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 A002061 Central polygonal numbers: a(n) = n^2 - n + 1. (Formerly M2638 N1049) 276

%I M2638 N1049

%S 1,1,3,7,13,21,31,43,57,73,91,111,133,157,183,211,241,273,307,343,381,

%T 421,463,507,553,601,651,703,757,813,871,931,993,1057,1123,1191,1261,

%U 1333,1407,1483,1561,1641,1723,1807,1893,1981,2071,2163,2257,2353,2451,2551,2653

%N Central polygonal numbers: a(n) = n^2 - n + 1.

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

%C ...2....

%C ....P...

%C ...2.n..

%C (P with three attachments).

%C 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 a(n) = Phi_6(n) = Phi_3(n-1), where Phi_k is the k-th cyclotomic polynomial.

%C Maximal number of interior regions formed by n intersecting circles, for n >= 1. - _Amarnath Murthy_, Jul 07 2001

%C 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_, May 19 2001

%C (n*a(n+1)+1)/(n^2+1) is the smallest integer of the form (n*k+1)/(n^2+1). - _Benoit Cloitre_, May 02 2002

%C 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 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_, Aug 23 2002

%C 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_, Oct 20 2002

%C Arithmetic mean of next 2n - 1 numbers. - _Amarnath Murthy_, Feb 16 2004

%C 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_, Mar 25 2004

%C 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_, Apr 01 2004

%C 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_, Apr 25 2006

%C 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 Also n^3 mod (n^2+1). - _Zak Seidov_, Aug 31 2006

%C Also, omitting the first 1, the main diagonal of A081344. - _Zak Seidov_, Oct 05 2006

%C 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_, Nov 10 2006

%C 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_, Jan 25 2007

%C Complement of A135668. - _Kieren MacMillan_, Dec 16 2007

%C From _William A. Tedeschi_, Feb 29 2008: (Start)

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

%C .

%C 7---8---9--10

%C | |

%C 6 1---2 11

%C | | |

%C 5---4---3 12

%C |

%C 16--15--14--13

%C .

%C Cf. A137928. (End)

%C a(n) = AlexanderPolynomial[n] defined as Det[Transpose[S]-n S] where S is Seifert matrix {{-1, 1}, {0, -1}}. - _Artur Jasinski_, Mar 31 2008

%C 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_, May 10 2008

%C Starting (1, 3, 7, 13, ...) = triangle A158821 * [1, 2, 3, ...]. - _Gary W. Adamson_, Mar 28 2009

%C Starting with offset 1 = triangle A128229 * [1,2,3,...]. - _Gary W. Adamson_, Mar 26 2009

%C 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. - _Jaroslav Krizek_, Jun 21 2009

%C For n > 0: a(n) = A170950(A002522(n-1)), A170950(a(n)) = A174114(n), A170949(a(n)) = A002522(n-1). - _Reinhard Zumkeller_, Mar 08 2010

%C a(n) = A176271(n,1) for n > 0. - _Reinhard Zumkeller_, Apr 13 2010

%C a(n) == 3 (mod n+1). - _Bruno Berselli_, Jun 03 2010

%C From _Emeric Deutsch_, Sep 23 2010: (Start)

%C 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.

%C (End)

%C 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

%C a(n) such that A002522(n-1) * A002522(n) = A002522(a(n)) where A002522(n) = n^2 + 1. - _Michel Lagneau_, Feb 10 2012

%C Left edge of the triangle in A214661: a(n) = A214661(n, 1), for n > 0. - _Reinhard Zumkeller_, Jul 25 2012

%C a(n) = A215630(n, 1), for n > 0; a(n) = A215631(n-1, 1), for n > 1. - _Reinhard Zumkeller_, Nov 11 2012

%C Sum_{n > 0} arccot(a(n)) = Pi/2. - _Franz Vrabec_, Dec 02 2012

%C If you draw a triangle with one side of unit length and one side of length n, with an angle of Pi/3 radians between them, then the length of the third side of the triangle will be the square root of a(n). - _Elliott Line_, Jan 24 2013

