A367204 -id:A367204 - cp's OEIS frontend

A002378 Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).

0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550

Offset: 0

N. J. A. Sloane

4*a(n) + 1 are the odd squares A016754(n).
The word "pronic" (used by Dickson) is incorrect. - Michael Somos
According to the 2nd edition of Webster, the correct word is "promic". - R. K. Guy
a(n) is the number of minimal vectors in the root lattice A_n (see Conway and Sloane, p. 109).
Let M_n denote the n X n matrix M_n(i, j) = (i + j); then the characteristic polynomial of M_n is x^(n-2) * (x^2 - a(n)*x - A002415(n)). - Benoit Cloitre, Nov 09 2002
The greatest LCM of all pairs (j, k) for j < k <= n for n > 1. - Robert G. Wilson v, Jun 19 2004
First differences are a(n+1) - a(n) = 2*n + 2 = 2, 4, 6, ... (while first differences of the squares are (n+1)^2 - n^2 = 2*n + 1 = 1, 3, 5, ...). - Alexandre Wajnberg, Dec 29 2005
25 appended to these numbers corresponds to squares of numbers ending in 5 (i.e., to squares of A017329). - Lekraj Beedassy, Mar 24 2006
A rapid (mental) multiplication/factorization technique -- a generalization of Lekraj Beedassy's comment: For all bases b >= 2 and positive integers n, c, d, k with c + d = b^k, we have (n*b^k + c)*(n*b^k + d) = a(n)*b^(2*k) + c*d. Thus the last 2*k base-b digits of the product are exactly those of c*d -- including leading 0(s) as necessary -- with the preceding base-b digit(s) the same as a(n)'s. Examples: In decimal, 113*117 = 13221 (as n = 11, b = 10 = 3 + 7, k = 1, 3*7 = 21, and a(11) = 132); in octal, 61*67 = 5207 (52 is a(6) in octal). In particular, for even b = 2*m (m > 0) and c = d = m, such a product is a square of this type. Decimal factoring: 5609 is immediately seen to be 71*79. Likewise, 120099 = 301*399 (k = 2 here) and 99990000001996 = 9999002*9999998 (k = 3). - Rick L. Shepherd, Jul 24 2021
Number of circular binary words of length n + 1 having exactly one occurrence of 01. Example: a(2) = 6 because we have 001, 010, 011, 100, 101 and 110. Column 1 of A119462. - Emeric Deutsch, May 21 2006
The sequence of iterated square roots sqrt(N + sqrt(N + ...)) has for N = 1, 2, ... the limit (1 + sqrt(1 + 4*N))/2. For N = a(n) this limit is n + 1, n = 1, 2, .... For all other numbers N, N >= 1, this limit is not a natural number. Examples: n = 1, a(1) = 2: sqrt(2 + sqrt(2 + ...)) = 1 + 1 = 2; n = 2, a(2) = 6: sqrt(6 + sqrt(6 + ...)) = 1 + 2 = 3. - Wolfdieter Lang, May 05 2006
Nonsquare integers m divisible by ceiling(sqrt(m)), except for m = 0. - Max Alekseyev, Nov 27 2006
The number of off-diagonal elements of an (n + 1) X (n + 1) matrix. - Artur Jasinski, Jan 11 2007
a(n) is equal to the number of functions f:{1, 2} -> {1, 2, ..., n + 1} such that for a fixed x in {1, 2} and a fixed y in {1, 2, ..., n + 1} we have f(x) <> y. - Aleksandar M. Janjic and Milan Janjic, Mar 13 2007
Numbers m >= 0 such that round(sqrt(m+1)) - round(sqrt(m)) = 1. - Hieronymus Fischer, Aug 06 2007
Numbers m >= 0 such that ceiling(2*sqrt(m+1)) - 1 = 1 + floor(2*sqrt(m)). - Hieronymus Fischer, Aug 06 2007
Numbers m >= 0 such that fract(sqrt(m+1)) > 1/2 and fract(sqrt(m)) < 1/2 where fract(x) is the fractional part (fract(x) = x - floor(x), x >= 0). - Hieronymus Fischer, Aug 06 2007
X values of solutions to the equation 4*X^3 + X^2 = Y^2. To find Y values: b(n) = n(n+1)(2n+1). - Mohamed Bouhamida, Nov 06 2007
Nonvanishing diagonal of A132792, the infinitesimal Lah matrix, so "generalized factorials" composed of a(n) are given by the elements of the Lah matrix, unsigned A111596, e.g., a(1)*a(2)*a(3) / 3! = -A111596(4,1) = 24. - Tom Copeland, Nov 20 2007
If Y is a 2-subset of an n-set X then, for n >= 2, a(n-2) is the number of 2-subsets and 3-subsets of X having exactly one element in common with Y. - Milan Janjic, Dec 28 2007
a(n) coincides with the vertex of a parabola of even width in the Redheffer matrix, directed toward zero. An integer p is prime if and only if for all integer k, the parabola y = kx - x^2 has no integer solution with 1 < x < k when y = p; a(n) corresponds to odd k. - Reikku Kulon, Nov 30 2008
The third differences of certain values of the hypergeometric function 3F2 lead to the squares of the oblong numbers i.e., 3F2([1, n + 1, n + 1], [n + 2, n + 2], z = 1) - 3*3F2([1, n + 2, n + 2], [n + 3, n + 3], z = 1) + 3*3F2([1, n + 3, n + 3], [n + 4, n + 4], z = 1) - 3F2([1, n + 4, n + 4], [n + 5, n + 5], z = 1) = (1/((n+2)*(n+3)))^2 for n = -1, 0, 1, 2, ... . See also A162990. - Johannes W. Meijer, Jul 21 2009
Generalized factorials, [a.(n!)] = a(n)*a(n-1)*...*a(0) = A010790(n), with a(0) = 1 are related to A001263. - Tom Copeland, Sep 21 2011
For n > 1, a(n) is the number of functions f:{1, 2} -> {1, ..., n + 2} where f(1) > 1 and f(2) > 2. Note that there are n + 1 possible values for f(1) and n possible values for f(2). For example, a(3) = 12 since there are 12 functions f from {1, 2} to {1, 2, 3, 4, 5} with f(1) > 1 and f(2) > 2. - Dennis P. Walsh, Dec 24 2011
a(n) gives the number of (n + 1) X (n + 1) symmetric (0, 1)-matrices containing two ones (see [Cameron]). - L. Edson Jeffery, Feb 18 2012
a(n) is the number of positions of a domino in a rectangled triangular board with both legs equal to n + 1. - César Eliud Lozada, Sep 26 2012
a(n) is the number of ordered pairs (x, y) in [n+2] X [n+2] with |x-y| > 1. - Dennis P. Walsh, Nov 27 2012
a(n) is the number of injective functions from {1, 2} into {1, 2, ..., n + 1}. - Dennis P. Walsh, Nov 27 2012
a(n) is the sum of the positive differences of the partition parts of 2n + 2 into exactly two parts (see example). - Wesley Ivan Hurt, Jun 02 2013
a(n)/a(n-1) is asymptotic to e^(2/n). - Richard R. Forberg, Jun 22 2013
Number of positive roots in the root system of type D_{n + 1} (for n > 2). - Tom Edgar, Nov 05 2013
Number of roots in the root system of type A_n (for n > 0). - Tom Edgar, Nov 05 2013
From Felix P. Muga II, Mar 18 2014: (Start)
