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A001844 Centered square numbers: a(n) = 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, Z=Y+1) ordered by increasing Z; then sequence gives Z values.

%I A001844 M3826 N1567 #596 Jun 21 2025 23:33:37
%S A001844 1,5,13,25,41,61,85,113,145,181,221,265,313,365,421,481,545,613,685,
%T A001844 761,841,925,1013,1105,1201,1301,1405,1513,1625,1741,1861,1985,2113,
%U A001844 2245,2381,2521,2665,2813,2965,3121,3281,3445,3613,3785,3961,4141,4325,4513
%N A001844 Centered square numbers: a(n) = 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, Z=Y+1) ordered by increasing Z; then sequence gives Z values.
%C A001844 These are Hogben's central polygonal numbers denoted by
%C A001844 ...2...
%C A001844 ....P..
%C A001844 ...4.n.
%C A001844 Numbers of the form (k^2+1)/2 for k odd.
%C A001844 (y(2x+1))^2 + (y(2x^2+2x))^2 = (y(2x^2+2x+1))^2. E.g., let y = 2, x = 1; (2(2+1))^2 + (2(2+2))^2 = (2(2+2+1))^2, (2(3))^2 + (2(4))^2 = (2(5))^2, 6^2 + 8^2 = 10^2, 36 + 64 = 100. - Glenn B. Cox (igloos_r_us(AT)canada.com), Apr 08 2002
%C A001844 a(n) is also the number of 3 X 3 magic squares with sum 3(n+1). - Sharon Sela (sharonsela(AT)hotmail.com), May 11 2002
%C A001844 For n > 0, a(n) is the smallest k such that zeta(2) - Sum_{i=1..k} 1/i^2 <= zeta(3) - Sum_{i=1..n} 1/i^3. - _Benoit Cloitre_, May 17 2002
%C A001844 Number of convex polyominoes with a 2 X (n+1) minimal bounding rectangle.
%C A001844 The prime terms are given by A027862. - _Lekraj Beedassy_, Jul 09 2004
%C A001844 First difference of a(n) is 4n = A008586(n). Any entry k of the sequence is followed by k + 2*(1 + sqrt(2k - 1)). - _Lekraj Beedassy_, Jun 04 2006
%C A001844 Integers of the form 1 + x + x^2/2 (generating polynomial is Schur's polynomial as in A127876). - _Artur Jasinski_, Feb 04 2007
%C A001844 If X is an n-set and Y and Z disjoint 2-subsets of X then a(n-4) is equal to the number of 4-subsets of X intersecting both Y and Z. - _Milan Janjic_, Aug 26 2007
%C A001844 Row sums of triangle A132778. - _Gary W. Adamson_, Sep 02 2007
%C A001844 Binomial transform of [1, 4, 4, 0, 0, 0, ...]; = inverse binomial transform of A001788: (1, 6, 24, 80, 240, ...). - _Gary W. Adamson_, Sep 02 2007
%C A001844 Narayana transform (A001263) of [1, 4, 0, 0, 0, ...]. Equals A128064 (unsigned) * [1, 2, 3, ...]. - _Gary W. Adamson_, Dec 29 2007
%C A001844 k such that the Diophantine equation x^3 - y^3 = x*y + k has a solution with y = x-1. If that solution is (x,y) = (m+1,m) then m^2 + (m+1)^2 = k. Note that this Diophantine equation is an elliptic curve and (m+1,m) is an integer point on it. - _James R. Buddenhagen_, Aug 12 2008
%C A001844 Numbers k such that (k, k, 2*k-2) are the sides of an isosceles triangle with integer area. Also, k such that 2*k-1 is a square. - _James R. Buddenhagen_, Oct 17 2008
%C A001844 a(n) is also the least weight of self-conjugate partitions having n+1 different odd parts. - _Augustine O. Munagi_, Dec 18 2008
%C A001844 Prefaced with a "1": (1, 1, 5, 13, 25, 41, ...) = A153869 * (1, 2, 3, ...). - _Gary W. Adamson_, Jan 03 2009
%C A001844 Prefaced with a "1": (1, 1, 5, 13, 25, 41, ...) where a(n) = 2n*(n-1)+1, all tuples of square numbers (X-Y, X, X+Y) are produced by ((m*(a(n)-2n))^2, (m*a(n))^2, (m*(a(n)+2n-2))^2) where m is a whole number. - _Doug Bell_, Feb 27 2009
%C A001844 Equals (1, 2, 3, ...) convolved with (1, 3, 4, 4, 4, ...). E.g., a(3) = 25 = (1, 2, 3, 4) dot (4, 4, 3, 1) = (4 + 8 + 9 + 4). - _Gary W. Adamson_, May 01 2009
%C A001844 The running sum of squares taken two at a time. - Al Hakanson (hawkuu(AT)gmail.com), May 18 2009
%C A001844 Equals the odd integers convolved with (1, 2, 2, 2, ...). - _Gary W. Adamson_, May 25 2009
