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 (maxal(AT)cs.ucsd.edu), 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

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.

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.

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

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.

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 410

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

Index entries for "core" sequences

Eric Weisstein's World of Mathematics, Wiener Index

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)=n^2-n for n>=2. - 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

[ 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; [Conjectured by 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

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

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.

Cf. A005563, A046092, A001082.

Sequence in context: A077253 A098734 A103505 this_sequence A005991 A003274 A121315

Adjacent sequences: A002375 A002376 A002377 this_sequence A002379 A002380 A002381

nonn,easy,core,nice

njas

Additional comments from Michael Somos