A110660 -id:A110660 - 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
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

A109613 Odd numbers repeated.

Original entry on oeis.org

1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, 17, 19, 19, 21, 21, 23, 23, 25, 25, 27, 27, 29, 29, 31, 31, 33, 33, 35, 35, 37, 37, 39, 39, 41, 41, 43, 43, 45, 45, 47, 47, 49, 49, 51, 51, 53, 53, 55, 55, 57, 57, 59, 59, 61, 61, 63, 63, 65, 65, 67, 67, 69, 69, 71, 71, 73

Offset: 0

Reinhard Zumkeller, Aug 01 2005

The number of rounds in a round-robin tournament with n competitors. - A. Timothy Royappa, Aug 13 2011
Diagonal sums of number triangle A113126. - Paul Barry, Oct 14 2005
When partitioning a convex n-gon by all the diagonals, the maximum number of sides in resulting polygons is 2*floor(n/2)+1 = a(n-1) (from Moscow Olympiad problem 1950). - Tanya Khovanova, Apr 06 2008
The inverse values of the coefficients in the series expansion of f(x) = (1/2)*(1+x)*log((1+x)/(1-x)) lead to this sequence; cf. A098557. - Johannes W. Meijer, Nov 12 2009
From Reinhard Zumkeller, Dec 05 2009: (Start)
First differences: A010673; partial sums: A000982;
A059329(n) = Sum_{k = 0..n} a(k)*a(n-k);
A167875(n) = Sum_{k = 0..n} a(k)*A005408(n-k);
A171218(n) = Sum_{k = 0..n} a(k)*A005843(n-k);
A008794(n+2) = Sum_{k = 0..n} a(k)*A059841(n-k). (End)
Dimension of the space of weight 2n+4 cusp forms for Gamma_0(5). - Michael Somos, May 29 2013
For n > 4: a(n) = A230584(n) - A230584(n-2). - Reinhard Zumkeller, Feb 10 2015
The arithmetic function v+-(n,2) as defined in A290988. - Robert Price, Aug 22 2017
For n > 0, also the chromatic number of the (n+1)-triangular (Johnson) graph. - Eric W. Weisstein, Nov 17 2017
a(n-1), for n >= 1, is also the upper bound a_{up}(b), where b = 2*n + 1, in the first (top) row of the complete coach system Sigma(b) of Hilton and Pedersen [H-P]. All odd numbers <= a_{up}(b) of the smallest positive restricted residue system of b appear once in the first rows of the c(2*n+1) = A135303(n) coaches. If b is an odd prime a_{up}(b) is the maximum. See a comment in the proof of the quasi-order theorem of H-P, on page 263 ["Furthermore, every possible a_i < b/2 ..."]. For an example see below. - Wolfdieter Lang, Feb 19 2020
Satisfies the nested recurrence a(n) = a(a(n-2)) + 2*a(n-a(n-1)) with a(0) = a(1) = 1. Cf. A004001. - Peter Bala, Aug 30 2022
The binomial transform is 1, 2, 6, 16, 40, 96, 224, 512, 1152, 2560,.. (see A057711). - R. J. Mathar, Feb 25 2023

			G.f. = 1 + x + 3*x^2 + 3*x^3 + 5*x^4 + 5*x^5 + 7*x^6 + 7*x^7 + 9*x^8 + 9*x^9 + ...
Complete coach system for (a composite) b = 2*n + 1 = 33: Sigma(33) ={[1; 5], [5, 7, 13; 2, 1, 2]} (the first two rows are here 1 and 5, 7, 13), a_{up}(33) = a(15) = 15. But 15 is not in the reduced residue system modulo 33, so the maximal (odd) a number is 13. For the prime b = 31, a_{up}(31) = a(14) = 15 appears as maximum of the first rows. - _Wolfdieter Lang_, Feb 19 2020

Peter Hilton and Jean Pedersen, A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, 2010, 3rd printing 2012, pp. (260-281).