a(m), for m >= 1, are the only positive integer values t for which the Binet-de Moivre formula for the recurrence b(n) = b(n-1) + t*b(n-2) with b(0) = 0 and b(1) = 1 has a root of a square. PROOF (as suggested by Wolfdieter Lang, Mar 26 2014): The sqrt(1 + 4t) appearing in the zeros r1 and r2 of the characteristic equation is (a positive) integer for positive integer t precisely if 4t + 1 = (2m + 1)^2, that is t = a(m), m >= 1. Thus, the characteristic roots are integers: r1 = m + 1 and r2 = -m.
Let m > 1 be an integer. If b(n) = b(n-1) + a(m)*b(n-2), n >= 2, b(0) = 0, b(1) = 1, then lim_{n->oo} b(n+1)/b(n) = m + 1. (End)
Cf. A130534 for relations to colored forests, disposition of flags on flagpoles, and colorings of the vertices (chromatic polynomial) of the complete graphs (here simply K_2). - Tom Copeland, Apr 05 2014
The set of integers k for which k + sqrt(k + sqrt(k + sqrt(k + sqrt(k + ...) ... is an integer. - Leslie Koller, Apr 11 2014
a(n-1) is the largest number k such that (n*k)/(n+k) is an integer. - Derek Orr, May 22 2014
Number of ways to place a domino and a singleton on a strip of length n - 2. - Ralf Stephan, Jun 09 2014
With offset 1, this appears to give the maximal number of crossings between n nonconcentric circles of equal radius. - Felix Fröhlich, Jul 14 2014
For n > 1, the harmonic mean of the n values a(1) to a(n) is n + 1. The lowest infinite sequence of increasing positive integers whose cumulative harmonic mean is integral. - Ian Duff, Feb 01 2015
a(n) is the maximum number of queens of one color that can coexist without attacking one queen of the opponent's color on an (n+2) X (n+2) chessboard. The lone queen can be placed in any position on the perimeter of the board. - Bob Selcoe, Feb 07 2015
With a(0) = 1, a(n-1) is the smallest positive number not in the sequence such that Sum_{i = 1..n} 1/a(i-1) has a denominator equal to n. - Derek Orr, Jun 17 2015
The positive members of this sequence are a proper subsequence of the so-called 1-happy couple products A007969. See the W. Lang link there, eq. (4), with Y_0 = 1, with a table at the end. - Wolfdieter Lang, Sep 19 2015
For n > 0, a(n) is the reciprocal of the area bounded above by y = x^(n-1) and below by y = x^n for x in the interval [0, 1]. Summing all such areas visually demonstrates the formula below giving Sum_{n >= 1} 1/a(n) = 1. - Rick L. Shepherd, Oct 26 2015
It appears that, except for a(0) = 0, this is the set of positive integers n such that x*floor(x) = n has no solution. (For example, to get 3, take x = -3/2.) - Melvin Peralta, Apr 14 2016
If two independent real random variables, x and y, are distributed according to the same exponential distribution: pdf(x) = lambda * exp(-lambda * x), lambda > 0, then the probability that n - 1 <= x/y < n is given by 1/a(n). - Andres Cicuttin, Dec 03 2016
a(n) is equal to the sum of all possible differences between n different pairs of consecutive odd numbers (see example). - Miquel Cerda, Dec 04 2016
a(n+1) is the dimension of the space of vector fields in the plane with polynomial coefficients up to order n. - Martin Licht, Dec 04 2016
It appears that a(n) + 3 is the area of the largest possible pond in a square (A268311). - Craig Knecht, May 04 2017
Also the number of 3-cycles in the (n+3)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jul 27 2017
Also the Wiener index of the (n+2)-wheel graph. - Eric W. Weisstein, Sep 08 2017
The left edge of a Floyd's triangle that consists of even numbers: 0; 2, 4; 6, 8, 10; 12, 14, 16, 18; 20, 22, 24, 26, 28; ... giving 0, 2, 6, 12, 20, ... The right edge generates A028552. - Waldemar Puszkarz, Feb 02 2018
a(n+1) is the order of rowmotion on a poset obtained by adjoining a unique minimal (or maximal) element to a disjoint union of at least two chains of n elements. - Nick Mayers, Jun 01 2018
From Juhani Heino, Feb 05 2019: (Start)
For n > 0, 1/a(n) = n/(n+1) - (n-1)/n.
For example, 1/6 = 2/3 - 1/2; 1/12 = 3/4 - 2/3.
Corollary of this:
Take 1/2 pill.
Next day, take 1/6 pill. 1/2 + 1/6 = 2/3, so your daily average is 1/3.
Next day, take 1/12 pill. 2/3 + 1/12 = 3/4, so your daily average is 1/4.
And so on. (End)
From Bernard Schott, May 22 2020: (Start)
For an oblong number m >= 6 there exists a Euclidean division m = d*q + r with q < r < d which are in geometric progression, in this order, with a common integer ratio b. For b >= 2 and q >= 1, the Euclidean division is m = qb*(qb+1) = qb^2 * q + qb where (q, qb, qb^2) are in geometric progression.
Some examples with distinct ratios and quotients:
   6 | 4            30 | 25          42 | 18
     -----             -----            -----
   2 | 1      ,      5 |  1   ,       6 |  2    ,
and also:
  42 | 12          420 | 100
     -----             -----
   6 |  3     ,     20 |   4  .
Some oblong numbers also satisfy a Euclidean division m = d*q + r with q < r < d that are in geometric progression in this order but with a common noninteger ratio b > 1 (see A335064). (End)
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [n; {2, 2n}]. For n=1, this collapses to [1; {2}]. - Magus K. Chu, Sep 09 2022
a(n-2) is the maximum irregularity over all trees with n vertices. The extremal graphs are stars. (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
For n > 0, number of diagonals in a regular 2*(n+1)-gon that are not parallel to any edge (cf. A367204). - Paolo Xausa, Mar 30 2024
a(n-1) is the maximum Zagreb index over all trees with n vertices. The extremal graphs are stars. (The Zagreb index of a graph is the sum of the squares of the degrees over all vertices of the graph.) - Allan Bickle, Apr 11 2024
For n >= 1, a(n) is the determinant of the distance matrix of a cycle graph on 2*n + 1 vertices (if the length of the cycle is even such a determinant is zero). - Miquel A. Fiol, Aug 20 2024
For n > 1, the continued fraction expansion of sqrt(16*a(n)) is [2n+1; {1, 2n-1, 1, 8n+2}]. - Magus K. Chu, Nov 20 2024
For n>=2, a(n) is the number of faces on a n+1-zone rhombic zonohedron. Each pair of a collection of great circles on a sphere intersects at two points, so there are 2*binomial(n+1,2) intersections. The dual of the implied polyhedron is a rhombic zonohedron, its faces corresponding to the intersections. - Shel Kaphan, Aug 12 2025