%C A001844 Equals the triangular numbers convolved with [1, 2, 1, 0, 0, 0, ...]. - _Gary W. Adamson_ & _Alexander R. Povolotsky_, May 29 2009
%C A001844 When the positive integers are written in a square array by diagonals as in A038722, a(n) gives the numbers appearing on the main diagonal. - _Joshua Zucker_, Jul 07 2009
%C A001844 The finite continued fraction [n,1,1,n] = (2n+1)/(2n^2 + 2n + 1) = (2n+1)/a(n); and the squares of the first two denominators of the convergents = a(n). E.g., the convergents and value of [4,1,1,4] = 1/4, 1/5, 2/9, 9/41 where 4^2 + 5^2 = 41. - _Gary W. Adamson_, Jul 15 2010
%C A001844 From _Keith Tyler_, Aug 10 2010: (Start)
%C A001844 Running sum of A008574.
%C A001844 Square open pyramidal number; that is, the number of elements in a square pyramid of height (n) with only surface and no bottom nodes. (End)
%C A001844 For k>0, x^4 + x^2 + k factors over the integers iff sqrt(k) is in this sequence. - _James R. Buddenhagen_, Aug 15 2010
%C A001844 Create the simple continued fraction from Pythagorean triples to get [2n + 1; 2n^2 + 2n, 2n^2 + 2n + 1]; its value equals the rational number 2n + 1 + a(n) / (4n^4 + 8n^3 + 6n^2 + 2n + 1). - _J. M. Bergot_, Sep 30 2011
%C A001844 a(n), n >= 1, has in its prime number factorization only primes of the form 4*k+1, i.e., congruent to 1 (mod 4) (see A002144). This follows from the fact that a(n) is a primitive sum of two squares and odd. See Theorem 3.20, p. 164, in the given Niven-Zuckerman-Montgomery reference. E.g., a(3) = 25 = 5^2, a(6) = 85 = 5*17. - _Wolfdieter Lang_, Mar 08 2012
%C A001844 From _Ant King_, Jun 15 2012: (Start)
%C A001844 a(n) is congruent to 1 (mod 4) for all n.
%C A001844 The digital roots of the a(n) form a purely periodic palindromic 9-cycle 1, 5, 4, 7, 5, 7, 4, 5, 1.
%C A001844 The units' digits of the a(n) form a purely periodic palindromic 5-cycle 1, 5, 3, 5, 1.
%C A001844 (End)
%C A001844 Number of integer solutions (x,y) of |x| + |y| <= n. Geometrically: number of lattice points inside a square with vertices (n,0), (0,-n), (-n,0), (0,n). - _César Eliud Lozada_, Sep 18 2012
%C A001844 (a(n)-1)/a(n) = 2*x / (1+x^2) where x = n/(n+1). Note that in this form, this is the velocity-addition formula according to the special theory of relativity (two objects traveling at 1/(n+1) slower than c relative to each other appear to travel at 1/a(n) less than c to a stationary observer). - _Christian N. K. Anderson_, May 20 2013 [Corrected by _Rémi Guillaume_, May 22 2025]
%C A001844 A geometric curiosity: the envelope of the circles x^2 + (y-a(n)/2)^2 = ((2n+1)/2)^2 is the parabola y = x^2, the n=0 circle being the osculating circle at the parabola vertex. - _Jean-François Alcover_, Dec 02 2013
%C A001844 Draw n ellipses in the plane (n>0), any 2 meeting in 4 points; a(n-1) gives the number of internal regions into which the plane is divided (cf. A051890, A046092); a(n-1) = A051890(n) - 1 = A046092(n-1) + 1. - _Jaroslav Krizek_, Dec 27 2013
%C A001844 a(n) is also, of course, the scalar product of the 2-vector (n, n+1) (or (n+1, n)) with itself. The unique inverse of (n, n+1) as vector in the Clifford algebra over the Euclidean 2-space is (1/a(n))(0, n, n+1, 0) (similarly for the other vector). In general the unique inverse of such a nonzero vector v (odd element in Cl_2) is v^(-1) = (1/|v|^2) v. Note that the inverse with respect to the scalar product is not unique for any nonzero vector. See the P. Lounesto reference, sects. 1.7 - 1.12, pp. 7-14. See also the Oct 15 2014 comment in A147973. - _Wolfdieter Lang_, Nov 06 2014
%C A001844 Subsequence of A004431, for n >= 1. - _Bob Selcoe_, Mar 23 2016