%C a(n) = A228643(n, 1), for n > 0. - _Reinhard Zumkeller_, Aug 29 2013

%C a(n+1) is the number j such that j^2 = j + m + sqrt(j*m), with corresponding number m given by A100019(n). Also: sqrt(j*m) = A027444(n) = n * a(n+1). - _Richard R. Forberg_, Sep 03 2013

%C Let p(x) the interpolating polynomial of degree n-1 passing through the n points (n,n) and (1,1), (2,1), ..., (n-1,1). Then p(n+1) = a(n). - _Giovanni Resta_, Feb 09 2014

%C The number of square roots >= sqrt(n) and < n+1 (n >= 0) gives essentially the same sequence, 1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, ... . - _Michael G. Kaarhus_, May 21 2014

%C For n>1: a(n) is the maximum total number of queens that can coexist without attacking each other on an [n+1] X [n+1] chessboard. Specifically, this will be a lone queen of one color placed in any position on the perimeter of the board, facing an opponent's "army" of size a(n)-1 == A002378(n-1). - _Bob Selcoe_, Feb 07 2015

%C A256188(a(n)) = 1. - _Reinhard Zumkeller_, Mar 26 2015

%C a(n+1) is, for n >= 1, the number of points as well as the number of lines of a finite projective plane of order n (cf. Hughes and Piper, 1973, Theorem 3.5., pp. 79-80). For n = 3, a(4) = 13, see the 'Finite example' in the Wikipedia link, section 2.3, for the point-line matrix. - _Wolfdieter Lang_, Nov 20 2015

%C Sum_{n>=0} 1/a(n) = 1 + Pi*tanh(Pi*sqrt(3)/2)/sqrt(3) = 2.79814728056269018... . - _Vaclav Kotesovec_, Apr 10 2016

%C Denominators of the solution to the generalization of the Feynman triangle problem. If each vertex of a triangle is joined to the point (1/p) along the opposite side (measured say clockwise), then the area of the inner triangle formed by these lines is equal to (p - 2)^2/(p^2 - p + 1) times the area of the original triangle, p>2. For example, when p = 3, the ratio of the areas is 1/7. The numerators of the ratio of the areas is given by A000290 with an offset of 2. [Cook & Wood, 2004]. - _Joe Marasco_, Feb 20 2017

%C n^2 equal triangular tiles with side lengths 1 X 1 X 1 may be put together to form an n X n X n triangle. For n>=2 a(n-1) is the number of different 2 X 2 X 2 triangles being contained. - _Heinrich Ludwig_, Mar 13 2017

%C For n >= 0, the continued fraction [n, n+1, n+2] = (n^3 + 3n^2 + 4n + 2)/(n^2 + 3n + 3) = A034262(n+1)/a(n+2) = n + (n+2)/a(n+2); e.g., [2, 3, 4] = A034262(3)/a(4) = 30/13 = 2 + 4/13. - _Rick L. Shepherd_, Apr 06 2017

%C Starting with b(1) = 1 and not allowing the digit 0, let b(n) = smallest nonnegative integer not yet in the sequence such that the last digit of b(n-1) plus the first digit of b(n) is equal to k for k = 1, ..., 9. This defines 9 finite sequences, each of length equal to a(k), k = 1, ..., 9. (See A289283-A289287 for the cases k = 5..9.) For k = 10, the sequence is infinite (A289288). For example, for k = 4, b(n) = 1,3,11,31,32,2,21,33,12,22,23,13,14. These terms can be ordered in the following array of size k*(k-1)+1:

%C 1 2 3

%C 21 22 23

%C 31 32 33

%C 11 12 13 14

%C .

