CO], 2025. See p. 3.
Milan Janjic, Two Enumerative Functions
Ron Knott, Pythagorean Triples and Online Calculators
Tanya Khovanova, A Miracle Equation.
Augustine O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501. [From _Augustine O. Munagi_, Dec 18 2008]
Enrique Navarrete and Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Rusliansyah D. Suprijanto, Observation on Sums of Powers of Integers Divisible by Four, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2219 - 2226.
Leo Tavares, Illustration: Diamond Rows
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).
Eric Weisstein's World of Mathematics, Aztec Diamond.
Eric Weisstein's World of Mathematics, Cocktail Party Graph.
Eric Weisstein's World of Mathematics, Connected Dominating Set.
Eric Weisstein's World of Mathematics, Gear Graph.
Eric Weisstein's World of Mathematics, Hamiltonian Path.
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A045943, A028895, A002943, A054000, A000330, A007290, A002378, A033996, A124080, A028896, A049598, A005563, A000217, A033586, A085250.
Main diagonal of array in A001477.
Equals A033996/2. Cf. A001844. - Augustine O. Munagi, Dec 18 2008
Cf. A078371, A141530 (see Librandi's comment in A078371).
Cf. A097080, A001845.
Cf. similar sequences listed in A299645.
Cf. A005408.
Cf. A016754.
Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).

a046092 = (* 2) . a002378  -- Reinhard Zumkeller, Dec 15 2013

[2*n*(n+1): n in [0..50]]; // Vincenzo Librandi, Oct 04 2011

Table[2 n (n + 1), {n, 0, 50}] (* Stefan Steinerberger, Apr 03 2006 *)
LinearRecurrence[{3, -3, 1}, {0, 4, 12}, 50] (* Harvey P. Dale, Jul 25 2011 *)
4*Binomial[Range[50], 2] (* Harvey P. Dale, Jul 25 2011 *)

A046092(n):=2*n*(n+1)$
makelist(A046092(n),n,0,30); /* Martin Ettl, Nov 08 2012 */

a(n)=binomial(n+1,2)<<2 \\ Charles R Greathouse IV, Jun 10 2011

def A046092(n): return n*(n+1)<<1 # Chai Wah Wu, Mar 11 2025

a(n) = A100345(n+1, n-1) for n>0.
a(n) = 2*A002378(n) = 4*A000217(n). - Lekraj Beedassy, May 25 2004
a(n) = C(2n, 2) - n = 4*C(n, 2). - Zerinvary Lajos, Feb 15 2005
From Lekraj Beedassy, Jun 04 2006: (Start)
a(n) - a(n-1)=4*n.
Let k=a(n). Then a(n+1) = k + 2*(1 + sqrt(2k + 1)). (End)
Array read by rows: row n gives A033586(n), A085250(n+1). - Omar E. Pol, May 03 2008
O.g.f.:4*x/(1-x)^3; e.g.f.: exp(x)*(2*x^2+4*x). - Geoffrey Critzer, May 17 2009
From Stephen Crowley, Jul 26 2009: (Start)
a(n) = 1/int(-(x*n+x-1)*(step((-1+x*n)/n)-1)*n*step((x*n+x-1)/(n+1)),x=0..1) where step(x)=piecewise(x<0,0,0<=x,1) is the Heaviside step function.
Sum_{n>=1} 1/a(n) = 1/2. (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=4, a(2)=12. - Harvey P. Dale, Jul 25 2011
For n > 0, a(n) = 1/(Integral_{x=0..Pi/2} (sin(x))^(2*n-1)*(cos(x))^3). - Francesco Daddi, Aug 02 2011
a(n) = A001844(n) - 1. - Omar E. Pol, Oct 03 2011
(a(n) - A000217(k))^2  = A000217(2n-k)*A000217(2n+1+k) - (A002378(n) - A000217(k)), for all k. See also A001105. - Charlie Marion, May 09 2013
From Ivan N. Ianakiev, Aug 30 2013: (Start)
a(n)*(2m+1)^2 + a(m) = a(n*(2m+1)+m), for any nonnegative integers n and m.
t(k)*a(n) + t(k-1)*a(n+1) = a((n+1)*(t(k)-t(k-1)-1)), where k>=2, n>=1, t(k)=A000217(k). (End)
a(n) = A245300(n,n). - Reinhard Zumkeller, Jul 17 2014
2*a(n)+1 = A016754(n) = A005408(n)^2, the odd squares. - M. F. Hasler, Oct 02 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) - 1/2 = A187832. - Ilya Gutkovskiy, Mar 16 2017
a(n) = lcm(2*n,2*n+2). - Enrique Navarrete, Aug 30 2017
a(n)*a(n+k) + k^2 = m^2 (a perfect square), n >= 1, k >= 0. - Ezhilarasu Velayutham, May 13 2019
From Amiram Eldar, Jan 29 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = cosh(Pi/2)/(Pi/2).
Product_{n>=1} (1 - 1/a(n)) = -2*cos(sqrt(3)*Pi/2)/Pi. (End)
a(n) = A016754(n) - A001844(n). - Leo Tavares, Sep 20 2022