%C A001844 Numbers k such that 2k - 1 is a perfect square. - _Juri-Stepan Gerasimov_, Apr 06 2016
%C A001844 The number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 574", based on the 5-celled von Neumann neighborhood. - _Robert Price_, May 13 2016
%C A001844 a(n) is the first integer in a sum of (2*n + 1)^2 consecutive integers that equals (2*n + 1)^4. - _Patrick J. McNab_, Dec 24 2016
%C A001844 Central elements of odd-length rows of the triangular array of positive integers. a(n) is the mean of the numbers in the (2*n + 1)-th row of this triangle. - _David James Sycamore_, Aug 01 2018
%C A001844 Intersection of A000982 and A080827. - _David James Sycamore_, Aug 07 2018
%C A001844 An off-diagonal of the array of Delannoy numbers, A008288, (or a row/column when the array is shown as a square). As such, this is one of the crystal ball sequences. - _Jack W Grahl_, Feb 15 2021 and _Shel Kaphan_, Jan 18 2023
%C A001844 a(n) appears as a solution to a "Riddler Express" puzzle on the FiveThirtyEight website. The Jan 21 2022 issue (problem) and the Jan 28 2022 issue (solution) present the following puzzle and include a proof. - Fold a square piece of paper in half, obtaining a rectangle. Fold again to obtain a square with 1/4 the size of the original square. Then make n cuts through the folded paper. a(n) is the greatest number of pieces of the unfolded paper after the cutting. - _Manfred Boergens_, Feb 22 2022
%C A001844 a(n) is (1/6) times the number of 2 X 2 triangles in the n-th order hexagram with 12*n^2 cells. - _Donghwi Park_, Feb 06 2024
%C A001844 If k is a centered square number, its index in this sequence is n = (sqrt(2k-1)-1)/2. - _Rémi Guillaume_, Mar 30 2025.
%C A001844 Row sums of the symmetric triangle of odd numbers [1]; [1, 3, 1]; [1, 3, 5, 3, 1]; [1, 3, 5, 7, 5, 3, 1]; .... - _Marco Zárate_, Jun 15 2025
%D A001844 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.
%D A001844 A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, p. 125, 1964.
%D A001844 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
%D A001844 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 50.
%D A001844 Pertti Lounesto, Clifford Algebras and Spinors, second edition, Cambridge University Press, 2001.
%D A001844 S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.
%D A001844 Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991.
%D A001844 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001844 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A001844 Travers et al., The Mysterious Lost Proof, Using Advanced Algebra, (1976), pp. 27.
%H A001844 T. D. Noe, <a href="/A001844/b001844.txt">Table of n, a(n) for n = 0..1000</a>
%H A001844 M. Ahmed, J. De Loera and R. Hemmecke, <a href="http://arxiv.org/abs/math/0201108">Polyhedral Cones of Magic Cubes and Squares</a>, arXiv:math/0201108 [math.CO], 2002.
%H A001844 U. Alfred, <a href="http://www.jstor.org/stable/2688938">n and n+1 consecutive integers with equal sums of squares</a>, Math. Mag., 35 (1962), 155-164.
%H A001844 Bela Bajnok, <a href="https://arxiv.org/abs/1705.07444">Additive Combinatorics: A Menu of Research Problems</a>, arXiv:1705.07444 [math.NT], May 2017. See Section 2.3.
%H A001844 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 A001844 Matthias Beck, Moshe Cohen, Jessica Cuomo, and Paul Gribelyuk, <a href="https://arxiv.org/abs/math/0201013">The number of "magic" squares and hypercubes</a>, arXiv:math/0201013 [math.CO], 2002-2005.
%H A001844 Arthur T. Benjamin and Doron Zeilberger, <a href="http://www.emis.de/journals/INTEGERS/papers/f30/f30.Abstract.html">Pythagorean Primes and Palindromic Continued Fractions</a>, Electronic Journal of Combinatorial Number Theory, 5(1) 2005, #A30.