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Chromatic Number
Eric Weisstein's World of Mathematics, Johnson Graph
Eric Weisstein's World of Mathematics, Triangular Graph
Wikipedia, Round-robin tournament
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000217, A063196, A110660, A186421, A186422, A211955, A230584, A290988.
Complement of A052928 with respect to the universe A004526. - Guenther Schrack, Aug 21 2018
First differences of A000982, A061925, A074148, A105343, A116940, and A179207. - Guenther Schrack, Aug 21 2018

a109613 = (+ 1) . (* 2) . (`div` 2)
a109613_list = 1 : 1 : map (+ 2) a109613_list
-- Reinhard Zumkeller, Oct 27 2012, Feb 21 2011

A109613:=n->2*floor(n/2)+1; seq(A109613(k), k=0..100); # Wesley Ivan Hurt, Oct 22 2013

Flatten@ Array[{2# - 1, 2# - 1} &, 37] (* Robert G. Wilson v, Jul 07 2012 *)
(# - Boole[EvenQ[#]] &) /@ Range[80] (* Alonso del Arte, Sep 11 2019 *)
With[{c=2*Range[0,40]+1},Riffle[c,c]] (* Harvey P. Dale, Jan 02 2020 *)

A109613(n)=n>>1<<1+1 \\ Charles R Greathouse IV, Feb 24 2011

def a(n) : return( len( CuspForms( Gamma0( 5), 2*n + 4, prec=1). basis())); # Michael Somos, May 29 2013

((1 to 49) by 2) flatMap { List.fill(2)() } // _Alonso del Arte, Sep 11 2019

a(n) = 2*floor(n/2) + 1.
a(n) = A052928(n) + 1 = 2*A004526(n) + 1.
a(n) = A028242(n) + A110654(n).
a(n) = A052938(n-2) + A084964(n-2) for n > 1. - Reinhard Zumkeller, Aug 27 2005
G.f.: (1 + x + x^2 + x^3)/(1 - x^2)^2. - Paul Barry, Oct 14 2005
a(n) = 2*a(n-2) - a(n-4), a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 3. - Philippe Deléham, Nov 03 2008
a(n) = A001477(n) + A059841(n). - Philippe Deléham, Mar 31 2009
a(n) = 2*n - a(n-1), with a(0) = 1. - Vincenzo Librandi, Nov 13 2010
a(n) = R(n, -2), where R(n, x) is the n-th row polynomial of A211955. a(n) = (-1)^n + 2*Sum_{k = 1..n} (-1)^(n - k - 2)*4^(k-1)*binomial(n+k, 2*k). Cf. A084159. - Peter Bala, May 01 2012
a(n) = A182579(n+1, n). - Reinhard Zumkeller, May 06 2012
G.f.: ( 1 + x^2 ) / ( (1 + x)*(x - 1)^2 ). - R. J. Mathar, Jul 12 2016
E.g.f.: x*exp(x) + cosh(x). - Ilya Gutkovskiy, Jul 12 2016
From Guenther Schrack, Sep 10 2018: (Start)
a(-n) = -a(n-1).
a(n) = A047270(n+1) - (2*n + 2).
a(n) = A005408(A004526(n)). (End)
a(n) = A000217(n) / A004526(n+1), n > 0. - Torlach Rush, Nov 10 2023

A110654 a(n) = ceiling(n/2), or: a(2k) = k, a(2k+1) = k+1.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38

Offset: 0

Reinhard Zumkeller, Aug 05 2005

The number of partitions of 2n into exactly 2 odd parts. - Wesley Ivan Hurt, Jun 01 2013
Number of nonisomorphic outer planar graphs of order n >= 3 and size n+1. - Christian Barrientos and Sarah Minion, Feb 27 2018
Also the clique covering number of the n-dipyramidal graph for n >= 3. - Eric W. Weisstein, Jun 27 2018

			G.f. = x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 4*x^8 + 5*x^9 + ...