A054000 a(n) = 2*n^2 - 2.

Original entry on oeis.org

0, 6, 16, 30, 48, 70, 96, 126, 160, 198, 240, 286, 336, 390, 448, 510, 576, 646, 720, 798, 880, 966, 1056, 1150, 1248, 1350, 1456, 1566, 1680, 1798, 1920, 2046, 2176, 2310, 2448, 2590, 2736, 2886, 3040, 3198, 3360, 3526, 3696, 3870, 4048, 4230, 4416

Offset: 1

Asher Auel, Jan 12 2000

a(n) is the number of edges in (n+1) X (n+1) square grid with all horizontal, vertical and great diagonal segments filled in.
Nonnegative X values of integer solutions to the equation 2*X^3 + 4*X^2 = Y^2. To find Y values: b(n) = 2*n*(2*n^2 - 2). - Mohamed Bouhamida, Nov 06 2007
Second term of an arithmetic progression of 5 numbers with common difference 2n+1. The sum of squares of such 5 terms equals the sum of squares of 5 consecutive numbers starting a(n) + 2n + 1. - Carmine Suriano, Oct 16 2013
For m > 2, a(m-1) = 2*m*(m-2) is the number of Hamiltonian circuits on an m-gonal bipyramid with labeled vertices. - Stanislav Sykora, Jul 22 2014
a(n+1), n >= 0, appears also as the third member of the quartet [p0(n), p1(n), a(n+1), p3(n)] of the square of [n, n+1, n+2, n+3] in the Clifford algebra Cl_2 for n >= 0. p0(n) = -A147973(n+3), p1(n) = A046092(n) and p3(n) = A139570(n). See a comment on A147973, also with a reference. - Wolfdieter Lang, Oct 15 2014
From Bui Quang Tuan, Mar 31 2015: (Start)
For n >= 2, a(n) is the total sum of all numbers on the perimeter of a square consisting of n columns, each of which contains n numbers 1, 2, 3, ..., n.
Here is an example with n = 5:
  1 1 1 1 1
  2 2 2 2 2
  3 3 3 3 3
  4 4 4 4 4
  5 5 5 5 5
where 1+1+1+1+1 + 2+2 + 3+3 + 4+4 + 5+5+5+5+5 = 48 = a(5).
(End)
Nonnegative k such that k/2+1 is a square. - Bruno Berselli, Apr 10 2018

			For n=5, a(5)=48 and 37^2 + 48^2 + 59^2 + 70^2 + 81^2 = 59^2 + 60^2 + 61^2 + 62^2 + 63^2. - _Carmine Suriano_, Oct 16 2013

G. C. Greubel, Table of n, a(n) for n = 1..5000
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

a(n) = A100345(n+1, n-4) for n>2.

Cf. A000217, A001082, A002378, A002943, A005563, A028347, A036666, A046092, A056220, A062717, A067725, A087475.

