|
| |
|
|
A002378
|
|
Oblong (or promic, pronic, or heteromecic) numbers: n(n+1).
(Formerly M1581 N0616)
|
|
368
|
|
|
|
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 denotes 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,... (whilst 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 (i.e. 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" comprised 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. [From 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. [From Jaroslav Krizek, Sep 13 2009]
a(j) = number of non-zero values of floor (j^2/n) taken over all n >= 1 for each j, with 1 <= j <= n-1.
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. [From 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 Walsh, Nov 27 2012]
a(n) is the number of injective functions from {1,2} into {1,2,...,n+1}. [Dennis Walsh, Nov 27 2012]
|
|
|
REFERENCES
|
W. W. Berman and D. E. Smith, A Brief History of Mathematics, 1910, Open Court, page 67.
P. Cameron, T. Prellberg and D. Stark, Asymptotics for incidence matrix classes, Electron. J. Combin. 13 (2006), #R85, p. 11.
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.
S. Crowley, Some Fractal String and Hypergeometric Aspects of the Riemann Zeta Function, http://www.vixra.org/pdf/1202.0066v1.pdf, 2012. - From N. J. A. Sloane, Jun 14 2012
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. [From Mohammad K. Azarian, October 27 2011.]
C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford University Press, 1966, pp. 61-62.
A. Petojevic and N. Dapic, The vAm(a,b,c;z) function, Preprint 2013, http://www.mi.sanu.ac.rs/~gvm/radovi/AP-Budva.pdf
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).
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
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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. [From 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 (zerinvarylajos(AT)yahoo.com), 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 8 2013
|
|
|
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). [N. J. A. Sloane, Dec 11 1999]
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)
A036689 is a subsequence.
Cf. A078358 (complement).
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
|
| |
|
|
