%I M2638 N1049 #574 Oct 04 2024 11:23:38
%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 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), that is A000037(a(n)) = A002522(n) = n^2 + 1, for n >= 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 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+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 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 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
%C Observed to be the squares visited by bishop moves on a spirally numbered board and moving to the lowest available unvisited square at each step, beginning at the second term (cf. A316667). It should be noted that the bishop will only travel to squares along the first diagonal of the spiral. - _Benjamin Knight_, Jan 30 2019
%C From _Ed Pegg Jr_, May 16 2019: (Start)
%C Bound for n-subset coverings. Values in A138077 covered by difference sets.
%C C(7,3,2), {1,2,4}
%C C(13,4,2), {0,1,3,9}
%C C(21,5,2), {3,6,7,12,14}
%C C(31,6,2), {1,5,11,24,25,27}
%C C(43,7,2), existence unresolved
%C C(57,8,2), {0,1,6,15,22,26,45,55}
%C Next unresolved cases are C(111,11,2) and C(157,13,2). (End)
%C "In the range we explored carefully, the optimal packings were substantially irregular only for n of the form n = k(k+1)+1, k = 3, 4, 5, 6, 7, i.e., for n = 13, 21, 31, 43, and 57." (cited from Lubachevsky, Graham link, Introduction). - _Rainer Rosenthal_, May 27 2020
%C From _Bernard Schott_, Dec 31 2020: (Start)
%C For n >= 1, a(n) is the number of solutions x in the interval 1 <= x <= n of the equation x^2 - [x^2] = (x - [x])^2, where [x] = floor(x). For n = 3, the a(3) = 7 solutions in the interval [1, 3] are 1, 3/2, 2, 9/4, 5/2, 11/4 and 3.
%C This sequence is the answer to the 4th problem proposed during the 20th British Mathematical Olympiad in 1984 (see link B.M.O 1984. and Gardiner reference). (End)
%C Called "Hogben numbers" after the British zoologist, statistician and writer Lancelot Thomas Hogben (1895-1975). - _Amiram Eldar_, Jun 24 2021
%C Minimum Wiener index of 2-degenerate graphs with n+1 vertices (n>0). A maximal 2-degenerate graph can be constructed from a 2-clique by iteratively adding a new 2-leaf (vertex of degree 2) adjacent to two existing vertices. The extremal graphs are maximal 2-degenerate graphs with diameter at most 2. - _Allan Bickle_, Oct 14 2022
%C a(n) is the number of parking functions of size n avoiding the patterns 123, 213, and 312. - _Lara Pudwell_, Apr 10 2023
%C Repeated iteration of a(k) starting with k=2 produces Sylvester's sequence, i.e., A000058(n) = a^n(2), where a^n is the n-th iterate of a(k). - _Curtis Bechtel_, Apr 04 2024
%D Archimedeans Problems Drive, Eureka, 22 (1959), 15.
%D Steve Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 1 of the British Mathematical Olympiad 2007, page 160.
%D Anthony Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Problem 4 pp. 64 and 173 (1984).
%D Paul R. Halmos, Linear Algebra Problem Book, MAA, 1995, pp. 75-6, 242-4.
%D Ross Honsberger, Ingenuity in Mathematics, Random House, 1970, p. 87.
%D Daniel R. Hughes and Frederick Charles 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, <a href="/A002061/b002061.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms from T. D. Noe)
%H Ayomikun Adeniran and Lara Pudwell, <a href="https://doi.org/10.54550/ECA2023V3S3R17">Pattern avoidance in parking functions</a>, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
%H Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%H Richard Bean and Ebadollah S. Mahmoodian, <a href="https://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 mathematics, Vol. 267, No. 1-3 (2003), pp. 13-21, <a href="http://arXiv.org/abs/math/0107159">arXiv preprint</a>, arXiv:math/0107159 [math.CO], 2001.
%H Allan Bickle and Zhongyuan Che, <a href="https://arxiv.org/abs/1908.09202">Wiener indices of maximal k-degenerate graphs</a>, arXiv:1908.09202 [math.CO], 2019.
%H Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/degeneratediam2-2.pdf">Wiener indices of maximal k-degenerate graphs</a>, International Journal of Mathematical Combinatorics 2 (2021) 68-79.
%H Allan Bickle, <a href="https://doi.org/10.20429/tag.2024.000105">A Survey of Maximal k-degenerate Graphs and k-Trees</a>, Theory and Applications of Graphs 0 1 (2024) Article 5.
%H Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, <a href="https://arxiv.org/abs/2108.04302">Restricted generating trees for weak orderings</a>, arXiv:2108.04302 [math.CO], 2021.
%H British Mathematical Olympiad, <a href="https://bmos.ukmt.org.uk/home/bmo-1984.pdf">1984 - Problem 4</a>.
%H British Mathematical Olympiad, <a href="https://bmos.ukmt.org.uk/home/bmo1-2008.pdf">2007 - Problem 1</a>.