Maple
```
[ seq(2*n^2 - 2, n=1..60) ];
```

2 Range[50]^2 - 2 (* or *) LinearRecurrence[{3, -3, 1}, {0, 6, 16}, 50] (* Harvey P. Dale, Feb 03 2012 *)
CoefficientList[Series[2 x (3 - x) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Apr 01 2015 *)

a(n)=2*n^2-2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 4*n + a(n-1) - 2, with n>1, a(1)=0. - Vincenzo Librandi, Aug 06 2010
a(1)=0, a(2)=6, a(3)=16; for n>3, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Feb 03 2012
a(n) = (n+i)^2 + (n-i)^2, where i=sqrt(-1). - Bruno Berselli, Jan 23 2014
a(n) = 1*A000290(n-1) + 2*A000217(n-1) + 3*A001477(n-1). - J. M. Bergot, Apr 23 2014
G.f.: 2*x^2*(3 - x)/(1 - x)^3. - Vincenzo Librandi, Apr 01 2015
E.g.f.: 2*(x^2 + x -1)*exp(x) + 2. - G. C. Greubel, Jul 13 2017
a(n) + a(n+2) = A005843(n+1)^2. - Ezhilarasu Velayutham, May 30 2019
From Amiram Eldar, Dec 09 2021: (Start)
Sum_{n>=2} 1/a(n) = 3/8.
Sum_{n>=2} (-1)^n/a(n) = 1/8. (End)

A147973 a(n) = -2n^2 + 12n - 14.

Original entry on oeis.org

-4, 2, 4, 2, -4, -14, -28, -46, -68, -94, -124, -158, -196, -238, -284, -334, -388, -446, -508, -574, -644, -718, -796, -878, -964, -1054, -1148, -1246, -1348, -1454, -1564, -1678, -1796, -1918, -2044, -2174, -2308, -2446, -2588, -2734, -2884, -3038, -3196, -3358

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 18 2008

-a(n+3) = 2*n^2 - 4, n >= 0, [-4,-2, 4, 14, ...] appears as the first member of the quartet for the square of [n, n+1, n+2, n+3], for n >= 0, in the Clifford algebra Cl_2. The other members are given in A046092(n), A054000(n+1) and A139570(n). The basis of Cl_2 is <1, s1, s2, s12> with s1.s1 = s2.s2 = 1, s12.s12 = -1, s1.s2 = -s2.s1 = s12. See e.g., pp. 5-6, eqs. (2.4)-(2.13) of the S. Gull et al. reference. - Wolfdieter Lang, Oct 15 2014
Related to the previous comment: if one uses the exterior (Grassmann) product with s1.s1 = s2.s2 = s12.s12 = 0 and s1.s2 = -s2.s1 = s12, then the four components of the square of [n, n+1, n+2, n+3] are [A000290(n), A046092(n), A054000(n+1), A139570(n)], n >= 0. - Wolfdieter Lang, Nov 13 2014
2 - a(n)/2 is a square. - Bruno Berselli, Apr 10 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
S. Gull, A. Lasenby and C. Doran, Imaginary Numbers are not Real - the Geometric Algebra of Spacetime, Found. Phys., Vol. 23(9) (1993), pp. 1175-1201.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A008865, A046092, A054000, A139570.

[-2*n^2+12*n-14: n in [1..50]]; // Vincenzo Librandi, Jul 10 2012

[-2*n^2+12*n-14$n=1..50]; # Muniru A Asiru, Feb 12 2019

lst={};Do[k=n^2-((n-1)^2+(n-2)^2+(n-3)^2);AppendTo[lst,k],{n,5!}];lst
Table[-2n^2+12n-14,{n,1,50}] (* Vincenzo Librandi, Jul 10 2012 *)
LinearRecurrence[{3,-3,1},{-4,2,4},50] (* Harvey P. Dale, Mar 02 2020 *)

a(n)=-2*n^2+12*n-14 \\ Charles R Greathouse IV, Sep 24 2015

Vec(-2*x*(2 - 7*x + 7*x^2) / (1 - x)^3 + O(x^40)) \\ Colin Barker, Feb 12 2019

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 10 2012
a(n) = -2*A008865(n-3). - J. M. Bergot, Jun 25 2018
G.f.: -2*x*(2 - 7*x + 7*x^2)/(1 - x)^3. - Colin Barker, Feb 12 2019
E.g.f.: -2*(exp(x)*(x^2 - 5*x + 7) - 7). - Elmo R. Oliveira, Nov 17 2024

A059993 Pinwheel numbers: a(n) = 2n^2 + 6n + 1.