%C The sequence ends with the term 1k, which lies outside the rectangular array and gives the term +1 (see link).- _Enrique Navarrete_, Jul 02 2017

%C The central polygonal numbers are the delimiters (in parenthesis below) when you write the natural numbers in groups of odd size 2*n+1 starting with the group {2} of size 1: (1) 2 (3) 4,5,6 (7) 8,9,10,11,12 (13) 14,15,16,17,18,19,20 (21) 22,23,24,25,26,27,28,29,30 (31) 32,33,34,35,36,37,38,39,40,41,42 (43) ... - _Enrique Navarrete_, Jul 11 2017

%C Also the number of (non-null) connected induced subgraphs in the n-cycle graph. - _Eric W. Weisstein_, Aug 09 2017

%C Since (n+1)^2 - (n+1) + 1 = n^2 + n + 1 then from 7 onwards these are also exactly the numbers that are represented as 111 in all number bases: 111(2)=7, 111(3)=13, ... - _Ron Knott_, Nov 14 2017

%C Number of binary 2 X (n-1) matrices such that each row and column has at most one 1. - _Dmitry Kamenetsky_, Jan 20 2018

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

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

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

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

%D D. R. Hughes and F. C. Piper. Projective Planes. Springer, 1973.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H N. J. A. Sloane and T. D. Noe, <a href="/A002061/b002061.txt">Table of n, a(n) for n = 0..10000</a> (First 1000 terms from T. D. Noe)

%H 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>, arXiv:math/0107159 [math.CO], 2001.

%H Richard Bean and E. S. Mahmoodian, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00599-X">A new bound on the size of the largest critical set in a Latin square</a>, Discrete Math., 267 (2003), 13-21.

%H R. J. Cook and G. V. Wood, <a href="http://www.jstor.org/stable/3620856"> Feynman's Triangle</a>, Mathematical Gazette, 88:299-302 (2004).

%H Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~guoniu/papers/p77puzzle.pdf">Enumeration of Standard Puzzles</a>

%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>

%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary Equations</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.

%H Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Kimberling/kimber67.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.

%H Craig Knecht, <a href="/A002061/a002061.png">Maximum number of elementary hexagons in an order-n hexagon.</a> (2018)

%H Markus Kuba and Alois Panholzer, <a href="http://info.tuwien.ac.at/panholzer/Papers/S01_100928.pdf">Enumeration formulas for pattern restricted Stirling permutations</a> Discrete Math. 312 (2012), no. 21, 3179--3194. MR2957938. - From _N. J. A. Sloane_, Sep 25 2012

%H R. J. Mathar, <a href="http://vixra.org/abs/1608.0380">Tiling hexagons with smaller hexagons and unit triangles</a>, vixra:1608.0380 (2016) eq. (11).

%H R. Munafo, <a href="http://mrob.com/pub/math/seq-a002061.html">Sequence A002061, Hogben's Centered Polygonal Numbers</a>

%H Enrique Navarrete, <a href="/A002061/a002061_2.pdf">Central Polygonal Numbers in Finite Sequences</a>

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="http://dx.doi.org/10.1002/(SICI)1097-461X(1996)60:5&lt;959::AID-QUA2&gt;3.0.CO;2-W">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60 (1996) 959-969.

%H A. Umar, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Umar/umar2.html">Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations</a>, Journal of Integer Sequences, 14 (2011), #11.7.5.

%H S. H. Weintraub, <a href="http://www.jstor.org/stable/4145074">An interesting recursion</a>, Amer. Math. Monthly, 111 (No. 6, 2004), 528-530.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Projective_plane">Projective Plane</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlexanderPolynomial.html">Alexander Polynomial</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConnectedGraph.html">Connected Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FanGraph.html">Fan Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Vertex-InducedSubgraph.html">Vertex-Induced Subgraph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WheelGraph.html">Wheel Graph</a>

%H <a href="/index/Cy#CyclotomicPolynomialsValuesAtX">Index to values of cyclotomic polynomials of integer argument</a>

%H <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: (1-2*x+3*x^2)/(1-x)^3. - _Simon Plouffe_ in his 1992 dissertation

%F a(n) = -(n-5)*a(n-1) + (n-2)*a(n-2).

%F a(1-n) = a(n). - _Michael Somos_, Sep 04 2006

%F a(n) = a(n-1) + 2*(n-1) = 2*a(n-1) - a(n-2) + 2 = 1+A002378(n-1) = 2*A000124(n-1) - 1. - _Henry Bottomley_, Oct 02 2000 [Corrected by _N. J. A. Sloane_, Jul 18 2010]

%F a(n) = A000217(n) + A000217(n-2) (sum of two triangular numbers).