			a(3) = 12, since 2(3)+2 = 8 has 4 partitions with exactly two parts: (7,1), (6,2), (5,3), (4,4). Taking the positive differences of the parts in each partition and adding, we get: 6 + 4 + 2 + 0 = 12. - _Wesley Ivan Hurt_, Jun 02 2013
G.f. = 2*x + 6*x^2 + 12*x^3 + 20*x^4 + 30*x^5 + 42*x^6 + 56*x^7 + ... - _Michael Somos_, May 22 2014
From _Miquel Cerda_, Dec 04 2016: (Start)
a(1) = 2, since 45-43 = 2;
a(2) = 6, since 47-45 = 2 and 47-43 = 4, then 2+4 = 6;
a(3) = 12, since 49-47 = 2, 49-45 = 4, and 49-43 = 6, then 2+4+6 = 12. (End)

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Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.
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Index entries for "core" sequences
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Partial sums of A005843 (even numbers). Twice triangular numbers (A000217).
1/beta(n, 2) in A061928.
Cf. A035106, A087811, A119462, A127235, A049598, A124080, A033996, A028896, A046092, A000217, A005563, A046092, A001082, A059300, A059297, A059298, A166373, A002943 (bisection), A002939 (bisection), A078358 (complement).
A036689 and A036690 are subsequences. Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488. - Bruno Berselli, Jun 10 2013
Row n=2 of A185651.
Cf. A007745, A169810, A213541, A005369 (characteristic function).
Cf. A281026. - Bruno Berselli, Jan 16 2017
Cf. A045943 (4-cycles in triangular honeycomb acute knight graph), A028896 (5-cycles), A152773 (6-cycles).
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
A335064 is a subsequence.
Second column of A003506.
Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).
Cf. A347213 (Dgf at s=4).
Cf. A002378, A152811, A371912 (Zagreb indices of maximal k-degenerate graphs).
Cf. A016742, A049450.

a002378 n = n * (n + 1)
a002378_list = zipWith (*) [0..] [1..]
-- Reinhard Zumkeller, Aug 27 2012, Oct 12 2011