Original entry on oeis.org

1, 9, 21, 37, 57, 81, 109, 141, 177, 217, 261, 309, 361, 417, 477, 541, 609, 681, 757, 837, 921, 1009, 1101, 1197, 1297, 1401, 1509, 1621, 1737, 1857, 1981, 2109, 2241, 2377, 2517, 2661, 2809, 2961, 3117, 3277, 3441, 3609, 3781, 3957, 4137, 4321, 4509, 4701, 4897

Offset: 0

Naohiro Nomoto, Mar 14 2001

Nonnegative integers m such that 2*m + 7 is a square. - Vincenzo Librandi, Mar 01 2013
Numbers of the form 4*(h+1)*(2*h-1) + 1, where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... . - Bruno Berselli, Feb 03 2017
a(n) is also the number of vertices of the Aztec diamond AZ(n) (see Lemma 2.1 of the Imran et al. paper). - Emeric Deutsch, Sep 23 2017

M. Imran and S. Hayat, On computation of topological indices of Aztec diamonds, Sci. Int. (Lahore), Vol. 26(4), 2014, pp. 1407-1412. - Emeric Deutsch, Sep 23 2017

Harry J. Smith, Table of n, a(n) for n = 0..1000
Author?, figure. [Wayback Machine link]
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. numbers n such that 2*n + 2*k + 1 is a square: A046092 (k=0), A142463 (k=1), A090288 (k=2), this sequence (k=3), A139570 (k=4), A222182 (k=5), A181510 (k=6).

[2*n^2+6*n+1: n in [0..50]]; // Vincenzo Librandi, Mar 01 2013

I:=[1,9]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2)+4: n in [1..50]]; // Vincenzo Librandi, Mar 01 2013

Table[2 n^2 + 6 n + 1, {n, 0, 46}] (* Zerinvary Lajos, Jul 10 2009 *)
LinearRecurrence[{3,-3,1},{1,9,21},50] (* Harvey P. Dale, Oct 01 2018 *)

a(n) = { 2*n^2 + 6*n + 1 } \\ Harry J. Smith, Jul 01 2009

a(n) = 4*n + a(n-1) + 4 for n > 0, a(0)=1. - Vincenzo Librandi, Aug 07 2010
G.f.: (1 + 6*x - 3*x^2)/(1-x)^3. - Arkadiusz Wesolowski, Dec 24 2011
a(n) = 2*a(n-1) - a(n-2) + 4. - Vincenzo Librandi, Mar 01 2013
a(n) = Hyper2F1([-2, n], [1], -2). - Peter Luschny, Aug 02 2014
Sum_{n>=0} 1/a(n) = 1/3 + Pi*tan(sqrt(7)*Pi/2)/(2*sqrt(7)). - Amiram Eldar, Dec 13 2022
From Elmo R. Oliveira, Nov 16 2024: (Start)
E.g.f.: exp(x)*(1 + 8*x + 2*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A222182 Numbers m such that 2*m + 11 is a square.

Original entry on oeis.org

-5, -1, 7, 19, 35, 55, 79, 107, 139, 175, 215, 259, 307, 359, 415, 475, 539, 607, 679, 755, 835, 919, 1007, 1099, 1195, 1295, 1399, 1507, 1619, 1735, 1855, 1979, 2107, 2239, 2375, 2515, 2659, 2807, 2959, 3115, 3275, 3439, 3607, 3779, 3955, 4135, 4319, 4507, 4699

Offset: 1

Bruno Berselli, Mar 01 2013

Except the first term, main diagonal of A155546. - Vincenzo Librandi, Mar 04 2013

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. numbers n such that 2*n + 2*k + 1 is a square: A046092 (k=0), A142463 (k=1), A090288 (k=2), A059993 (k=3), A139570 (k=4), this sequence (k=5), A181510 (k=6).
Cf. A005408 (square roots of 2*a(n)+11), A155546.
After a(2), subsequence of A168489.

