0,3

These are Hogben's central polygonal numbers denoted by the symbol

...2....

....P...

...2.n..

(P with three attachments).

Also the maximal number of 1's that an n X n invertible {0,1} matrix can have. (See Halmos for proof). - Felix Goldberg (felixg(AT)tx.technion.ac.il), Jul 07 2001

Let Phi_k(x) be the k-th cyclotomic polynomial and form the sequence Phi_k(0), Phi_k(1), Phi_k(2), ... This gives A000027 (k=2), A002061 (k=3), A002522 (k=4), A053699 (k=5), A002061 (k=6), A053716 (k=7), A002523 (k=8), A060883 (k=9), A060884 (k=10), A060885 (k=11), A060886 (k=12), A060887 (k=13), A060888 (k=14), A060889 (k=15), A060890 (k=16), A060891 (k=18), A060892 (k=20), A060893 (k=24), A060894 (k=30), A060895 (k=32), A060896 (k=36).

Maximal number of parts into which n intersecting circles can divide themselves, for n >= 1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 07 2001

The terms are the smallest of n consecutive odd numbers whose sum is n^3: 1, 3+5 = 8 = 2^3, 7+9+11 = 27 = 3^3, etc. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 19 2001

(n*a(n+1)+1)/(n^2+1) is the smallest integer of the form (nk+1)/(n^2+1) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 02 2002

For n>=3 a(n) is also the number of cycles in the wheel graph W(n) of order n. - Sharon Sela (sharonsela(AT)hotmail.com), May 17 2002

Let b(k) be defined as follows: b(1)=1 and b(k+1)>b(k) is the smallest integer such that sum(i=b(k),b(k+1), 1/sqrt(i)) > 2; then b(n)=a(n) for n>0. - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 23 2002

Drop the first three terms. Then n*a(n) + 1 = (n+1)^3. E.g. 7*1 +1 = 8 = 2^3, 13*2 +1 = 27 = 3^3, 21*3+1 = 64 = 4^3. etc. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 20 2002

Arithmetic mean of next 2n-1 numbers. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Feb 16 2004

The n-th term of an arithmetic progression with first term 1 and common difference n: a(1) = 1 -> 1,2,3,4,5... a(2) = 3 -> 1,3,... a(3) = 7 -> 1,4,7,... a(4) = 13 -> 1,5,9,13,... - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 25 2004

Number of walks of length 3 between any two distinct vertices of the complete graph K_{n+1} (n>=1). Example: a(2)=3 because in the complete graph ABC we have the following walks of length 3 between A and B: ABAB, ACAB, and ABCB. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 01 2004

The sequence 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, ... is the trajectory of 3 under repeated application of the map n -> n + 2 * square excess of n, cf. A094765.

Also n^3 mod (n^2+1) - Zak Seidov (zakseidov(AT)yahoo.com), Aug 31 2006

Also, omitting the first 1, the main diagonal of A081344. - Zak Seidov (zakseidov(AT)yahoo.com), Oct 5 2006

Ignoring the first ones, these are rectangular parallelepipeds with integer dimensions that have integer interior diagonals. Using Pythagoras: sqrt[a^2+b^2+c^2] = d, an integer; then this sequence: sqrt[n^2+(n+1)^2+(n(n+1))^2]= 2T_n+1 is the first and most simple example. Problem: Are there any integer diagonals which do not satisfy the following general formula? sqrt[(kn)^2+(k(n+(2m+1)))^2+(k(n(n+(2m+1))+4T_m))^2]=kd where (m=0,1,2...),(k=1,2,3...) and T is a triangular number. - Marco Matosic (marcomatosic(AT)hotmail.com), Nov 10 2006

Numbers n such that a(n) is prime are listed in A055494 = {2,3,4,6,7,9,13,15,16,18,21,22,25,28,34,39,42,51,55,58,60,63,67,70,72,76,78,79, 81,90,91,100,...}. Prime a(n) are listed in A002383 = {3, 7, 13, 31, 43, 73, 157, 211, 241, 307, 421, 463, ...}. All terms are odd. Prime factors of a(n) are listed in A007645 = {3, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, ...} Cuban primes: of form x^2+xy+y^2; or: primes of form x^2+3*y^2; or: primes == 0 or 1 mod 3. 3 divides a(3k-1). 7 divides a(7k-4) and a(7k-2). 7^2 divides a(7^2*k-18) and a(7^2*k+19). 7^3 divides a(7^3*k-18) and a(7^3*k+19). 7^4 divides a(7^4*k+1048) and a(7^4*k-1047). 7^5 divides a(7^5*k+1354) and a(7^5*k-1353). 13 divides a(13k-9) and a(13k-3). 13^2 divides a(13^2*k+23) and a(13^2*k-22). 13^3 divides a(13^3*k+1037) and a(13^3*k-1036). - Alexander Adamchuk (alex(AT)kolmogorov.com), Jan 25 2007

Complement of A135668. - Kieren MacMillan (kieren(AT)alumni.rice.edu), Dec 16 2007

Numbers (sorted) on the main diagonal of a 2n X 2n spiral. For example, when n=2:

7...8...9...10

6...1...2...11

5...4...3...12

16..15..14..13 - cf. A137928. (William A. Tedeschi (fynmun(AT)hotmail.com), Feb 29 2008)

a(n)=AlexanderPolynomial[n] defined as Det[Transpose[S]-n S] where S is Seifert matrix {{-1, 1}, {0, -1}} - Artur Jasinski (grafix(AT)csl.pl), Mar 31 2008

Archimedeans Problems Drive, Eureka, 22 (1959), 15.