[n*(n+1) : n in [0..100]]; // Wesley Ivan Hurt, Oct 26 2015

A002378 := proc(n)
        n*(n+1) ;
end proc:
seq(A002378(n),n=0..100) ;

Table[n(n + 1), {n, 0, 50}] (* Robert G. Wilson v, Jun 19 2004 *)
oblongQ[n_] := IntegerQ @ Sqrt[4 n + 1]; Select[Range[0, 2600], oblongQ] (* Robert G. Wilson v, Sep 29 2011 *)
2 Accumulate[Range[0, 50]] (* Harvey P. Dale, Nov 11 2011 *)
LinearRecurrence[{3, -3, 1}, {2, 6, 12}, {0, 20}] (* Eric W. Weisstein, Jul 27 2017 *)

PARI
```
{a(n) = n*(n+1)};
```

concat(0, Vec(2*x/(1-x)^3 + O(x^100))) \\ Altug Alkan, Oct 26 2015

is(n)=my(m=sqrtint(n)); m*(m+1)==n \\ Charles R Greathouse IV, Nov 01 2018

is(n)=issquare(4*n+1) \\ Charles R Greathouse IV, Mar 16 2022

def a(n): return n*(n+1)
print([a(n) for n in range(51)]) # Michael S. Branicky, Jan 13 2022

(2 to 100 by 2).scanLeft(0)( + ) // Alonso del Arte, Sep 12 2019

G.f.: 2*x/(1-x)^3. - Simon Plouffe in his 1992 dissertation.
a(n) = a(n-1) + 2*n, a(0) = 0.
Sum_{n >= 1} a(n) = n*(n+1)*(n+2)/3 (cf. A007290, partial sums).
Sum_{n >= 1} 1/a(n) = 1. (Cf. Tijdeman)
Sum_{n >= 1} (-1)^(n+1)/a(n) = log(4) - 1 = A016627 - 1 [Jolley eq (235)].
1 = 1/2 + Sum_{n >= 1} 1/(2*a(n)) = 1/2 + 1/4 + 1/12 + 1/24 + 1/40 + 1/60 + ... with partial sums: 1/2, 3/4, 5/6, 7/8, 9/10, 11/12, 13/14, ... - Gary W. Adamson, Jun 16 2003
a(n)*a(n+1) = a(n*(n+2)); e.g., a(3)*a(4) = 12*20 = 240 = a(3*5). - Charlie Marion, Dec 29 2003
Sum_{k = 1..n} 1/a(k) = n/(n+1). - Robert G. Wilson v, Feb 04 2005
a(n) = A046092(n)/2. - Zerinvary Lajos, Jan 08 2006
Log 2 = Sum_{n >= 0} 1/a(2n+1) = 1/2 + 1/12 + 1/30 + 1/56 + 1/90 + ... = (1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + (1/7 - 1/8) + ... = Sum_{n >= 0} (-1)^n/(n+1) = A002162. - Gary W. Adamson, Jun 22 2003
a(n) = A110660(2*n). - N. J. A. Sloane, Sep 21 2005
a(n-1) = n^2 - n = A000290(n) - A000027(n) for n >= 1. a(n) is the inverse (frequency distribution) sequence of A000194(n). - Mohammad K. Azarian, Jul 26 2007
(2, 6, 12, 20, 30, ...) = binomial transform of (2, 4, 2). - Gary W. Adamson, Nov 28 2007
a(n) = 2*Sum_{i=0..n} i = 2*A000217(n). - Artur Jasinski, Jan 09 2007, and Omar E. Pol, May 14 2008
a(n) = A006503(n) - A000292(n). - Reinhard Zumkeller, Sep 24 2008
a(n) = A061037(4*n) = (n+1/2)^2 - 1/4 = ((2n+1)^2 - 1)/4 = (A005408(n)^2 - 1)/4. - Paul Curtz, Oct 03 2008 and Klaus Purath, Jan 13 2022
a(0) = 0, a(n) = a(n-1) + 1 + floor(x), where x is the minimal positive solution to fract(sqrt(a(n-1) + 1 + x)) = 1/2. - Hieronymus Fischer, Dec 31 2008
E.g.f.: (x+2)*x*exp(x). - Geoffrey Critzer, Feb 06 2009
Product_{i >= 2} (1-1/a(i)) = -2*sin(Pi*A001622)/Pi = -2*sin(A094886)/A000796 = 2*A146481. - R. J. Mathar, Mar 12 2009, Mar 15 2009
E.g.f.: ((-x+1)*log(-x+1)+x)/x^2 also Integral_{x = 0..1} ((-x+1)*log(-x+1) + x)/x^2 = zeta(2) - 1. - Stephen Crowley, Jul 11 2009
a(A007018(n)) = A007018(n+1), i.e., A007018(n+1) = A007018(n)-th oblong numbers. - Jaroslav Krizek, Sep 13 2009
a(n) = floor((n + 1/2)^2). a(n) = A035608(n) + A004526(n+1). - Reinhard Zumkeller, Jan 27 2010
a(n) = 2*(2*A006578(n) - A035608(n)). - Reinhard Zumkeller, Feb 07 2010
a(n-1) = floor(n^5/(n^3 + n^2 + 1)). - Gary Detlefs, Feb 11 2010
For n > 1: a(n) = A173333(n+1, n-1). - Reinhard Zumkeller, Feb 19 2010
a(n) = A004202(A000217(n)). - Reinhard Zumkeller, Feb 12 2011
a(n) = A188652(2*n+1) + 1. - Reinhard Zumkeller, Apr 13 2011
For n > 0 a(n) = 1/(Integral_{x=0..Pi/2} 2*(sin(x))^(2*n-1)*(cos(x))^3). - Francesco Daddi, Aug 02 2011
a(n) = A002061(n+1) - 1. - Omar E. Pol, Oct 03 2011
a(0) = 0, a(n) = A005408(A034856(n)) - A005408(n-1). - Ivan N. Ianakiev, Dec 06 2012
a(n) = A005408(A000096(n)) - A005408(n). - Ivan N. Ianakiev, Dec 07 2012
a(n) = A001318(n) + A085787(n). - Omar E. Pol, Jan 11 2013
Sum_{n >= 1} 1/(a(n))^(2s) = Sum_{t = 1..2*s} binomial(4*s - t - 1, 2*s - 1) * ( (1 + (-1)^t)*zeta(t) - 1). See Arxiv:1301.6293. - R. J. Mathar, Feb 03 2013
a(n)^2 + a(n+1)^2 = 2 * a((n+1)^2), for n > 0. - Ivan N. Ianakiev, Apr 08 2013
a(n) = floor(n^2 * e^(1/n)) and a(n-1) = floor(n^2 / e^(1/n)). - Richard R. Forberg, Jun 22 2013
a(n) = 2*C(n+1, 2), for n >= 0. - Felix P. Muga II, Mar 11 2014
A005369(a(n)) = 1. - Reinhard Zumkeller, Jul 05 2014
Binomial transform of [0, 2, 2, 0, 0, 0, ...]. - Alois P. Heinz, Mar 10 2015
a(2n) = A002943(n) for n >= 0, a(2n-1) = A002939(n) for n >= 1. - M. F. Hasler, Oct 11 2015
For n > 0, a(n) = 1/(Integral_{x=0..1} (x^(n-1) - x^n) dx). - Rick L. Shepherd, Oct 26 2015
a(n) = A005902(n) - A007588(n). - Peter M. Chema, Jan 09 2016
For n > 0, a(n) = lim_{m -> oo} (1/m)*1/(Sum_{i=m*n..m*(n+1)} 1/i^2), with error of ~1/m. - Richard R. Forberg, Jul 27 2016
From Ilya Gutkovskiy, Jul 28 2016: (Start)
Dirichlet g.f.: zeta(s-2) + zeta(s-1).
Convolution of nonnegative integers (A001477) and constant sequence (A007395).
Sum_{n >= 0} a(n)/n! = 3*exp(1). (End)
From Charlie Marion, Mar 06 2020: (Start)
a(n)*a(n+2k-1) + (n+k)^2 = ((2n+1)*k + n^2)^2.
a(n)*a(n+2k) + k^2 = ((2n+1)*k + a(n))^2. (End)
Product_{n>=1} (1 + 1/a(n)) = cosh(sqrt(3)*Pi/2)/Pi. - Amiram Eldar, Jan 20 2021
A generalization of the Dec 29 2003 formula, a(n)*a(n+1) = a(n*(n+2)), follows. a(n)*a(n+k) = a(n*(n+k+1)) + (k-1)*n*(n+k+1). - Charlie Marion, Jan 02 2023
a(n) = A016742(n) - A049450(n). - Leo Tavares, Mar 15 2025