[m: m in [-5..5000] | IsSquare(2*m+11)];

I:=[-5,-1,7]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Mar 04 2013

Mathematica
```
Table[2 n^2 - 2 n - 5, {n, 50}]
```

makelist(coeff(taylor(-(5-14*x+5*x^2)/(1-x)^3, x, 0, n), x, n), n, 0, 50);

a(n)=2*n^2-2*n-5 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: -x*(5 - 14*x + 5*x^2)/(1-x)^3.
a(n) = a(-n+1) = 2*n^2 - 2*n - 5.
a(n) = A046092(n-1) - 5.
Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(11)*Pi/2)/(2*sqrt(11)). - Amiram Eldar, Dec 23 2022
From Elmo R. Oliveira, Nov 17 2024: (Start)
E.g.f.: exp(x)*(2*x^2 - 5) + 5.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A268581 a(n) = 2n^2 + 8n + 5.

Original entry on oeis.org

5, 15, 29, 47, 69, 95, 125, 159, 197, 239, 285, 335, 389, 447, 509, 575, 645, 719, 797, 879, 965, 1055, 1149, 1247, 1349, 1455, 1565, 1679, 1797, 1919, 2045, 2175, 2309, 2447, 2589, 2735, 2885, 3039, 3197, 3359, 3525, 3695, 3869, 4047, 4229, 4415, 4605

Offset: 0

Juri-Stepan Gerasimov, Apr 10 2016

Also, numbers m such that 2*m + 6 is a square.

All the terms end with a digit in {5, 7, 9}, or equivalently, are congruent to {5, 7, 9} mod 10. - Stefano Spezia, Aug 05 2021

Stefano Spezia, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. numbers n such that 2*n + k is a perfect square: A093328 (k=-6), A097080 (k=-5), no sequence (k=-4), A051890 (k=-3), A058331 (k=-2), A001844 (k=-1), A001105 (k=0), A046092 (k=1), A056222 (k=2), A142463 (k=3), A054000 (k=4), A090288 (k=5), this sequence (k=6), A059993 (k=7), A147973 (k=8), A139570 (k=9), no sequence (k=10), A222182 (k=11), A152811 (k=12), A181570 (k=13).

Magma
```
[2*n^2+8*n+5: n in [0..60]];
```
Magma
```
[n: n in [0..6000] | IsSquare(2*n+6)];
```

Table[2 n^2 + 8 n + 5, {n, 0, 50}] (* Vincenzo Librandi, Apr 13 2016 *)
LinearRecurrence[{3,-3,1},{5,15,29},50] (* Harvey P. Dale, Jan 18 2017 *)

lista(nn) = for(n=0, nn, print1(2*n^2+8*n+5, ", ")); \\ Altug Alkan, Apr 10 2016

[2*n^2 + 8*n + 5 for n in [0..46]] # Stefano Spezia, Aug 04 2021

From Vincenzo Librandi, Apr 13 2016: (Start)
G.f.: (5-x^2)/(1-x)^3.
a(n) = 2*(n+2)^2 - 3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
E.g.f.: exp(x)*(5 + 10*x + 2*x^2). - Stefano Spezia, Aug 03 2021

Changed offset from 1 to 0, adapted formulas and programs by Bruno Berselli, Apr 13 2016

A271625 a(n) = = 2*(n+1)^2 - 5.

Original entry on oeis.org

3, 13, 27, 45, 67, 93, 123, 157, 195, 237, 283, 333, 387, 445, 507, 573, 643, 717, 795, 877, 963, 1053, 1147, 1245, 1347, 1453, 1563, 1677, 1795, 1917, 2043, 2173, 2307, 2445, 2587, 2733, 2883, 3037, 3195, 3357, 3523, 3693, 3867, 4045, 4227, 4413, 4603, 4797, 4995, 5197, 5403, 5613, 5827

Offset: 1

Juri-Stepan Gerasimov, Apr 11 2016

Numbers n such that 2*n + 10 is a perfect square.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A201713.