Richard Bean and E. S. Mahmoodian, A new bound on the size of the largest critical set in a Latin square, Discrete Math., 267 (2003), 13-21.

Paul R. Halmos, Linear Algebra Problem Book. MAA: 1995. pp. 75-6, 242-4.

L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer Press, NY, 1950, p. 22.

R. Honsberger, Ingenuity in Math., Random House, 1970, p. 87.

Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.

S. H. Weintraub, An interesting recursion, Amer. Math. Monthly, 111 (No. 6, 2004), 528-530.

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

Richard Bean and E. S. Mahmoodian, A new bound on the size of the largest critical set in a Latin square

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

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.

Eric Weisstein's World of Mathematics, Graph Cycle

Eric Weisstein's World of Mathematics, Wheel Graph

Index entries for sequences related to centered polygonal numbers

E.W. Weisstein, "Alexander Polynomial."

G.f.: (1-2x+3x^2)/(1-x)^3. a(n)=-(n-5)a(n-1)+(n-2)a(n-2).

a(n) = a(n - 1) + 2n = 2a(n - 1) - a(n - 2) + 2 = A002378(n - 1) + 1 = 2*A000124(n - 1) - 1 - Henry Bottomley (se16(AT)btinternet.com), Oct 02 2000

Sum of two triangular numbers t_n and t_{n-2}.

(x(1+x^2))/(1-x)^3 is g.f. for 0, 1, 3, 7, 13, ... a(n)=2C(n, 2)+C(n-1, 0). E.g.f. (1+x^2)exp(x). - Paul Barry (pbarry(AT)wit.ie), Mar 13 2003

a(n) = ceiling((n-1/2)^2) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 16 2003. Hence the terms are about midway between successive square, and so so (except for 1) are not squares. - njas, Nov 01, 2005

a(n)= 1+ sum (2*n) - Xavier Acloque Oct 08 2003

a(n)=1 + A002378(n-1). - Lekraj Beedassy (blekraj(AT)yahoo.com), Oct 17 2003

a(n)=floor(t(n^2)/t(n)), where t(n)=n*(n+1)/2 - Jon Perry (perry(AT)globalnet.co.uk), Feb 14 2004

a(n) = leftmost term in M^(n-1) * [1 1 1], where M = the 3 X 3 matrix [1 1 1 / 0 1 2 / 0 0 1]. E.g. a(6) = 31 since M^5 * [1 1 1] = [31 11 1] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 11 2004

a(n+1) = n^2 + n + 1. a(n+1)*a(n)=(n^6-1)/(n^2-1)=n^4+n^2+1=a(n^2+1) - a product of two consecutive numbers from this sequence belongs to this sequence too. (a(n+1)+a(n))/2=n^2+1. (a(n+1)-a(n))/2=n. a((a(n+1)+a(n))/2)=a(n+1)*a(n). - Alexander Adamchuk (alex(AT)kolmogorov.com), Apr 13 2006

Narayana transform of [1, 2, 0, 0, 0...] = [1, 3, 7, 13, 21...]. Let M = the infinite lower triangular matrix of A001263 and let V = the Vector [1, 2, 0, 0, 0...]. Then A002061 starting (1, 3, 7...) = M * V. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 25 2006

binomial(n+4,n+2)+binomial(n+2,n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2006

a(1-n)=a(n). - Michael Somos Sep 04 2006

a(n+3)=Numerator of ((n + 1)! + (n - 1)!)/(n!) - Artur Jasinski (grafix(AT)csl.pl), Jan 09 2007

a(n) = A132111(n-1,1) for n>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 10 2007

a(n)=Det[Transpose[{{-1, 1}, {0, -1}}] - n {{-1, 1}, {0, -1}}] - Artur Jasinski (grafix(AT)csl.pl), Mar 31 2008

A002061:=-(1-2*z+3*z**2)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]

with(combinat, fibonacci):seq(fibonacci(3, i)-i, i=0..52); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 20 2008

Table[Numerator[((n + 1)! + (n - 1)!)/(n!)], {n, 1, 30}] - Artur Jasinski (grafix(AT)csl.pl), Jan 09 2007

S = {{-1, 1}, {0, -1}}; Table[Det[Transpose[S] - n S], {n, 0, 30}] (*or*) a = {1, 1}; Do[AppendTo[a, n ((n + 1)! + (n - 1)!)/n! ], {n, 1, 30}]; a (*Artur Jasi�ński*) - Artur Jasinski (grafix(AT)csl.pl), Mar 31 2008

(PARI) a(n)=n^2-n+1

Cf. A001263, A001844, A051890, A000124, A091776, A014206, A055494, A002383, A007645.

Cf. A132014, A132382, A135668.

Cf. A137928, A000217, A004273, A005408.

Adjacent sequences: A002058 A002059 A002060 this_sequence A002062 A002063 A002064

Sequence in context: A115298 A025728 A084537 this_sequence A063541 A011898 A098577

nonn,easy,nice

njas