Additional comments from Michael Somos

Comment and cross-reference added by Christopher Hunt Gribble, Oct 13 2009

A005563 a(n) = n*(n+2) = (n+1)^2 - 1.

Original entry on oeis.org

0, 3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, 195, 224, 255, 288, 323, 360, 399, 440, 483, 528, 575, 624, 675, 728, 783, 840, 899, 960, 1023, 1088, 1155, 1224, 1295, 1368, 1443, 1520, 1599, 1680, 1763, 1848, 1935, 2024, 2115, 2208, 2303, 2400, 2499, 2600

Offset: 0

N. J. A. Sloane

Erdős conjectured that n^2 - 1 = k! has a solution if and only if n is 5, 11 or 71 (when k is 4, 5 or 7).
Second-order linear recurrences y(m) = 2y(m-1) + a(n)*y(m-2), y(0) = y(1) = 1, have closed form solutions involving only powers of integers. - Len Smiley, Dec 08 2001
Number of edges in the join of two cycle graphs, both of order n, C_n * C_n. - Roberto E. Martinez II, Jan 07 2002
Let k be a positive integer, M_n be the n X n matrix m_(i,j) = k^abs(i-j) then det(M_n) = (-1)^(n-1)*a(k-1)^(n-1). - Benoit Cloitre, May 28 2002
Also numbers k such that 4*k + 4 is a square. - Cino Hilliard, Dec 18 2003
For each term k, the function sqrt(x^2 + 1), starting with 1, produces an integer after k iterations. - Gerald McGarvey, Aug 19 2004
a(n) mod 3 = 0 if and only if n mod 3 > 0: a(A008585(n)) = 2; a(A001651(n)) = 0; a(n) mod 3 = 2*(1-A079978(n)). - Reinhard Zumkeller, Oct 16 2006
a(n) is the number of divisors of a(n+1) that are not greater than n. - Reinhard Zumkeller, Apr 09 2007
Nonnegative X values of solutions to the equation X^3 + X^2 = Y^2. To find Y values: b(n) = n(n+1)(n+2). - Mohamed Bouhamida, Nov 06 2007
Sequence allows us to find X values of the equation: X + (X + 1)^2 + (X + 2)^3 = Y^2. To prove that X = n^2 + 2n: Y^2 = X + (X + 1)^2 + (X + 2)^3 = X^3 + 7*X^2 + 15X + 9 = (X + 1)(X^2 + 6X + 9) = (X + 1)*(X + 3)^2 it means: (X + 1) must be a perfect square, so X = k^2 - 1 with k>=1. we can put: k = n + 1, which gives: X = n^2 + 2n and Y = (n + 1)(n^2 + 2n + 3). - Mohamed Bouhamida, Nov 12 2007
From R. K. Guy, Feb 01 2008: (Start)
Toads and Frogs puzzle:
This is also the number of moves that it takes n frogs to swap places with n toads on a strip of 2n + 1 squares (or positions, or lily pads) where a move is a single slide or jump, illustrated for n = 2, a(n) = 8 by
   T T - F F
   T - T F F
   T F T - F
   T F T F -
   T F - F T
   - F T F T
   F - T F T
   F F T - T
   F F - T T
I was alerted to this by the Holton article, but on consulting Singmaster's sources, I find that the puzzle goes back at least to 1867.
Probably the first to publish the number of moves for n of each animal was Edouard Lucas in 1883. (End)
a(n+1) = terms of rank 0, 1, 3, 6, 10 = A000217 of A120072 (3, 8, 5, 15). - Paul Curtz, Oct 28 2008
Row 3 of array A163280, n >= 1. - Omar E. Pol, Aug 08 2009
Final digit belongs to a periodic sequence: 0, 3, 8, 5, 4, 5, 8, 3, 0, 9. - Mohamed Bouhamida, Sep 04 2009 [Comment edited by N. J. A. Sloane, Sep 24 2009]
Let f(x) be a polynomial in x. Then f(x + n*f(x)) is congruent to 0 (mod f(x)); here n belongs to N. There is nothing interesting in the quotients f(x + n*f(x))/f(x) when x belongs to Z. However, when x is irrational these quotients consist of two parts, a) rational integers and b) integer multiples of x. The present sequence represents the non-integer part when the polynomial is x^2 + x + 1 and x = sqrt(2), f(x+n*f(x))/f(x) = A056108(n) + a(n)*sqrt(2). - A.K. Devaraj, Sep 18 2009
For n >= 1, a(n) is the number for which 1/a(n) = 0.0101... (A000035) in base (n+1). - Rick L. Shepherd, Sep 27 2009
For n > 0, continued fraction [n, 1, n] = (n+1)/a(n); e.g., [6, 1, 6] = 7/48. - Gary W. Adamson, Jul 15 2010
Starting (3, 8, 15, ...) = binomial transform of [3, 5, 2, 0, 0, 0, ...]; e.g., a(3) = 15 = (1*3 + 2*5 +1*2) = (3 + 10 + 2). - Gary W. Adamson, Jul 30 2010
a(n) is essentially the case 0 of the polygonal numbers. The polygonal numbers are defined as P_k(n) = Sum_{i=1..n} ((k-2)*i-(k-3)). Thus P_0(n) = 2*n-n^2 and a(n) = -P_0(n+2). See also A067998 and for the case k=1 A080956. - Peter Luschny, Jul 08 2011
a(n) is the maximal determinant of a 2 X 2 matrix with integer elements from {1, ..., n+1}, so the maximum determinant of a 2x2 matrix with integer elements from {1, ..., 5} = 5^2 - 1 = a(4) = 24. - Aldo González Lorenzo, Oct 12 2011
