0,2

Consider all Pythagorean triples (X,Y,Z=Y+1) ordered by increasing Z; sequence gives Y values. X values are 1, 3, 5, 7, 9, ... (A005408), Z values are A001844.

In the triple (X, Y, Z) we have X^2=Y+Z. Actually, the triple is given by {x, (x^2 -+ 1)/2}, where x runs over the odd numbers (A005408) and x^2 over the odd squares (A016754). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 11 2004

a(n) = number of edges in (n+1) X (n+1) square grid with all horizontal and vertical segments filled in - Asher Auel (asher.auel(AT)reed.edu) Jan 12, 2000.

a(n) is the only number satisfying an inequality related to zeta(2) and zeta(3): sum(i>a(n)+1,1/i^2) < sum(i>n,1/i^3) < sum(i>a(n),1/i^2). - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 02 2001

Number of right triangles made from vertices of a regular n-gon when n is even - Sen-Peng You (giawgwan(AT)single.url.com.tw), Apr 05 2001

Number of ways to change two non-identical letters in the word aabbccdd..., where there are n type of letters. - Zerinvary Lajos (zlaja(AT)freemail.hu), Feb 15 2005

a(n) is the number of (n-1)-dimensional sides of an (n+1)-dimensional hypercube (e.g. squares have 4 corners, cubes have 12 edges, etc.). - Freek van Walderveen (freek_is(AT)vanwal.nl), Nov 11 2005

Comments from Nikolaos Diamantis (nikos7am(AT)yahoo.com), May 23 2006: "Consider a triangle, a pentagon, an eptagon, ..., a k-gon where k is odd. We label a triangle with n=1, a pentagon with n=2, .., a k-gon with n = floor(k/2). Imagine every player standing on every vertex of the k-gon.

"Initially there are 2 frisbees on two neighboring players. Every time they throw the frisbee to their neighbor with equal probability. Then a(n) gives the average number of steps needed so that the frisbees meet.

"I verified it by simulating the processes with a computer program. For example a(2) = 12 because in a pentagon that's the expected number of trials we need to perform. That is an exercise in Concrete Mathematics and it can be done using generating functions."

First difference of a(n) is 4n = A008586(n). Any entry k of the sequence is followed by k + 2*{1 + sqrt(2k + 1)}. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 04 2006

A diagonal of A059056. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 18 2007

If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n-1) is equal to the number of 2-subsets of X containing none of X_i, (i=1,...n). - Milan R. Janjic (agnus(AT)blic.net), Jul 16 2007

Sequence allows us to find X values of the equation: 2*X^3 + X^2 = Y^2. To find Y values: b(n)=2n(n+1)(2n+1). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Nov 06 2007

Number of (n+1)-permutations of 3 objects u,v,w, with repetition allowed, containing n-1 u's. Example: a(1)=4 because we have vv, vw, wv and ww; a(2)=12 because we can place u in each of the previous four 2-permutations either in front, or in the middle, or at the end. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 27 2007

Sequence found by reading the line from 0, in the direction 0, 4,... and the same line from 0, in the direction 0, 12,..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol (info(AT)polprimos.com), May 03 2008

Twice oblong numbers. - Omar E. Pol (info(AT)polprimos.com), May 03 2008

a(n) is also the least weight of self-conjugate partitions having n different even parts. [From Augustine O. Munagi (amunagi(AT)yahoo.com), Dec 18 2008]

Except for the two terms of [A141530] and the first term of [A046092[, if X=[A141530], A=[A078371], Y=[A046092], we have, for all others terms, Pell's equation: X^2-A*Y^2=1. Example: 9^2-5*4^2=1; 55^2-21*12^2=1; 161^2-45*24^2=1. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 13 2009]

Contribution from Peter Luschny (peter(AT)luschny.de), Jul 12 2009: (Start)

The general formula for alternating sums of powers of even integers is in terms of the Swiss-Knife polynomials P(n,x) A153641 (P(n,1)-(-1)^k P(n,2k+1))/2. Here n=2, thus

a(k) = |(P(2,1)-(-1)^k P(2,2k+1))/2|. (End)

The sum of squares of n+1 consecutive numbers between a(n)-n and a(n) inclusive equals the sum of squares of n consecutive numbers following a(n). For example, for n = 2, a(2) = 12, and the corresponding equation is 10^2+11^2+12^2=13^2+14^2. [From Tanya Khovanova (tanyakh(AT)yahoo.com), Jul 20 2009]

Number of units of a(n) belongs to a periodic sequence: 0, 4, 2, 4, 0. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009]

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.