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Clique Covering Number
Eric Weisstein's World of Mathematics, Dipyramidal Graph
Wikipedia, Floor and ceiling functions
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Essentially the same sequence as A008619 and A123108.
Cf. A014557, A275416 (multisets).
Cf. A298648 (number of smallest coverings of dipyramidal graphs by maximal cliques).

a110654 = (`div` 2) . (+ 1)
a110654_list = tail a004526_list  -- Reinhard Zumkeller, Jul 27 2012

[Ceiling(n/2): n in [0..80]]; // Vincenzo Librandi, Nov 04 2014

a[ n_] := Ceiling[ n / 2]; (* Michael Somos, Jun 15 2014 *)
a[ n_] := Quotient[ n, 2, -1]; (* Michael Somos, Jun 15 2014 *)
a[0] = 0; a[n_] := a[n] = n - a[n - 1]; Table[a[n], {n, 0, 100}] (* Carlos Eduardo Olivieri, Dec 22 2014 *)
CoefficientList[Series[x^/(1 - x - x^2 + x^3), {x, 0, 75}], x] (* Robert G. Wilson v, Feb 05 2015 *)
LinearRecurrence[{1, 1, -1}, {0, 1, 1}, 75] (* Robert G. Wilson v, Feb 05 2015 *)

PARI
```
a(n)=n\2+n%2;
```

a(n)=(n+1)\2; \\ M. F. Hasler, Nov 17 2008

def A110654(n): return n+1>>1 # Chai Wah Wu, Jun 27 2025

[floor(n/2) + 1 for n in range(-1,75)] # Zerinvary Lajos, Dec 01 2009

a(n) = floor(n/2) + n mod 2.
a(n) = A004526(n+1) = A001057(n)*(-1)^(n+1).
For n > 0: a(n) = A008619(n-1).
A110655(n) = a(a(n)), A110656(n) = a(a(a(n))).
a(n) = A109613(n) - A028242(n) = A110660(n) / A028242(n).
a(n) = A001222(A029744(n)). - Reinhard Zumkeller, Feb 16 2006
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 2, a(2) = a(1) = 1, a(0) = 0. - Reinhard Zumkeller, May 22 2006
First differences of quarter-squares: a(n) = A002620(n+1) - A002620(n). - Reinhard Zumkeller, Aug 06 2009
a(n) = A007742(n) - A173511(n). - Reinhard Zumkeller, Feb 20 2010
a(n) = A000217(n) / A008619(n). - Reinhard Zumkeller, Aug 24 2011
From Michael Somos, Sep 19 2006: (Start)
Euler transform of length 2 sequence [1, 1].
G.f.: x/((1-x)*(1-x^2)).
a(-1-n) = -a(n). (End)
a(n) = floor((n+1)/2) = |Sum_{m=1..n} Sum_{k=1..m} (-1)^k|, where |x| is the absolute value of x. - William A. Tedeschi, Mar 21 2008
a(n) = A065033(n) for n > 0. - R. J. Mathar, Aug 18 2008
a(n) = ceiling(n/2) = smallest integer >= n/2. - M. F. Hasler, Nov 17 2008
If n is zero then a(n) is zero, else a(n) = a(n-1) + (n mod 2). - R. J. Cano, Jun 15 2014
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + x) * u * v - (u^2 - v) / 2. - Michael Somos, Jun 15 2014
Given g.f. A(x) then 2 * x^3 * (1 + x) * A(x) * A(x^2) is the g.f. of A014557. - Michael Somos, Jun 15 2014
a(n) = (n + (n mod 2)) / 2. - Fred Daniel Kline, Jun 08 2016
E.g.f.: (sinh(x) + x*exp(x))/2. - Ilya Gutkovskiy, Jun 08 2016
Satisfies the nested recurrence a(n) = a(a(n-2)) + a(n-a(n-1)) with a(1) = a(2) = 1. Cf. A004001. - Peter Bala, Aug 30 2022

Deleted wrong formula and added formula. - M. F. Hasler, Nov 17 2008

A028242 Follow n+1 by n. Also (essentially) Molien series of 2-dimensional quaternion group Q_8.