Using four consecutive triangular numbers t1, t2, t3 and t4, plot the points (0, 0), (t1, t2), and (t3, t4) to create a triangle. Twice the area of this triangle are the numbers in this sequence beginning with n = 1 to give 8. - J. M. Bergot, May 03 2012
Given a particle with spin S = n/2 (always a half-integer value), the quantum-mechanical expectation value of the square of the magnitude of its spin vector evaluates to  = S(S+1) = n(n+2)/4, i.e., one quarter of a(n) with n = 2S. This plays an important role in the theory of magnetism and magnetic resonance. - Stanislav Sykora, May 26 2012
Twice the harmonic mean [H(x, y) = (2*x*y)/(x + y)] of consecutive triangular numbers A000217(n) and A000217(n+1). - Raphie Frank, Sep 28 2012
Number m such that floor(sqrt(m)) = floor(m/floor(sqrt(m))) - 2 for m > 0. - Takumi Sato, Oct 10 2012
The solutions of equation 1/(i - sqrt(j)) = i + sqrt(j), when i = (n+1), j = a(n). For n = 1, 2 + sqrt(3) = 3.732050.. = A019973. For n = 2, 3 + sqrt(8) = 5.828427... = A156035. - Kival Ngaokrajang, Sep 07 2013
The integers in the closed form solution of a(n) = 2*a(n-1) + a(m-2)*a(n-2), n >= 2, a(0) = 0, a(1) = 1 mentioned by Len Smiley, Dec 08 2001, are m and -m + 2 where m >= 3 is a positive integer. - Felix P. Muga II, Mar 18 2014
Let m >= 3 be a positive integer. If a(n) = 2*a(n-1) + a(m-2) * a(n-2), n >= 2, a(0) = 0, a(1) = 1, then lim_{n->oo} a(n+1)/a(n) = m. - Felix P. Muga II, Mar 18 2014
For n >= 4 the Szeged index of the wheel graph W_n (with n + 1 vertices). In the Sarma et al. reference, Theorem 2.7 is incorrect. - Emeric Deutsch, Aug 07 2014
If P_{k}(n) is the n-th k-gonal number, then a(n) = t*P_{s}(n+2) - s*P_{t}(n+2) for s=t+1. - Bruno Berselli, Sep 04 2014
For n >= 1, a(n) is the dimension of the simple Lie algebra A_n. - Wolfdieter Lang, Oct 21 2015
Finding all positive integers (n, k) such that n^2 - 1 = k! is known as Brocard's problem, (see A085692). - David Covert, Jan 15 2016
For n > 0, a(n) mod (n+1) = a(n) / (n+1) = n. - Torlach Rush, Apr 04 2016
Conjecture: When using the Sieve of Eratosthenes and sieving (n+1..a(n)), with divisors (1..n) and n>0, there will be no more than a(n-1) composite numbers. - Fred Daniel Kline, Apr 08 2016
a(n) mod 8 is periodic with period 4 repeating (0,3,0,7), that is a(n) mod 8 = 5/2 - (5/2) cos(n*Pi) - sin(n*Pi/2) + sin(3*n*Pi/2). - Andres Cicuttin, Jun 02 2016
Also for n > 0, a(n) is the number of times that n-1 occurs among the first (n+1)! terms of A055881. - R. J. Cano, Dec 21 2016
The second diagonal of composites (the only prime is number 3) from the right on the Klauber triangle (see Kival Ngaokrajang link), which is formed by taking the positive integers and taking the first 1, the next 3, the following 5, and so on, each centered below the last. - Charles Kusniec, Jul 03 2017
Also the number of independent vertex sets in the n-barbell graph. - Eric W. Weisstein, Aug 16 2017
Interleaving of A000466 and A033996. - Bruce J. Nicholson, Nov 08 2019
a(n) is the number of degrees of freedom in a triangular cell for a Raviart-Thomas or Nédélec first kind finite element space of order n. - Matthew Scroggs, Apr 22 2020
From Muge Olucoglu, Jan 19 2021: (Start)
For n > 1, a(n-2) is the maximum number of elements in the second stage of the Quine-McCluskey algorithm whose minterms are not covered by the functions of n bits. At n=3, we have a(3-2) = a(1) = 1*(1+2) = 3 and f(A,B,C) = sigma(0,1,2,5,6,7).
.
          0 1 2   5 6 7
        +---------------
  *(0,1)| X X
   (0,2)| X   X
   (1,5)|   X     X
  *(2,6)|     X     X
  *(5,7)|         X   X
   (6,7)|           X X
.
*: represents the elements that are covered. (End)
1/a(n) is the ratio of the sum of the first k odd numbers and the sum of the next n*k odd numbers. - Melvin Peralta, Jul 15 2021
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [n; {1, 2n}]. - Magus K. Chu, Sep 09 2022
Number of diagonals parallel to an edge in a regular (2*n+4)-gon (cf. A367204). - Paolo Xausa, Nov 21 2023
For n >= 1, also the number of minimum cyclic edge cuts in the (n+2)-trapezohedron graph. - Eric W. Weisstein, Nov 21 2024
For n >= 1, a(n) is the sum of the interior angles of a polygon with n+2 sides, in radians, multiplied by (n+2)/Pi. - Stuart E Anderson, Aug 06 2025

