%I A002061 M2638 N1049
%S A002061 1,1,3,7,13,21,31,43,57,73,91,111,133,157,183,211,241,273,307,343,381,
421,
%T A002061 463,507,553,601,651,703,757,813,871,931,993,1057,1123,1191,1261,1333,
1407,
%U A002061 1483,1561,1641,1723,1807,1893,1981,2071,2163,2257,2353,2451,2551,2653
%N A002061 Central polygonal numbers: n^2 - n + 1.
%C A002061 These are Hogben's central polygonal numbers denoted by the symbol
%C A002061 ...2....
%C A002061 ....P...
%C A002061 ...2.n..
%C A002061 (P with three attachments).
%C A002061 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
%C A002061 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).
%C A002061 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
%C A002061 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
%C A002061 (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
%C A002061 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
%C A002061 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
%C A002061 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
%C A002061 Arithmetic mean of next 2n-1 numbers. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com),
Feb 16 2004
%C A002061 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
%C A002061 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
%C A002061 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.
%C A002061 Also n^3 mod (n^2+1) - Zak Seidov (zakseidov(AT)yahoo.com), Aug 31 2006
%C A002061 Also, omitting the first 1, the main diagonal of A081344. - Zak Seidov
(zakseidov(AT)yahoo.com), Oct 5 2006
%C A002061 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
%C A002061 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
%C A002061 Complement of A135668. - Kieren MacMillan (kieren(AT)alumni.rice.edu),
Dec 16 2007
%C A002061 Numbers (sorted) on the main diagonal of a 2n X 2n spiral. For example,
when n=2:
%C A002061 7...8...9...10
%C A002061 6...1...2...11
%C A002061 5...4...3...12
%C A002061 16..15..14..13 - cf. A137928. (William A. Tedeschi (fynmun(AT)hotmail.com),
Feb 29 2008)
%C A002061 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
%C A002061 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
%C A002061 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]
%C A002061 Starting (1, 3, 7, 13,...) = triangle A158821 * [1, 2, 3,...]. [From
Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 28 2009]
%C A002061 Starting with offset 1 = triangle A128229 * [1,2,3,...]. [From Gary W.
Adamson (qntmpkt(AT)yahoo.com), Mar 26 2009]
%C A002061 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]
%D A002061 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A002061 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A002061 Archimedeans Problems Drive, Eureka, 22 (1959), 15.
%D A002061 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.
%D A002061 Paul R. Halmos, Linear Algebra Problem Book. MAA: 1995. pp. 75-6, 242-4.
%D A002061 L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer
Press, NY, 1950, p. 22.
%D A002061 R. Honsberger, Ingenuity in Math., Random House, 1970, p. 87.
%D A002061 Clark Kimberling, Complementary Equations, Journal of Integer Sequences,
Vol. 10 (2007), Article 07.1.4.
%D A002061 S. H. Weintraub, An interesting recursion, Amer. Math. Monthly, 111 (No.
6, 2004), 528-530.
%H A002061 T. D. Noe, <a href="b002061.txt">Table of n, a(n) for n=0..1000</a>
%H A002061 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%H A002061 Richard Bean and E. S. Mahmoodian, <a href="http://arXiv.org/abs/math/
0107159">A new bound on the size of the largest critical set in a
Latin square</a>
%H A002061 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas
for Some Functions on Finite Sets</a>
%H A002061 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/
journals/JIS/index.html">Partial Complements and Transposable Dispersions</
a>, J. Integer Seqs., Vol. 7, 2004.
%H A002061 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A002061 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A002061 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
GraphCycle.html">Graph Cycle</a>
%H A002061 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
WheelGraph.html">Wheel Graph</a>
%H A002061 <a href="Sindx_Ce.html#CENTRALCUBE">Index entries for sequences related
to centered polygonal numbers</a>
%H A002061 E.W. Weisstein, <a href="http://mathworld.wolfram.com/AlexanderPolynomial.html">
"Alexander Polynomial."</a>
%F A002061 G.f.: (1-2x+3x^2)/(1-x)^3. a(n)=-(n-5)a(n-1)+(n-2)a(n-2).
%F A002061 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
%F A002061 Sum of two triangular numbers t_n and t_{n-2}.
%F A002061 (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
%F A002061 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
%F A002061 a(n)= 1+ sum (2*n) - Xavier Acloque Oct 08 2003
%F A002061 a(n)=1 + A002378(n-1). - Lekraj Beedassy (blekraj(AT)yahoo.com), Oct
17 2003
%F A002061 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
%F A002061 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
%F A002061 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
%F A002061 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
%F A002061 binomial(n+4,n+2)+binomial(n+2,n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
May 11 2006
%F A002061 a(1-n)=a(n). - Michael Somos Sep 04 2006
%F A002061 a(n+3)=Numerator of ((n + 1)! + (n - 1)!)/(n!) - Artur Jasinski (grafix(AT)csl.pl),
Jan 09 2007
%F A002061 a(n) = A132111(n-1,1) for n>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Aug 10 2007
%F A002061 a(n)=Det[Transpose[{{-1, 1}, {0, -1}}] - n {{-1, 1}, {0, -1}}] - Artur
Jasinski (grafix(AT)csl.pl), Mar 31 2008
%F A002061 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]
%F A002061 a(n)=2*n+a(n-1)-4 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it),
Nov 07 2009]
%e A002061 For n=2, a(2)=2*2+1-4=1; n=3, a(3)=2*3+1-4=3; n=4, a(4)=2*4+3-4=7 [From
Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 07 2009]
%p A002061 A002061:=-(1-2*z+3*z**2)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
%p A002061 with(combinat, fibonacci):seq(fibonacci(3, i)-i,i=0..52); - Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Mar 20 2008
%t A002061 Table[Numerator[((n + 1)! + (n - 1)!)/(n!)], {n, 1, 30}] - Artur Jasinski
(grafix(AT)csl.pl), Jan 09 2007
%t A002061 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 Jasinski*)
%t A002061 s=0;lst={};Do[s+=n;AppendTo[lst,s+1],{n,0,6!,2}];lst [From Vladimir Orlovsky
(4vladimir(AT)gmail.com), Feb 01 2009]
%o A002061 (PARI) a(n)=n^2-n+1
%Y A002061 Cf. A001263, A001844, A051890, A000124, A091776, A014206, A055494, A002383,
A007645.
%Y A002061 Cf. A132014, A132382, A135668.
%Y A002061 Cf. A137928, A000217, A004273, A005408.
%Y A002061 Sequence in context: A161206 A025728 A084537 this_sequence A063541 A011898
A098577
%Y A002061 Adjacent sequences: A002058 A002059 A002060 this_sequence A002062 A002063
A002064
%K A002061 nonn,easy,nice,new
%O A002061 0,3
%A A002061 N. J. A. Sloane (njas(AT)research.att.com).
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