Original entry on oeis.org

1, 0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 8, 7, 9, 8, 10, 9, 11, 10, 12, 11, 13, 12, 14, 13, 15, 14, 16, 15, 17, 16, 18, 17, 19, 18, 20, 19, 21, 20, 22, 21, 23, 22, 24, 23, 25, 24, 26, 25, 27, 26, 28, 27, 29, 28, 30, 29, 31, 30, 32, 31, 33, 32, 34, 33, 35, 34, 36, 35, 37, 36, 38

Offset: 0

N. J. A. Sloane

A two-way infinite sequences which is palindromic (up to sign). - Michael Somos, Mar 21 2003
Number of permutations of [n+1] avoiding the patterns 123, 132 and 231 and having exactly one fixed point. Example: a(0) because we have 1; a(2)=2 because we have 213 and 321; a(3)=1 because we have 3214. - Emeric Deutsch, Nov 17 2005
The ring of invariants for the standard action of Quaternions on C^2 is generated by x^4 + y^4, x^2 * y^2, and x * y * (x^4 - y^4). - Michael Somos, Mar 14 2011
A000027 and A001477 interleaved. - Omar E. Pol, Feb 06 2012
First differences are A168361, extended by an initial -1. (Or: a(n)-a(n-1) = A168361(n+1), for all n >= 1.) - M. F. Hasler, Oct 05 2017
Also the number of unlabeled simple graphs with n + 1 vertices and exactly n endpoints (vertices of degree 1). The labeled version is A327370. - Gus Wiseman, Sep 06 2019

			G.f. = 1 + 2*x^2 + x^3 + 3*x^4 + 2*x^5 + 4*x^6 + 3*x^7 + 5*x^8 + 4*x^9 + 6*x^10 + 5*x^11 + ...
Molien g.f. = 1 + 2*t^4 + t^6 + 3*t^8 + 2*t^10 + 4*t^12 + 3*t^14 + 5*t^16 + 4*t^18 + 6*t^20 + ...

D. Benson, Polynomial Invariants of Finite Groups, Cambridge, p. 23.
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 15.
M. D. Neusel and L. Smith, Invariant Theory of Finite Groups, Amer. Math. Soc., 2002; see p. 97.
L. Smith, Polynomial Invariants of Finite Groups, A K Peters, 1995, p. 90.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
H. W. Gould, The inverse of a finite series and a third-order recurrent sequence, Fibonacci Quart. 44 (2006), no. 4, 302-315. See page 311.
T. Mansour and A. Robertson, Refined restricted permutations avoiding subsets of patterns of length three, Annals of Combinatorics, 6, 2002, 407-418 (Theorem 3.3).
MathOverflow, A question about an application of Molien's formula to find the generators and relations of an invariant ring.
Gus Wiseman, The a(3) = 2 through a(7) = 4 graphs with exactly n - 1 endpoints.
Index entries for Molien series.
Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000124 (a=1, a=n+a), A028242 (a=1, a=n-a).
Partial sums give A004652. A030451(n)=a(n+1), n>0.
Cf. A052938 (same sequence except no leading 1,0,2).
Cf. A000027, A001477.
Column k = n - 1 of A327371.
Cf. A000035, A004526, A004110, A059167, A109613, A110654, A110657, A110658, A110660, A168361, A245797, A327227, A327369, A327370, A327377.

a:=[1];; for n in [2..80] do a[n]:=(n-1)-a[n-1]; od; a; # Muniru A Asiru, Dec 16 2018

import Data.List (transpose)
a028242 n = n' + 1 - m where (n',m) = divMod n 2
a028242_list = concat $ transpose [a000027_list, a001477_list]
-- Reinhard Zumkeller, Nov 27 2012

&cat[ [n+1, n]: n in [0..37] ]; // Klaus Brockhaus, Nov 23 2009

series((1+x^3)/(1-x^2)^2,x,80);
A028242:=n->floor((n+1+(-1)^n)/2): seq(A028242(n), n=0..100); # Wesley Ivan Hurt, Mar 17 2015

