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A002378 Oblong (or promic, pronic, or heteromecic) numbers: n(n+1).
(Formerly M1581 N0616)
434
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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 ceil(sqrt(m)), except 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

Sequence allows us to find X values of the equation: 4*X^3 + X^2 = Y^2. To find Y values: b(n) = n(n+1)(2n+1). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), 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 iff 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

a(A007018(n)) = A007018(n+1), see sequence A007018 (1,2,6,42,1806,...), i.e. A007018(n+1) = A007018(n) th oblong numbers. - Jaroslav Krizek, Sep 13 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) + A002378(m)*b(n-2), n>=2, b(0)=0, b(1)=1, then lim b(n+1)/b(n) = m+1 as n approaches infinity. (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 n for which n+sqrt(n+sqrt(n+sqrt(n+sqrt(n+...)... 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

REFERENCES

W. W. Berman and D. E. Smith, A Brief History of Mathematics, 1910, Open Court, page 67.

J. H. Conway and R. K. Guy, The Book of Numbers, 1996, p. 34.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.

L. E. Dickson, History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. 357, 1952.

L. E. Dickson, History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 6, 232-233, 350 and 407, 1952.

H. Eves, An Introduction to the History of Mathematics, revised, Holt, Rinehart and Winston, 1964, page 72.

Nicomachus of Gerasa, Introduction to Arithmetic, translation by Martin Luther D'Ooge, Ann Arbor, University of Michigan Press, 1938, p. 254.

Jolley, Summation of Series, Oxford (1961).

Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), pp. 980-981.  [Mohammad K. Azarian, Oct 27 2011.]

C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford University Press, 1966, pp. 61-62.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. Suprijanto and Rusliansyah, Observation on Sums of Powers of Integers Divisible by Four, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2219 - 2226; http://dx.doi.org/10.12988/ams.2014.4140.

F. J. Swetz, From Five Fingers to Infinity, Open Court, 1994, p. 219.

R. Tijdeman, Some applications of Diophantine approximation, pp. 261-284 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

H. Bottomley, Illustration of initial terms of A000217, A002378

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 410

P. Cameron, T. Prellberg and D. Stark, Asymptotics for incidence matrix classes, Electron. J. Combin. 13 (2006), #R85, p. 11.

S. Crowley, Some Fractal String and Hypergeometric Aspects of the Riemann Zeta Function, 2012. - N. J. A. Sloane, Jun 14 2012

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Refik Keskin and Olcay Karaatli, Some New Properties of Balancing Numbers and Square Triangular Numbers, Journal of Integer Sequences, Vol. 15 (2012), Article #12.1.4

A. Petojevic and N. Dapic, The vAm(a,b,c;z) function, Preprint 2013.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014

G. Villemin's Almanach of Numbers, Nombres Proniques

Eric Weisstein's World of Mathematics, Pronic Number

Eric Weisstein's World of Mathematics, Leibniz Harmonic Triangle

Eric Weisstein's World of Mathematics, Crown Graph

Eric Weisstein's World of Mathematics, Wiener Index

Wolfram Research, Hypergeometric Function 3F2, The Wolfram Functions site. [Johannes W. Meijer, Jul 21 2009]

Index entries for "core" sequences

Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

G.f.: 2*x/(1-x)^3.

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 through infinity) 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, inf.) 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, inf.) (-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) = 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. - Paul Curtz, Oct 03 2008

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..infinity} (1-1/a(i)) = -2*sin(Pi*A001622)/Pi = -2*sin(A094886)/A000796 = 2*A146481. - R. J. Mathar, Mar 12 2009

E.g.f.: ((-x+1)*ln(-x+1)+x)/x^2 also int(((-x+1)*ln(-x+1)+x)/x^2,x=0..1) = Zeta(2)-1. - Stephen Crowley, Jul 11 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) = floor(n^5/(n^3+n^2+1)) with offset 1...a(1) = 0. - 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 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

a(n) == 2 mod (n+2). - Jon Perry, Aug 07 2014

EXAMPLE

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 + ...

MAPLE

[ seq(n*(n+1), n = 0..100) ];

A002378 := -2*z/(z-1)**3; # Simon Plouffe in his 1992 dissertation

MATHEMATICA

Table[ n(n + 1), {n, 0, 50}] (* Robert G. Wilson v, Jun 19 2004 *)

fQ[n_] := IntegerQ@ Sqrt[4 n + 1]; Select[ Range[0, 2600], fQ] (* Robert G. Wilson v, Sep 29 2011 *)

2Accumulate[Range[0, 50]] (* Harvey P. Dale, Nov 11 2011 *)

PROG

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

(Haskell)

a002378 n = n * (n + 1)

a002378_list = zipWith (*) [0..] [1..]

-- Reinhard Zumkeller, Aug 27 2012, Oct 12 2011

CROSSREFS

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 is a subsequence. 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.

Cf. A005369.

Sequence in context: A160942 A160929 A103505 * A005991 A194110 A184432

Adjacent sequences:  A002375 A002376 A002377 * A002379 A002380 A002381

KEYWORD

nonn,easy,core,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

Additional comments from Michael Somos

Corrected l.h.s. of my formula - R. J. Mathar, Mar 15 2009

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

STATUS

approved

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Last modified August 27 23:36 EDT 2014. Contains 246151 sequences.