%H A001844 J. A. De Loera, D. C. Haws and M. Koppe, <a href="http://arxiv.org/abs/0710.4346">Ehrhart Polynomials of Matroid Polytopes and Polymatroids</a>, arXiv:0710.4346 [math.CO], 2007; Discrete Comput. Geom., 42 (2009), 670-702.
%H A001844 FiveThirtyEight, <a href="https://fivethirtyeight.com/features/can-you-tune-up-the-truck/">"Riddler Express" paper cutting problem and solution</a>, Jan 28 2022.
%H A001844 D. C. Haws, <a href="http://www.math.ucdavis.edu/~haws/Matroids/">Matroids</a> [Broken link, Oct 30 2017]
%H A001844 D. C. Haws, <a href="https://www.math.ucdavis.edu/~mkoeppe/art/Matroids/">Matroids</a> [Copy on website of Matthias Koeppe]
%H A001844 D. C. Haws, <a href="/A160747/a160747.pdf">Matroids</a> [Cached copy, pdf file only]
%H A001844 L. 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, pp. 22 and 36.
%H A001844 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>. [Broken link; <a href="https://web.archive.org/web/20110204173116/http://www.pmfbl.org:80/janjic/">WayBackMachine archive</a>.]
%H A001844 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Janjic/janjic19.html">On a class of polynomials with integer coefficients</a>, JIS 11 (2008) 08.5.
%H A001844 Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.
%H A001844 Clark Kimberling, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary Equations</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
%H A001844 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 A001844 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>.
%H A001844 G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1973__20__3_0">Sur les hiérarchies de segments</a>, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.
%H A001844 G. Kreweras, <a href="/A001844/a001844.pdf">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
%H A001844 A. O. Munagi, <a href="http://dx.doi.org/10.1016/j.disc.2007.05.022">Pairing conjugate partitions by residue classes</a>, Discrete Math., 308 (2008), 2492-2501.
%H A001844 Mitchell Paukner, Lucy Pepin, Manda Riehl, and Jarred Wieser, <a href="http://arxiv.org/abs/1511.00080">Pattern Avoidance in Task-Precedence Posets</a>, arXiv:1511.00080 [math.CO], 2015-2016.
%H A001844 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 A001844 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A001844 John A. Jr. Rochowicz, <a href="http://epublications.bond.edu.au/ejsie/vol8/iss2/4">Harmonic Numbers: Insights, Approximations and Applications</a>, Spreadsheets in Education (eJSiE), 2015, Vol. 8: Iss. 2, Article 4.
%H A001844 Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3252339">Groupoid of OEIS A001844 Numbers (centered square numbers)</a>, Politecnico di Torino, Italy (2019).
%H A001844 R. G. Stanton and D. D. Cowan, <a href="http://dx.doi.org/10.1137/1012049">Note on a "square" functional equation</a>, SIAM Rev., 12 (1970), 277-279.
%H A001844 David James Sycamore, <a href="/A001844/a001844.jpg">Triangular array</a>
%H A001844 Leo Tavares, <a href="/A001844/a001844_1.jpg">Illustration: Diamond Rows</a>
%H A001844 B. K. Teo and N. J. A. Sloane, <a href="http://dx.doi.org/10.1021/ic00220a025">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem. 24 (1985), 4545-4558.
%H A001844 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Number</a>, <a href="https://mathworld.wolfram.com/CenteredSquareNumber.html">Centered Square Number</a>, <a href="https://mathworld.wolfram.com/Diamond.html">Diamond</a>, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>, and <a href="https://mathworld.wolfram.com/vonNeumannNeighborhood.html">von Neumann Neighborhood</a>.
%H A001844 <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>
%H A001844 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>.
%H A001844 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%H A001844 Nicolay Avilov, <a href="/A001844/a001844_2.jpg">Graphical representation of the sequence members</a>
%F A001844 a(n) = 2*n^2 + 2*n + 1 = n^2 + (n+1)^2.
%F A001844 a(n) = 1 + 3 + 5 + ... + 2*n-1 + 2*n+1 + 2*n-1 + ... + 3 + 1. - _Amarnath Murthy_, May 28 2001
%F A001844 a(n) = 1/real(z(n+1)) where z(1)=i, (i^2=-1), z(k+1) = 1/(z(k)+2i). - _Benoit Cloitre_, Aug 06 2002
%F A001844 Nearest integer to 1/Sum_{k>n} 1/k^3. - _Benoit Cloitre_, Jun 12 2003
%F A001844 G.f.: (1+x)^2/(1-x)^3.
