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A002061
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Central polygonal numbers: n^2 - n + 1.
(Formerly M2638 N1049)
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143
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1, 1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, 273, 307, 343, 381, 421, 463, 507, 553, 601, 651, 703, 757, 813, 871, 931, 993, 1057, 1123, 1191, 1261, 1333, 1407, 1483, 1561, 1641, 1723, 1807, 1893, 1981, 2071, 2163, 2257, 2353, 2451, 2551, 2653
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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 (n*k+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((k*n)^2+(k*(n+(2*m+1)))^2+(k*(n*(n+(2*m+1))+4*T_m))^2)=k*d where m>=0, k>=1, 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. Prime a(n) are listed in A002383. All terms are odd. Prime factors of a(n) are listed in A007645. 3 divides a(3*k-1), 7 divides a(7*k-4) and a(7*k-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(13*k-9) and a(13*k-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
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 = (n^2+1). A000037(a(n)) = A002522(n) = n^2+1. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jun 21 2009]
For n>0: a(n)=A170950(A002522(n-1)), A170950(a(n))=A174114(n), A170949(a(n))=A002522(n-1). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 08 2010]
a(n) = A176271(n,1) for n>0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 13 2010]
Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 23 2010: (Start)
a(n) is also the Wiener index of the fan graph F(n). The fan graph F(n) is defined as the graph obtained by joining each node of an n-node path graph with an additional node. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph. The Wiener polynomial of the graph F(n) is (1/2)t[(n-1)(n-2)t + 2(2n-1)]. Example: a(2)=3 because the corresponding fan graph is a cycle on 3 nodes (a triangle), having distances 1, 1, and 1.
(End)
For all elements k = n^2-n+1 of the sequence, sqrt(4*(k-1)+1) is an integer because 4*(k-1)+1 = (2*n-1)^2 is a perfect square. Building the intersection of this sequence with A000225, k may in addition be of the form k = 2^x-1, which happens only for k = 1, 3, 7, 31, and 8191. [Proof: Still 4*(k-1)+1 = 2^(x+2)-7 must be a perfect square, which has the finite number of solutions provided by A060728: x= 1, 2, 3, 5, or 13.] In other words, the sequence A038198 defines all elements of the form 2^x-1 in this sequence. For example k = 31 = 6*6-6+1; sqrt((31-1)*4+1) = sqrt(121) = 11 = A038198(4). [Alzhekeyev Ascar M, Jun 01 2011]
a(n) such that A002522(n-1) * A002522(n) = A002522(a(n)) where A002522(n) = n^2+1. - Michel Lagneau, Feb 10 2012
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REFERENCES
| 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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Umar, Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations, Journal of Integer Sequences, 14 (2011), #11.7.5.
S. H. Weintraub, An interesting recursion, Amer. Math. Monthly, 111 (No. 6, 2004), 528-530.
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LINKS
| 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
Guo-Niu Han, Enumeration of Standard Puzzles
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.
R. Munafo, Sequence A002061, Hogben's Centered Polygonal Numbers
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.
B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 23 2010]
Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Wheel Graph
Eric Weisstein's World of Mathematics, Fan Graph. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 23 2010]
E.W. Weisstein, "Alexander Polynomial."
Index entries for sequences related to centered polygonal numbers
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
| G.f.: (1-2*x+3*x^2)/(1-x)^3.
a(n) = -(n-5)*a(n-1) + (n-2)*a(n-2).
a(1-n)=a(n). - Michael Somos Sep 04 2006
a(n) = a(n-1) + 2*(n-1) = 2*a(n-1) - a(n-2) + 2 = A002378(n-1) + 1 = 2*A000124(n-1) - 1. - Henry Bottomley (se16(AT)btinternet.com), Oct 02 2000. [Corrected by N. J. A. Sloane, Jul 18 2010]
a(n) = A000217(n) + A000217(n-2) (sum of two triangular numbers).
x*(1+x^2)/(1-x)^3 is g.f. for 0, 1, 3, 7, 13, ... a(n)=2*C(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_{j=0..n-1} (2*j). - 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)= A000217(n). - 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). (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
a(n+3) is the 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)=3*a(n-1)-3*a(n-2)+a(n-3), n>=3. [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 02 2008]
a(n) = 3 (mod n+1). [From Bruno Berselli (berselli.bruno(AT)yahoo.it), Jun 03 2010]
a(n)= n^2 - n + 1 = (n-1)^2 + (n-1) + 1 = 111 read in base n-1 (for n>2). - Jason Kimberley, Oct 18 2011
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EXAMPLE
| For n=1, a(1)=2*1+1-2=1; n=2, a(2)=2*2+1-2=3; n=3, a(3)=2*3+3-2=7 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jul 20 2010]
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MAPLE
| A002061:=-(1-2*z+3*z**2)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| 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]
FoldList[#1 + #2 &, 1, 2 Range[0, 50]] (*Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{3, -3, 1}, {1, 1, 3}, 60] (* From Harvey P. Dale, May 25 2011 *)
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PROG
| (PARI) a(n)=n^2-n+1
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CROSSREFS
| Cf. A001263, A001844, A051890, A000124, A091776, A014206, A055494, A002383, A007645, A132014, A132382, A135668, A137928, A000217, A004273, A005408.
Sequence in context: A161206 A025728 A084537 * A063541 A206246 A171965
Adjacent sequences: A002058 A002059 A002060 * A002062 A002063 A002064
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KEYWORD
| nonn,easy,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Partially edited by Joerg Arndt (arndt(AT)jjj.de), Mar 11 2010
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