Numbers h such that 2*h + k is a perfect square: A294774 (k=-9), A255843 (k=-8), A271649 (k=-7), A093328 (k=-6), A097080 (k=-5), A271624 (k=-4), A051890 (k=-3), A058331 (k=-2), A001844 (k=-1), A001105 (k=0), A046092 (k=1), A056222 (k=2), A142463 (k=3), A054000 (k=4), A090288 (k=5), A268581 (k=6), A059993 (k=7), (-1)*A147973 (k=8), A139570 (k=9), this sequence (k=10), A222182 (k=11), A152811 (k=12), A181510 (k=13), A161532 (k=14), no sequence (k=15).

Magma
```
[ 2*n^2 + 4*n - 3: n in [1..60]];
```

[ n: n in [1..6000] | IsSquare(2*n+10)];

Table[2 n^2 + 4 n - 3, {n, 53}] (* Michael De Vlieger, Apr 11 2016 *)
LinearRecurrence[{3,-3,1},{3,13,27},60] (* Harvey P. Dale, Jun 08 2023 *)
2*Range[2,60]^2 -5 (* G. C. Greubel, Jan 21 2025 *)

x='x+O('x^99); Vec(x*(3+4*x-3*x^2)/(1-x)^3) \\ Altug Alkan, Apr 11 2016

def A271625(n): return 2*pow(n+1,2) - 5
print([A271625(n) for n in range(1,61)]) # G. C. Greubel, Jan 21 2025

G.f.: x*(3 + 4*x - 3*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Apr 11 2016
Sum_{n>=1} 1/a(n) = 13/30 - Pi*cot(sqrt(5/2)*Pi)/(2*sqrt(10)) = 0.5627678459924... . - Vaclav Kotesovec, Apr 11 2016
From Elmo R. Oliveira, Nov 17 2024: (Start)
E.g.f.: exp(x)*(2*x^2 + 6*x - 3) + 3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
a(n) = 2*A000290(n+1) - 5. - G. C. Greubel, Jan 21 2025

Name simplified by G. C. Greubel, Jan 21 2025

A155724 Triangle read by rows: T(n, k) = 2nk + n + k - 4.

Original entry on oeis.org

0, 3, 8, 6, 13, 20, 9, 18, 27, 36, 12, 23, 34, 45, 56, 15, 28, 41, 54, 67, 80, 18, 33, 48, 63, 78, 93, 108, 21, 38, 55, 72, 89, 106, 123, 140, 24, 43, 62, 81, 100, 119, 138, 157, 176, 27, 48, 69, 90, 111, 132, 153, 174, 195, 216, 30, 53, 76, 99, 122, 145, 168, 191, 214, 237, 260

Offset: 1

Vincenzo Librandi, Jan 25 2009

			Triangle begins:
   0;
   3,  8;
   6, 13, 20;
   9, 18, 27, 36;
  12, 23, 34, 45,  56;
  15, 28, 41, 54,  67,  80;
  18, 33, 48, 63,  78,  93, 108;
  21, 38, 55, 72,  89, 106, 123, 140;
  24, 43, 62, 81, 100, 119, 138, 157, 176;
  27, 48, 69, 90, 111, 132, 153, 174, 195, 216;

Vincenzo Librandi, Rows n = 1..100, flattened

All terms are in A155723.
Cf. A008585, A016885, A017053, A020705, A154685, A155722.
Cf. A162261 (row sums).
Columns k: A008585 (k=1), A016885 (k=2), A017053 (k=3), 9*A020705 (k=4).
Diagonals include: A139570, A181510, A271625.