Table[(1 + 2 n + 3 (-1)^n)/4, {n, 0, 74}] (* or *)
LinearRecurrence[{1, 1, -1}, {1, 0, 2}, 75] (* or *)
CoefficientList[Series[(1 - x + x^2)/((1 - x) (1 - x^2)), {x, 0, 74}], x] (* Michael De Vlieger, May 21 2017 *)
Table[{n,n-1},{n,40}]//Flatten (* Harvey P. Dale, Jun 26 2017 *)
Table[3*floor(n/2)-n+1,{n,0,40}] (* Pierre-Alain Sallard, Dec 15 2018 *)

{a(n) = (n\2) - (n%2) + 1} \\ Michael Somos, Oct 02 1999

A028242(n)=n\2+!bittest(n,0) \\ M. F. Hasler, Oct 05 2017

s=((1+x^3)/(1-x^2)^2).series(x, 80); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 16 2018

Expansion of the Molien series for standard action of Quaternions on C^2: (1 + t^6) / (1 - t^4)^2 = (1 - t^12) / ((1 - t^4)^2 * (1 - t^6)) in powers of t^2.
Euler transform of length 6 sequence [0, 2, 1, 0, 0, -1]. - Michael Somos, Mar 14 2011
a(n) = n - a(n-1) [with a(0) = 1] = A000035(n-1) + A004526(n). - Henry Bottomley, Jul 25 2001
G.f.: (1 - x + x^2) / ((1 - x) * (1 - x^2)) = ( 1+x^2-x ) / ( (1+x)*(x-1)^2 ).
a(2*n) = n + 1, a(2*n + 1) = n, a(-1 - n) = -a(n).
a(n) = a(n - 1) + a(n - 2) - a(n - 3).
a(n) = floor(n/2) + 1 - n mod 2. a(2*k) = k+1, a(2*k+1) = k; A110657(n) = a(a(n)), A110658(n) = a(a(a(n))); a(n) = A109613(n)-A110654(n) = A110660(n)/A110654(n). - Reinhard Zumkeller, Aug 05 2005
a(n) = 2*floor(n/2) - floor((n-1)/2). - Wesley Ivan Hurt, Oct 22 2013
a(n) = floor((n+1+(-1)^n)/2). - Wesley Ivan Hurt, Mar 15 2015
a(n) = (1 + 2n + 3(-1)^n)/4. - Wesley Ivan Hurt, Mar 18 2015
a(n) = Sum_{i=1..floor(n/2)} floor(n/(n-i)) for n > 0. - Wesley Ivan Hurt, May 21 2017
a(2n) = n+1, a(2n+1) = n, for all n >= 0. - M. F. Hasler, Oct 05 2017
a(n) = 3*floor(n/2) - n + 1. - Pierre-Alain Sallard, Dec 15 2018
E.g.f.: ((2 + x)*cosh(x) + (x - 1)*sinh(x))/2. - Stefano Spezia, Aug 01 2022
Sum_{n>=2} (-1)^(n+1)/a(n) = 1. - Amiram Eldar, Oct 04 2022

First part of definition adjusted to match offset by Klaus Brockhaus, Nov 23 2009

cp's OEIS Frontend

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

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Extensions

A109613 Odd numbers repeated.

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Haskell

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Scala

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A110654 a(n) = ceiling(n/2), or: a(2*k) = k, a(2*k+1) = k+1.

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Haskell

Magma

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Extensions

A028242 Follow n+1 by n. Also (essentially) Molien series of 2-dimensional quaternion group Q_8.

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GAP

Haskell

Magma

Maple

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PARI

PARI

Sage

Formula

Extensions

A006584 If n mod 2 = 0 then n*(n^2-4)/12 else n*(n^2-1)/12.

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A110654 a(n) = ceiling(n/2), or: a(2k) = k, a(2k+1) = k+1.

A006584 If n mod 2 = 0 then n(n^2-4)/12 else n(n^2-1)/12.

A219191 Numbers of the form k(7k+1), where k = 0,-1,1,-2,2,-3,3,...

A219190 Numbers of the form k(5k+1), where k = 0,-1,1,-2,2,-3,3,...