0,2

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 (rkg(AT)cpsc.ucalgary.ca)

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 (benoit7848c(AT)orange.fr), 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 2 4 6 8 10 12 14... (whilst first differences of the squares are 1 3 5 7 9 11 13...) - Alexandre Wajnberg (alexandre.wajnberg(AT)skynet.be), Dec 29 2005

25 appended to these numbers corresponds to squares of numbers ending in 5 (i.e. to squares of A017329). - Lekraj Beedassy (blekraj(AT)yahoo.com), 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 (deutsch(AT)duke.poly.edu), 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. W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), May 05 2006.

Nonsquare integers m divisible by ceil(sqrt(m)), except m=0. - Max Alekseyev (maxale(AT)gmail.com), Nov 27 2006

a(n) = 2*sum(1..n-1). - Artur Jasinski (grafix(AT)csl.pl), Jan 09 2007

The number of off-diagonal elements of an n+1 X n+1 matrix. - Artur Jasinski (grafix(AT)csl.pl), 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 R. Janjic (agnus(AT)blic.net), Mar 13 2007

Numbers m>=0 such that round(sqrt(m+1))-round(sqrt(m))=1. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Aug 06 2007

Numbers m>=0 such that ceiling(2*sqrt(m+1))-1=1+floor(2*sqrt(m)). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 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 (Hieronymus.Fischer(AT)gmx.de), 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 (tcjpn(AT)msn.com), 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 R. Janjic (agnus(AT)blic.net), Dec 28 2007

Infinite Sum[1/a[n+1],{n,1,Infinity}] = 1 but series is very slowly convergent. [From Artur Jasinski (grafix(AT)csl.pl), Sep 28 2008]

a(n)=A061037(4n)=(n+1/2)^2-1/4=((2n+1)^2-1)/4.From Balmer spectrum of hydrogen. [From Paul Curtz (bpcrtz(AT)free.fr), Oct 03 2008]

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 (reikku(AT)gmail.com), Nov 30 2008]

Except for the first term of [A002378], if X=[A144396], Y=[A007395], A= [A002378], we have, for all other terms, Pell's equation: [A144396]^2 - [A002378]*[A007395]^2=1; (X^2-A*Y^2=1); example: 3^2-2*2^2=1; 5^2-6*2^2=1; 19^2-90*2^2=1, and so on. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 11 2009]

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 21 2009: (Start)

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.

(End)

Number of units of a(n) belongs to a periodic sequence: 0, 2, 6, 2, 0. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 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 (jaroslav.krizek(AT)atlas.cz), 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.

Except for the first term, a(n)=2*n+a(n-1), (with a(1)=2) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Oct 23 2009]

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.

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).

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.

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

S. 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.

S. 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, Link to a section of The World of Mathematics.

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 (meijgia(AT)hotmail.com), Jul 21 2009]

Index entries for "core" sequences

Index entries for sequences related to linear recurrences with constant coefficients

G.f.: (2*x)/(1-x)^3; a(n)=a(n-1)+2*n, a(0)=0.

Sum_{n >= 1} n*(n+1) = n(n+1)(n+2)/3 (cf. A007290).

Sum_{n >= 1} 1/(n*(n+1)) = 1. (Cf. Tijdeman)

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 (qntmpkt(AT)yahoo.com), 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 (charliemath(AT)verizon.net), Dec 29 2003

Sum_{k=1..n} 1/a(k) = n/(n+1). - Robert G. Wilson v Feb 04 2005.

a(n)=A046092/2. - 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/(Nn+1), with N=1. Log 2 = Integral_{0..1} 1/(1+x) dx = .69314718...; sum: 1/2 + 1/12 + 1/30 + 1/56 + 1/90 = 1627/2520 = .64563... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 22 2003

a(n)=A049598-A124080; a(n)=A124080-A033996: a(n)=A033996-A028896: a(n)=A028896-A046092. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 06 2007

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 (azarian(AT)evansville.edu), Jul 26 2007

(2, 6, 12, 20, 30,...) = binomial transform of (2, 4, 2). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 28 2007

a(n)=A000217(n)*2. - Omar E. Pol (info(AT)polprimos.com), May 14 2008

a(n) = A006503(n) - A000292(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 24 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 (Hieronymus.Fischer(AT)gmx.de), Dec 31 2008

E.g.f.:(x+2)*x*exp(x) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443), Feb 06 2009]

Product_{i=2..infinity} (1-1/a(i)) = -2*sin(Pi*A001622)/Pi = -2*sin(A094886)/A000796. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 12 2009]

egf: ((-x+1)*ln(-x+1)+x)/x^2 also int(((-x+1)*ln(-x+1)+x)/x^2,x=0..1)=Zeta(2)-1 [From Stephen Crowley (crow(AT)crowlogic.net), Jul 11 2009]

a(n)=2*n+a(n-1)-2 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 29 2009]

For n=2, a(2)=2*2+0-2=2; n=3, a(3)=2*3+2-2=6; n=4, a(4)=2*4+6-2=12 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 29 2009]

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

[seq(2*binomial(n, 2), n=1..51)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 24 2006

[seq(numbperm (n, 2), n=1..51)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2007

a:=n->sum(numbcomb (n, 1), j=0..n): seq(a(n), n=0..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2007

a:=n->sum(n+2*j, j=0..n)/2: seq(a(n), n=0..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 29 2007

seq(sum(sum(gcd(k, j), j=1..n), k=0..n), n=0..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 01 2007

A002378:=-2*z/(z-1)**3; [S. Plouffe in his 1992 dissertation.]

with (combinat):seq(fibonacci(3, n)+n-1, n=0..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 07 2008

with(finance):seq(add(futurevalue( k, 3, 2), k=0..n)/8, n=0..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 22 2008

with(finance):seq(add(cashflows([2, k, k], 0 ), k=0..n), n=-1..51); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 22 2008

restart: G(x):=x^2*exp(x): f[0]:=G(x): for n from 1 to 51 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=1..51); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 05 2009]

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

Table[(n^2 - n), {n, 51}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 21 2007

(PARI) je=[]; for(n=0, 5000, if(issquare(4*n+1), je=concat(je, n))); je

sage: [lucas_number1(3, n, n) for n in xrange(1, 52)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 16 2008

(PARI) j=[]; for (n=0, 40, j=concat(j, n^2+floor(((n+1)^2 - n^2)/2))); j [From Alexander R. Povolotsky (pevnev(AT)juno.com), Nov 09 2008]

Partial sums of A005843 (even numbers). Twice triangular numbers A000217. Partial sums give A007290.

1/beta(n, 2) in A061928.

a(n) = A110660(2*n).

Cf. A035106, A087811, A119462, A127235.

Cf. A049598, A124080, A033996, A028896, A046092.

Cf. A000217, A005563, A046092, A001082.

Cf. A059300, A059297, A059298 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 06 2008]

Cf. A059300, A059297, A059298.

Cf. A000040 [From Reikku Kulon (reikku(AT)gmail.com), Nov 30 2008]

Cf. A000217.

Cf. A144396, A007395 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 11 2009]

Cf. A166373

Sequence in context: A160942 A160929 A103505 this_sequence A005991 A003274 A121315

Adjacent sequences: A002375 A002376 A002377 this_sequence A002379 A002380 A002381

nonn,easy,core,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Michael Somos

Corrected l.h.s. of my formula - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 15 2009

Comment and cross-reference added by Christopher Hunt Gribble (chris.eveswell(AT)virgin.net), Oct 13 2009