/* Triangle: */ [[2*m*n+m+n-4: m in [1..n]]: n in [1..10]]; // Bruno Berselli, Aug 31 2012

Flatten[Table[2 n m + m + n - 4, {n, 10}, {m, n}]] (* Vincenzo Librandi, Mar 01 2012 *)

def A155724(n,k): return 2*n*k+n+k-4
print(flatten([[A155724(n,k) for k in range(1,n+1)] for n in range(1,16)])) # G. C. Greubel, Jan 21 2025

T(n, k) = A154685(n, k) - 8. - L. Edson Jeffery, Oct 12 2012
2*T(n, k) + 9 = (2*k+1)*(2*n+1). - Vincenzo Librandi, Nov 18 2012
From G. C. Greubel, Jan 21 2025: (Start)
T(2*n-1, n) = 4*n^2 + n - 5 (main diagonal).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (1/4)*( 4*(-1)^(n+1)*n^2 + 2*(2-3*(-1)^n)*n - 7*(1-(-1)^n)).
G.f.: x*y*(3*x + 3*y - 4*x*y)/((1-x)*(1-y))^2. (End)

Edited by N. J. A. Sloane, Jun 23 2010

A271624 a(n) = 2n^2 - 4n + 4.

Original entry on oeis.org

2, 4, 10, 20, 34, 52, 74, 100, 130, 164, 202, 244, 290, 340, 394, 452, 514, 580, 650, 724, 802, 884, 970, 1060, 1154, 1252, 1354, 1460, 1570, 1684, 1802, 1924, 2050, 2180, 2314, 2452, 2594, 2740, 2890, 3044, 3202, 3364, 3530, 3700, 3874, 4052, 4234, 4420, 4610, 4804, 5002, 5204, 5410, 5620

Offset: 1

Juri-Stepan Gerasimov, Apr 11 2016

Numbers n such that 2*n - 4 is a perfect square.

For n > 2, the number of square a(n)-gonal numbers is finite. - Muniru A Asiru, Oct 16 2016

			a(1) = 2*1^2 - 4*1 + 4 = 2.

Muniru A Asiru, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002522, numbers n such that 2*n + k is a perfect square: no sequence (k = -9), A255843 (k = -8), A271649 (k = -7), A093328 (k = -6), A097080 (k = -5), this sequence (k = -4), A051890 (k = -3), A058331 (k = -2), A001844 (k = -1), A001105 (k = 0), A046092 (k = 1), A056222 (k = 2), A142463 (k = 3), A054000 (k = 4), A090288 (k = 5), A268581 (k = 6), A059993 (k = 7), (-1)*A147973 (k = 8), A139570 (k = 9), A271625 (k = 10), A222182 (k = 11), A152811 (k = 12), A181510 (k = 13), A161532 (k = 14), no sequence (k = 15).

Cf. A005893, A160457.

Magma
```
[ 2*n^2 - 4*n + 4: n in [1..60]];
```

[ n: n in [1..6000] | IsSquare(2*n-4)];

Table[2 n^2 - 4 n + 4, {n, 54}] (* Michael De Vlieger, Apr 11 2016 *)
LinearRecurrence[{3,-3,1},{2,4,10},60] (* Harvey P. Dale, Jul 18 2023 *)

x='x+O('x^99); Vec(2*x*(1-x+2*x^2)/(1-x)^3) \\ Altug Alkan, Apr 11 2016

a(n)=2*n^2-4*n+4 \\ Charles R Greathouse IV, Apr 11 2016

a(n) = 2*A002522(n-1).
G.f.: 2*x*(1 - x + 2*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Apr 11 2016
Sum_{n>=1} 1/a(n) = (1 + Pi*coth(Pi))/4 = 1.038337023734290587067... . - Vaclav Kotesovec, Apr 11 2016
a(n) = A005893(n-1), n > 1. - R. J. Mathar, Apr 12 2016
a(n) = 2 + 2*(n-1)^2. - Tyler Skywalker, Jul 21 2016
From Elmo R. Oliveira, Nov 17 2024: (Start)
E.g.f.: 2*(exp(x)*(x^2 - x + 2) - 2).
a(n) = 2*A160457(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A245300 Triangle T(n,k) = (n+k)*(n+k+1)/2 + k, 0 <= k <= n, read by rows.