			G.f. = 3*x + 8*x^2 + 15*x^3 + 24*x^4 + 35*x^5 + 48*x^6 + 63*x^7 + 80*x^8 + ...

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see index under Toads and Frogs Puzzle.
Martin Gardner, Perplexing Puzzles and Tantalizing Teasers, p. 21 (for "The Dime and Penny Switcheroo").
R. K. Guy, Unsolved Problems in Theory of Numbers, Section D25.
Derek Holton, Math in School, 37 #1 (Jan 2008) 20-22.
Edouard Lucas, Récréations Mathématiques, Gauthier-Villars, Vol. 2 (1883) 141-143.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Classes of Gap Balancing Numbers, arXiv:1810.07895 [math.NT], 2018.
Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024. See p. 22.
R. K. Guy, Letter to N. J. A. Sloane, May 1990.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
Hsien-Kuei Hwang, S. Janson, and T.-H. 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, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
Kival Ngaokrajang, Klauber triangle
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.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
K. V. S. Sarma and I. V. N. Uma, On Szeged index of standard graphs, International J. of Math. Archive, 3(8), 2012, 3129-3135. - _Emeric Deutsch_, Aug 07 2014
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Leo Tavares, Illustration: Bounded Squares.
Leo Tavares, Illustration: Trapezoids.
Leo Tavares, Illustration: Trapagons.
Eric Weisstein's World of Mathematics, Barbell Graph.
Eric Weisstein's World of Mathematics, Cyclic Edge Cut.
Eric Weisstein's World of Mathematics, Independent Vertex Set.
Eric Weisstein's World of Mathematics, Near-Square Prime.
Eric Weisstein's World of Mathematics, Trapezohedral Graph.
Wikipedia, Quine-McCluskey Algorithm.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

A column of triangle A102537.
a(n+1), n>=2, first column of triangle A120070.
Cf. A000035, A000217, A000290, A000466, A001651, A002378, A005563.
Cf. A007531, A008585, A008833, A013468, A016742, A019973, A028560.
Cf. A033996, A046092, A053186, A054000, A055881, A056108, A062196.
Cf. A068310, A067725, A067728, A067998, A069817, A079978, A080956.
Cf. A085692, A120072, A123865, A123866, A123867, A123868, A134582.
Cf. A140091, A140681, A143053, A144396, A156035, A163280, A212331.
Cf. A253607, A367204.