A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, p. 125, 1964.

Ronald L. Graham, D. E. Knuth and Oren Patashnik, Concrete Mathematics, Reading, Massachusetts: Addison-Wesley, 1994.

A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501. [From Augustine O. Munagi (amunagi(AT)yahoo.com), Dec 18 2008]

Milan Janjic, Two Enumerative Functions

Ron Knott, Pythagorean Triples and Online Calculators

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Aztec Diamond

Eric Weisstein's World of Mathematics, Gear Graph

Eric Weisstein's World of Mathematics, Hamiltonian Path

O. E. Pol, Determinacion geometrica de los numeros primos y perfectos.

A Miracle Equation [From Tanya Khovanova (tanyakh(AT)yahoo.com), Jul 20 2009]

a(n) = 2*A002378(n) = 4*A000217(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), May 25 2004

a(n) = C(2n, 2) - n = 4*C(n, 2) - Zerinvary Lajos (zlaja(AT)freemail.hu), Feb 15 2005

a(n)=A028896-A002378; a(n)=A124080-A028896; a(n)=A049598-A033996. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 06 2007

Array read by rows: row n gives A033586(n), A085250(n+1). - Omar E. Pol (info(AT)polprimos.com), May 03 2008

a(n)=a(n-1)+4n O.g.f.:4x/(1-x)^3 E.g.f.:Exp(x)*(2x^2+4x) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 17 2009]

Contribution from Stephen Crowley (crow(AT)crowlogic.net), Jul 26 2009: (Start)

a(n)=1/int(-(x*n+x-1)*(step((-1+x*n)/n)-1)*n*step((x*n+x-1)/(n+1)),x=0..1) where step(x)=piecewise(x<0,0,0<=x,1) is the Heaviside step function

sum(1/a(n),n=1..inf)=sum(1/((2*n)*(n+1)),n=1..inf)=1/2 (End)

a(n)=4*n+a(n-1)-4 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]

a(7)=112 because 112 = 2*7*(7+1).

The first few triples are (1,0,1), (3,4,5), (5,12,13), (7,24,25),...

The first such partitions, corresponding to a(n)=1,2,3,4, are 2+2, 4+4+2+2, 6+6+4+4+2+2, 8+8+6+6+4+4+2+2. [From Augustine O. Munagi (amunagi(AT)yahoo.com), Dec 18 2008]

For n=2, a(2)=4*2+0-4=4; n=3, a(3)=4*3+4-4=12; n=4, a(4)=4*4+12-4=24 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]

a:=n->sum(n+2*j, j=0..n): seq(a(n), n=0..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 29 2007

a:=n->sum(n, k=0..n):seq(a(n)+sum(n, k=4..n), n=1..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008

with(finance):seq(add(futurevalue( k, 3, 2), k=0..n)/4, n=0..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008

with(finance):seq(add(cashflows([n, k, k], 0 ), k=0..n), n=0..51); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 30 2008

Table[2n(n + 1), {n, 0, 50}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 03 2006

Cf. A045943, A028895.

Cf. A002943, A054000, A000330, A007290.

Main diagonal of array in A001477.

a(n)=A100345(n+1, n-1) for n>0.

Equals A033996/2

Cf. A002378, A033996, A124080, A028896, A049598.

Cf. A005563.

Cf. A000217, A033586, A085250.

Cf. A001844 [From Augustine O. Munagi (amunagi(AT)yahoo.com), Dec 18 2008]

Cf. A078371, A141530 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 13 2009]

Sequence in context: A081937 A115228 A088557 this_sequence A008241 A008216 A008074

Adjacent sequences: A046089 A046090 A046091 this_sequence A046093 A046094 A046095

nonn,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com), Eric Weisstein (eric(AT)weisstein.com)