Original entry on oeis.org

0, 1, 4, 3, 7, 12, 6, 11, 17, 24, 10, 16, 23, 31, 40, 15, 22, 30, 39, 49, 60, 21, 29, 38, 48, 59, 71, 84, 28, 37, 47, 58, 70, 83, 97, 112, 36, 46, 57, 69, 82, 96, 111, 127, 144, 45, 56, 68, 81, 95, 110, 126, 143, 161, 180, 55, 67, 80, 94, 109, 125, 142, 160, 179, 199, 220

Offset: 0

Reinhard Zumkeller, Jul 17 2014

			First rows and their row sums (A245301):
   0                                                                  0;
   1,  4                                                              5;
   3,  7,  12                                                        22;
   6, 11,  17,  24                                                   58;
  10, 16,  23,  31,  40                                             120;
  15, 22,  30,  39,  49,  60                                        215;
  21, 29,  38,  48,  59,  71,  84                                   350;
  28, 37,  47,  58,  70,  83,  97, 112                              532;
  36, 46,  57,  69,  82,  96, 111, 127, 144                         768;
  45, 56,  68,  81,  95, 110, 126, 143, 161, 180                   1065;
  55, 67,  80,  94, 109, 125, 142, 160, 179, 199, 220              1430;
  66, 79,  93, 108, 124, 141, 159, 178, 198, 219, 241, 264         1870;
  78, 92, 107, 123, 140, 158, 177, 197, 218, 240, 263, 287, 312    2392.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A000217, A046092, A062725, A245301.

a245300 n k = (n + k) * (n + k + 1) `div` 2 + k
a245300_row n = map (a245300 n) [0..n]
a245300_tabl = map a245300_row [0..]
a245300_list = concat a245300_tabl

[k + Binomial(n+k+1,2): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 01 2021

Table[k + Binomial[n+k+1,2], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 01 2021 *)

flatten([[k + binomial(n+k+1,2) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 01 2021

T(n, 0) = A000217(n).
T(n, n) = A046092(n).
T(2*n, n) = A062725(n) (central terms).
Sum_{k=0..n} T(n, k) = A245301(n).
From G. C. Greubel, Apr 01 2021: (Start)
T(n, 1) = A000124(n+1) = A134869(n+1), n >= 1.
T(n, 2) = A152948(n+4), n >= 2.
T(n, 3) = A152950(n+4), n >= 3.
T(n, 4) = A145018(n+5), n >= 4.
T(n, 5) = A167499(n+4), n >= 5.
T(n, 6) = A166136(n+5), n >= 6.
T(n, 7) = A167487(n+6), n >= 7.
T(n, n-1) = A056220(n), n >= 1.
T(n, n-2) = A142463(n-1), n >= 2.
T(n, n-3) = A054000(n-1), n >= 3.
T(n, n-4) = A090288(n-3), n >= 4.
T(n, n-5) = A268581(n-4), n >= 5.
T(n, n-6) = A059993(n-4), n >= 6.
T(n, n-7) = (-1)*A147973(n), n >= 7.
T(n, n-8) = A139570(n-5), n >= 8.
T(n, n-9) = A271625(n-5), n >= 9.
T(n, n-10) = A222182(n-4), n >= 10.
T(2*n, n-1) = A081270(n-1), n >= 1.
T(2*n, n+1) = A117625(n+1), n >= 1. (End)

cp's OEIS Frontend

A046092 4 times triangular numbers: a(n) = 2*n*(n+1).

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A054000 a(n) = 2*n^2 - 2.

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A147973 a(n) = -2*n^2 + 12*n - 14.

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A059993 Pinwheel numbers: a(n) = 2*n^2 + 6*n + 1.

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A222182 Numbers m such that 2*m + 11 is a square.

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Magma

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Maxima

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A268581 a(n) = 2*n^2 + 8*n + 5.

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A046092 4 times triangular numbers: a(n) = 2n(n+1).

A147973 a(n) = -2n^2 + 12n - 14.

A059993 Pinwheel numbers: a(n) = 2n^2 + 6n + 1.

A268581 a(n) = 2n^2 + 8n + 5.

A155724 Triangle read by rows: T(n, k) = 2nk + n + k - 4.

A271624 a(n) = 2n^2 - 4n + 4.