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COMMENT
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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
Starting (1, 3, 7, 13, 21,...) = binomial transform of [1, 2, 2, 0, 0, 0]; example: a(4) = 13 = (1, 3, 3, 1) dot (1, 2, 2, 0) = (1 + 6 + 6 + 0). - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 10 2008
s=0;lst={};Do[s+=n;AppendTo[lst,s],{n,0,4!,1}];lst....0,1,3,6,10,15,21,28,36,45,55,66,78,91,... s=0;lst={};Do[s+=n;AppendTo[lst,s],{n,0,4!,2}];lst....0,2,6,12,20,30,42,56,72,90,110,132,156... s=0;lst={};Do[s+=n;AppendTo[lst,s+1],{n,0,4!,2}];lst..1,3,7,13,21,31,43,57,73,91,111,133,157... CentralPolygonalNumbers==DoubleTriangeNumbersPlusOne No need for: Factorials, Power,...etc, just add all Even numbers, plus One. [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 01 2009]
Starting (1, 3, 7, 13,...) = triangle A158821 * [1, 2, 3,...]. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 28 2009]
Starting with offset 1 = triangle A128229 * [1,2,3,...]. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 26 2009]
a(n) = k such that floor(1/2 *(1 + sqrt(4*k-3)))+ k is integer (n^2+1]. A000037(a(n)) = A002522(n) = n^2+1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jun 21 2009]
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FORMULA
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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. - N. J. A. Sloane (njas(AT)research.att.com), 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
a(n)=3a(n-1)-3a(n-2)+a(n-3), a(0)=1, a(1)=1, a(2)=3 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 02 2008]
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