a005563 n = n * (n + 2)
a005563_list = zipWith (*) [0..] [2..]  -- Reinhard Zumkeller, Dec 16 2012

[n*(n+2): n in [0..60]]; // G. C. Greubel, Mar 29 2024

Table[n^2 - 1, {n, 42}] (* Zerinvary Lajos, Mar 21 2007 *)
ListCorrelate[{1, 2}, Range[-1, 50], {1, -1}, 0, Plus, Times] (* Harvey P. Dale, Aug 29 2015 *)
Range[20]^2 - 1 (* Eric W. Weisstein, Aug 16 2017 *)
Table[n (n + 2), {n, 20}] (* Eric W. Weisstein, Nov 21 2024 *)
CoefficientList[Series[(-3 + x)/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 21 2024 *)
LinearRecurrence[{3, -3, 1}, {3, 8, 15}, 20] (* Eric W. Weisstein, Nov 21 2024 *)

makelist(n*(n+2), n, 0, 56); /* Martin Ettl, Oct 15 2012 */

a(n)=n*(n+2) \\ Charles R Greathouse IV, Dec 22 2011

concat(0, Vec(x*(3-x)/(1-x)^3 + O(x^90))) \\ Altug Alkan, Oct 22 2015

[n*(n+2) for n in range(61)] # G. C. Greubel, Mar 29 2024

G.f.: x*(3-x)/(1-x)^3. - Simon Plouffe in his 1992 dissertation
a(n) = A000290(n+1) - 1.
A002378(a(n)) = A002378(n)*A002378(n+1); e.g., A002378(15)=240=12*20. - Charlie Marion, Dec 29 2003
a(n) = A067725(n)/3. - Zerinvary Lajos, Mar 06 2007
a(n) = Sum_{k=1..n} A144396(k). - Zerinvary Lajos, May 11 2007
a(n) = A134582(n+1)/4. - Zerinvary Lajos, Feb 01 2008
A143053(a(n)) = A000290(n+1), for n > 0. - Reinhard Zumkeller, Jul 20 2008
a(n) = Real((n+1+i)^2). - Gerald Hillier, Oct 12 2008
A053186(a(n)) = 2*n. - Reinhard Zumkeller, May 20 2009
a(n) = (n! + (n+1)!)/(n-1)!, n > 0. - Gary Detlefs, Aug 10 2009
a(n) = floor(n^5/(n^3+1)) with offset 1 (a(1)=0). - Gary Detlefs, Feb 11 2010
a(n) = a(n-1) + 2*n + 1 (with a(0)=0). - Vincenzo Librandi, Nov 18 2010
Sum_{n>=1} 1/a(n) = 3/4. - Mohammad K. Azarian, Dec 29 2010
a(n) = 2/(Integral_{x=0..Pi/2} (sin(x))^(n-1)*(cos(x))^3), for n > 0. - Francesco Daddi, Aug 02 2011
a(n) = A002378(n) + floor(sqrt(A002378(n))); pronic number + its root. - Fred Daniel Kline, Sep 16 2011
a(n-1) = A008833(n) * A068310(n) for n > 1. - Reinhard Zumkeller, Nov 26 2011
G.f.: U(0) where U(k) = -1 + (k+1)^2/(1 - x/(x + (k+1)^2/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 19 2012
a(n) = 15*C(n+4,3)*C(n+4,5)/(C(n+4,2)*C(n+4,4)). - Gary Detlefs, Aug 05 2013
a(n) = (n+2)!/((n-1)! + n!), n > 0. - Ivan N. Ianakiev, Nov 11 2013
a(n) = 3*C(n+1,2) - C(n,2) for n >= 0. - Felix P. Muga II, Mar 11 2014
a(n) = (A016742(n+1) - 4)/4 for n >= 0. - Felix P. Muga II, Mar 11 2014
a(-2 - n) = a(n) for all n in Z. - Michael Somos, Aug 07 2014
A253607(a(n)) = 1. - Reinhard Zumkeller, Jan 05 2015
E.g.f.: x*(x + 3)*exp(x). - Ilya Gutkovskiy, Jun 03 2016
For n >= 1, a(n^2 + n - 2) = a(n-1) * a(n). - Miko Labalan, Oct 15 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/4. - Amiram Eldar, Nov 04 2020
From Amiram Eldar, Feb 17 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = 2.
Product_{n>=1} (1 - 1/a(n)) = -sqrt(2)*sin(sqrt(2)*Pi)/Pi. (End)
a(n) = A000290(n+2) - n*2. See Bounded Squares illustration. - Leo Tavares, Oct 05 2021
From Leo Tavares, Oct 10 2021: (Start)
a(n) = A008585(n) + 2*A000217(n-1). See Trapezoids illustration.
2*A005563 = A054000(n+1). See Trapagons illustration.
a(n) = 2*A000217(n) + n. (End)
a(n) = (n+2)!!/(n-2)!! for n > 1. - Jacob Szlachetka, Jan 02 2022

Partially edited by Joerg Arndt, Mar 11 2010

More terms from N. J. A. Sloane, Aug 01 2010

cp's OEIS Frontend

A002378 Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).

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A005563 a(n) = n*(n+2) = (n+1)^2 - 1.

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A211379 Number of pairs of parallel diagonals in a regular n-gon.

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