%F 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_, Mar 13 2003

%F a(n) = ceiling((n-1/2)^2). - _Benoit Cloitre_, Apr 16 2003. [Hence the terms are about midway between successive squares and so (except for 1) are not squares. - _N. J. A. Sloane_, Nov 01 2005]

%F a(n) = 1 + Sum_{j=0..n-1} (2*j). - Xavier Acloque, Oct 08 2003

%F a(n) = floor(t(n^2)/t(n)), where t(n) = A000217(n). - _Jon Perry_, Feb 14 2004

%F 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_, Nov 11 2004

%F 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_, Apr 13 2006

%F a(n+3) is the numerator of ((n + 1)! + (n - 1)!)/ n!. - _Artur Jasinski_, Jan 09 2007

%F a(n) = A132111(n-1, 1), for n > 1. - _Reinhard Zumkeller_, Aug 10 2007

%F a(n) = Det[Transpose[{{-1, 1}, {0, -1}}] - n {{-1, 1}, {0, -1}}]. - _Artur Jasinski_, Mar 31 2008

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 3. - _Jaume Oliver Lafont_, Dec 02 2008

%F a(n) = (n-1)^2 + (n-1) + 1 = 111 read in base n-1 (for n > 2). - _Jason Kimberley_, Oct 18 2011

%F a(n) = sqrt(A058031(n)). - _Richard R. Forberg_, Sep 03 2013

%F G.f.: 1 / (1 - x / (1 - 2*x / (1 + x / (1 - 2*x / (1 + x))))). - _Michael Somos_, Apr 03 2014

%F a(n) = A243201(n - 1) / A003215(n - 1), n > 0. - _Mathew Englander_, Jun 03 2014

%F For n >= 2, a(n) = ceiling(4/(Sum_{k = A000217(n-1)..A000217(n) - 1}, 1/k)). - _Richard R. Forberg_, Aug 17 2014

%F a(n) = A101321(2,n-1). - _R. J. Mathar_, Jul 28 2016

%F a(n) = A000217(n-1) + A000124(n-1), n > 0. - _Torlach Rush_, Aug 06 2018

%e G.f. = 1 + x + 3*x^2 + 7*x^3 + 13*x^4 + 21*x^5 + 31*x^6 + 43*x^7 + ...

%p A002061 := proc(n)

%p numtheory[cyclotomic](6,n) ;

%p end proc:

%p seq(A002061(n), n=0..20); # _R. J. Mathar_, Feb 07 2014

%t FoldList[#1 + #2 &, 1, 2 Range[0, 50]] (* _Robert G. Wilson v_, Feb 02 2011 *)

%t LinearRecurrence[{3, -3, 1}, {1, 1, 3}, 60] (* _Harvey P. Dale_, May 25 2011 *)

%t Table[n^2 - n + 1, {n, 0, 50}] (* _Wesley Ivan Hurt_, Jun 12 2014 *)

%t CoefficientList[Series[(1 - 2x + 3x^2)/(1 - x)^3, {x, 0, 52}], x] (* _Robert G. Wilson v_, Feb 18 2018 *)

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

%o (Maxima) makelist(n^2 - n + 1,n,0,55); /* _Martin Ettl_, Oct 16 2012 */

%o a002061 n = n * (n - 1) + 1 -- _Reinhard Zumkeller_, Dec 18 2013

%o (MAGMA) [ n^2 - n + 1 : n in [0..50] ]; // _Wesley Ivan Hurt_, Jun 12 2014

%o (GAP) List([0..50], n->n^2-*n+1); # _Muniru A Asiru_, May 27 2018

%Y Cf. A000124, A000217, A001263, A001844, A002383, A004273, A005408, A007645, A014206, A051890, A055494, A091776, A132014, A132382, A135668, A137928, A139250, A256188, A028387.

%Y Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.

%Y Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.

%Y Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.

%K nonn,easy,nice,changed

%O 0,3

%A _N. J. A. Sloane_

%E Partially edited by _Joerg Arndt_, Mar 11 2010

%E Partially edited by _Bruno Berselli_, Dec 19 2013

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Last modified January 21 19:08 EST 2019. Contains 319350 sequences. (Running on oeis4.)