%F A001844 E.g.f.: exp(x)*(1+4x+2x^2).
%F A001844 a(n) = a(n-1) + 4n.
%F A001844 a(-n) = a(n-1).
%F A001844 a(n) = A064094(n+3, n) (fourth diagonal).
%F A001844 a(n) = 1 + Sum_{j=0..n} 4*j. - Xavier Acloque, Oct 08 2003
%F A001844 a(n) = A046092(n)+1 = (A016754(n)+1)/2. - _Lekraj Beedassy_, May 25 2004
%F A001844 a(n) = Sum_{k=0..n+1} (-1)^k*binomial(n, k)*Sum_{j=0..n-k+1} binomial(n-k+1, j)*j^2. - _Paul Barry_, Dec 22 2004
%F A001844 a(n) = ceiling((2n+1)^2/2). - _Paul Barry_, Jul 16 2006
%F A001844 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0)=1, a(1)=5, a(2)=13. - _Jaume Oliver Lafont_, Dec 02 2008
%F A001844 a(n)*a(n-1) = 4*n^4 + 1 for n > 0. - _Reinhard Zumkeller_, Feb 12 2009
%F A001844 Prefaced with a "1" (1, 1, 5, 13, 25, 41, ...): a(n) = 2*n*(n-1)+1. - _Doug Bell_, Feb 27 2009
%F A001844 a(n) = sqrt((A056220(n)^2 + A056220(n+1)^2) / 2). - _Doug Bell_, Mar 08 2009
%F A001844 a(n) = floor(2*(n+1)^3/(n+2)). - _Gary Detlefs_, May 20 2010
%F A001844 a(n) = A000330(n) - A000330(n-2). - _Keith Tyler_, Aug 10 2010
%F A001844 a(n) = A069894(n)/2. - _J. M. Bergot_, Jun 11 2012
%F A001844 a(n) = 2*a(n-1) - a(n-2) + 4. - _Ant King_, Jun 12 2012
%F A001844 Sum_{n>=0} 1/a(n) = (Pi/2)*tanh(Pi/2) = 1.4406595199775... = A228048. - _Ant King_, Jun 15 2012
%F A001844 a(n) = A209297(2*n+1,n+1). - _Reinhard Zumkeller_, Jan 19 2013
%F A001844 a(n)^3 = A048395(n)^2 + A048395(-n-1)^2. - _Vincenzo Librandi_, Jan 19 2013
%F A001844 a(n) = A000217(2n+1) - n. - _Ivan N. Ianakiev_, Nov 08 2013
%F A001844 a(n) = A251599(3*n+1). - _Reinhard Zumkeller_, Dec 13 2014
%F A001844 a(n) = A101321(4,n). - _R. J. Mathar_, Jul 28 2016
%F A001844 From _Ilya Gutkovskiy_, Jul 30 2016: (Start)
%F A001844 a(n) = Sum_{k=0..n} A008574(k).
%F A001844 Sum_{n>=0} (-1)^(n+1)*a(n)/n! = exp(-1) = A068985. (End)
%F A001844 a(n) = 4 * A000217(n) + 1. - _Bruce J. Nicholson_, Jul 10 2017
%F A001844 a(n) = A002522(n) + A005563(n) = A002522(n+1) + A005563(n-1). - _Bruce J. Nicholson_, Aug 05 2017
%F A001844 Sum_{n>=0} a(n)/n! = 7*e. Sum_{n>=0} 1/a(n) = A228048. - _Amiram Eldar_, Jun 20 2020
%F A001844 a(n) = A000326(n+1) + A000217(n-1). - _Charlie Marion_, Nov 16 2020
%F A001844 a(n) = Integral_{x=0..2n+2} |1-x| dx. - _Pedro Caceres_, Dec 29 2020
%F A001844 From _Amiram Eldar_, Feb 17 2021: (Start)
%F A001844 Product_{n>=0} (1 + 1/a(n)) = cosh(sqrt(3)*Pi/2)*sech(Pi/2).
%F A001844 Product_{n>=1} (1 - 1/a(n)) = Pi*csch(Pi)*sinh(Pi/2). (End)
%F A001844 a(n) = A001651(n+1) + 1 - A028242(n). - _Charlie Marion_, Apr 05 2022
%F A001844 a(n) = A016754(n) - A046092(n). - _Leo Tavares_, Sep 16 2022
%F A001844 For n>0, a(n) = A101096(n+2) / 30. - _Andy Nicol_, Feb 06 2025
%F A001844 From _Rémi Guillaume_, Apr 21 2025: (Start)
%F A001844 a(n) = (2*A003215(n)+1)/3.