%H R. J. Cook and G. V. Wood, <a href="http://www.jstor.org/stable/3620856">Feynman's Triangle</a>, Mathematical Gazette, Vol. 88, No. 512 (2004), pp. 299-302.
%H Carlos M. da Fonseca and Anthony G. Shannon, <a href="https://doi.org/10.7546/nntdm.2024.30.3.491-498">A formal operator involving Fermatian numbers</a>, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
%H Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~guoniu/papers/p77puzzle.pdf">Enumeration of Standard Puzzles</a>, 2011.
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H Lancelot Hogben, <a href="https://archive.org/details/chanceandchoiceb029729mbp/page/n25">Choice and Chance by Cardpack and Chessboard</a>, Vol. 1, Max Parrish and Co., London, 1950, p. 22.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>.
%H Atabey Kaygun, <a href="https://arxiv.org/abs/2101.02299">Enumerating Labeled Graphs that Realize a Fixed Degree Sequence</a>, arXiv:2101.02299 [math.CO], 2021.
%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 Seq., Vol. 7 (2004), Article 04.1.6.
%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., Vol. 312, No. 21 (2012), pp. 3179-3194. MR2957938. - From _N. J. A. Sloane_, Sep 25 2012
%H Boris D. Lubachevsky and Ronald L. Graham, <a href="https://arxiv.org/abs/math/0412443">Minimum perimeter rectangles that enclose congruent non-overlapping circles</a>, arXiv:math/0412443 [math.MG], 2004-2008.
%H Boris D. Lubachevsky and Ronald L. Graham, <a href="https://doi.org/10.1016/j.disc.2008.03.017">Minimum perimeter rectangles that enclose congruent non-overlapping circles</a>, Discrete Mathematics, Vol. 309, No. 8, (28 April 2009), pp. 1947-1962.
%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 Robert 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="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Bruce E. Sagan, Yeong-Nan Yeh, and Ping Zhang, <a href="http://dx.doi.org/10.1002/(SICI)1097-461X(1996)60:5<959::AID-QUA2>3.0.CO;2-W">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., Vol. 60 (1996), pp. 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, Vol. 14 (2011), #11.7.5.
%H Steven H. Weintraub, <a href="http://www.jstor.org/stable/4145074">An interesting recursion</a>, Amer. Math. Monthly, Vol. 111, No. 6 (2004), pp. 528-530.
%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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Projective_plane">Projective Plane</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).
%H <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.
%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(n) = Phi_6(n) = Phi_3(n-1), where Phi_k is the k-th cyclotomic polynomial.
%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 From _Paul Barry_, Mar 13 2003: (Start)
%F x*(1+x^2)/(1-x)^3 is g.f. for 0, 1, 3, 7, 13, ...
%F a(n) = 2*C(n, 2) + C(n-1, 0).
%F E.g.f.: (1+x^2)*exp(x). (End)
%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+1) 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) = A176271(n,1) for n > 0. - _Reinhard Zumkeller_, Apr 13 2010
%F a(n) == 3 (mod n+1). - _Bruno Berselli_, Jun 03 2010
%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) = A228643(n, 1), for n > 0. - _Reinhard Zumkeller_, Aug 29 2013
%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 A256188(a(n)) = 1. - _Reinhard Zumkeller_, Mar 26 2015
%F Sum_{n>=0} 1/a(n) = 1 + Pi*tanh(Pi*sqrt(3)/2)/sqrt(3) = 2.79814728056269018... . - _Vaclav Kotesovec_, Apr 10 2016
%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
%F Sum_{n>=1} arctan(1/a(n)) = Pi/2. - _Amiram Eldar_, Nov 01 2020
%F Sum_{n=1..M} arctan(1/a(n)) = arctan(M). - _Lee A. Newberg_, May 08 2024
%F From _Amiram Eldar_, Jan 20 2021: (Start)
%F Product_{n>=1} (1 + 1/a(n)) = cosh(sqrt(7)*Pi/2)*sech(sqrt(3)*Pi/2).
%F Product_{n>=2} (1 - 1/a(n)) = Pi*sech(sqrt(3)*Pi/2). (End)
%F For n > 1, sqrt(a(n)+sqrt(a(n)-sqrt(a(n)+sqrt(a(n)- ...)))) = n. - _Diego Rattaggi_, Apr 17 2021
%F a(n) = (1 + (n-1)^4 + n^4) / (1 + (n-1)^2 + n^2) [see link B.M.O. 2007 and Steve Dinh reference]. - _Bernard Schott_, Dec 27 2021
%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 *)
%t Cyclotomic[6, Range[0, 100]] (* _Paolo Xausa_, Feb 09 2024 *)
%o (PARI) a(n) = n^2 - n + 1
%o (Maxima) makelist(n^2 - n + 1,n,0,55); /* _Martin Ettl_, Oct 16 2012 */
%o (Haskell)
%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. A000037, A000124, A000217, A001263, A001844, A002383, A004273, A005408, A005563, 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.
%Y Cf. A010000 (minimum Weiner index of 3-degenerate graphs).
%K nonn,easy,nice
%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