%F A001844 a(n) = (4*A005448(n+1)-1)/3.
%F A001844 a(n) + a(n-1) = A001845(n) - A001845(n-1), for n >= 1.
%F A001844 a(n) = (A005917(n+1))/(2n+1). (End)
%e A001844 G.f.: 1 + 5*x + 13*x^2 + 25*x^3 + 41*x^4 + 61*x^5 + 85*x^6 + 113*x^7 + 145*x^8 + ...
%e A001844 The first few triples are (1,0,1), (3,4,5), (5,12,13), (7,24,25), ...
%e A001844 The first four such partitions, corresponding to n = 0,1,2,3, i.e., to a(n) = 1,5,13,25, are 1, 3+1+1, 5+3+3+1+1, 7+5+5+3+3+1+1. - _Augustine O. Munagi_, Dec 18 2008
%p A001844 A001844:=-(z+1)**2/(z-1)**3; # _Simon Plouffe_ in his 1992 dissertation
%t A001844 Table[2n(n + 1) + 1, {n, 0, 50}]
%t A001844 FoldList[#1 + #2 &, 1, 4 Range@ 50] (* _Robert G. Wilson v_, Feb 02 2011 *)
%t A001844 maxn := 47; Flatten[Table[SeriesCoefficient[Series[(n + (n - 1)*x)/(1 - x)^2, {x, 0, maxn}], k], {n, maxn}, {k, n - 1, n - 1}]] (* _L. Edson Jeffery_, Aug 24 2014 *)
%t A001844 CoefficientList[ Series[-(x^2 + 2x + 1)/(x - 1)^3, {x, 0, 48}], x] (* or *)
%t A001844 LinearRecurrence[{3, -3, 1}, {1, 5, 13}, 48] (* _Robert G. Wilson v_, Aug 01 2018 *)
%t A001844 Total/@Partition[Range[0,50]^2,2,1] (* _Harvey P. Dale_, Dec 05 2020 *)
%t A001844 Table[ j! Coefficient[Series[Exp[x]*(1 + 4*x + 2*x^2), {x, 0, 20}], x,
%t A001844 j], {j, 0, 20}] (* _Nikolaos Pantelidis_, Feb 07 2023 *)
%o A001844 (PARI) {a(n) = 2*n*(n+1) + 1};
%o A001844 (PARI) x='x+O('x^200); Vec((1+x)^2/(1-x)^3) \\ _Altug Alkan_, Mar 23 2016
%o A001844 (Sage) [i**2 + (i + 1)**2 for i in range(46)] # _Zerinvary Lajos_, Jun 27 2008
%o A001844 (Haskell)
%o A001844 a001844 n = 2 * n * (n + 1) + 1
%o A001844 a001844_list = zipWith (+) a000290_list $ tail a000290_list
%o A001844 -- _Reinhard Zumkeller_, Dec 04 2012
%o A001844 (Magma) [2*n^2 + 2*n + 1: n in [0..50]]; // _Vincenzo Librandi_, Jan 19 2013
%o A001844 (Magma) [n: n in [0..4400] | IsSquare(2*n-1)]; // _Juri-Stepan Gerasimov_, Apr 06 2016
%o A001844 (Python) print([2*n*(n+1)+1 for n in range(48)]) # _Michael S. Branicky_, Jan 05 2021
%Y A001844 X values are A005408; Y values are A046092.
%Y A001844 Cf. A008586 (first differences), A005900 (partial sums), A254373 (digital roots).
%Y A001844 Cf. A069894, A000217, A000290, A001263, A001788, A001845, A002061, A002144, A003215, A005448, A005891, A005917, A008844 (square terms), A027862 (prime terms), A048395, A051890, A056106, A101096, A127876, A128064, A132778, A147973, A153869, A240876, A251599, A000982, A080827.
%Y A001844 Subsequence of A004431.
%Y A001844 Right edge of A055096; main diagonal of A069480, A078475, A129312.
%Y A001844 Row n=2 (or column k=2) of A008288.
%Y A001844 Cf. A016754.
%K A001844 nonn,easy,nice
%O A001844 0,2
%A A001844 _N. J. A. Sloane_
%E A001844 Partially edited by _Joerg Arndt_, Mar 11 2010