A001844 - OEIS

%I M3826 N1567 #525 Jul 30 2024 01:51:43

%S 1,5,13,25,41,61,85,113,145,181,221,265,313,365,421,481,545,613,685,

%T 761,841,925,1013,1105,1201,1301,1405,1513,1625,1741,1861,1985,2113,

%U 2245,2381,2521,2665,2813,2965,3121,3281,3445,3613,3785,3961,4141,4325,4513

%N 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 These are Hogben's central polygonal numbers denoted by

%C ...2...

%C ....P..

%C ...4.n.

%C a(n) = 1 + 3 + 5 + ... + 2*n-1 + 2*n+1 + 2*n-1 + ... + 3 + 1. - _Amarnath Murthy_, May 28 2001

%C Numbers of the form (k^2+1)/2 for k odd.

%C (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 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 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 Number of convex polyominoes with a 2 X (n+1) minimal bounding rectangle.

%C The prime terms are given by A027862. - _Lekraj Beedassy_, Jul 09 2004

%C 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 Integers of the form 1 + x + x^2/2 (generating polynomial is Schur's polynomial as in A127876). - _Artur Jasinski_, Feb 04 2007

%C 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 Row sums of triangle A132778. - _Gary W. Adamson_, Sep 02 2007

%C 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 Narayana transform (A001263) of [1, 4, 0, 0, 0, ...]. Equals A128064 (unsigned) * [1, 2, 3, ...]. - _Gary W. Adamson_, Dec 29 2007

%C n such that the Diophantine equation x^3 - y^3 = x*y + n has a solution with y = x-1. If that solution is (x,y) = (m+1,m) then m^2 + (m+1)^2 = n. 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 Numbers n such that (n, n, 2*n-2) are the sides of an isosceles triangle with integer area. Also, n such that 2*n-1 is a square. - _James R. Buddenhagen_, Oct 17 2008

%C a(n) is also the least weight of self-conjugate partitions having n+1 different odd parts. - _Augustine O. Munagi_, Dec 18 2008

%C Prefaced with a "1": (1, 1, 5, 13, 25, 41, ...) = A153869 * (1, 2, 3, ...). - _Gary W. Adamson_, Jan 03 2009

%C 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 Equals (1, 2, 3, ...) convolved with (1, 3, 4, 4, 4, ...). a(3) = 25 = (1, 2, 3, 4) dot (4, 4, 3, 1) = (4 + 8 + 9 + 4). - _Gary W. Adamson_, May 01 2009

%C The running sum of squares taken two at a time. - Al Hakanson (hawkuu(AT)gmail.com), May 18 2009

%C Equals the odd integers convolved with (1, 2, 2, 2, ...). - _Gary W. Adamson_, May 25 2009

%C Equals the triangular numbers convolved with [1, 2, 1, 0, 0, 0, ...]. - _Gary W. Adamson_ & _Alexander R. Povolotsky_, May 29 2009

%C 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 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 From _Keith Tyler_, Aug 10 2010: (Start)

%C Running sum of A008574.

%C 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 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 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) / (4*n^4 + 8*n^3 + 6*n^2 + 2*n + 1). - _J. M. Bergot_, Sep 30 2011

%C a(n), n >= 1, has in its prime number factorization only primes of the form 4*k+1, i.e., congruent 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 From _Ant King_, Jun 15 2012: (Start)

%C a(n) is congruent to 1 (mod 4) for all n.

%C The digital roots of the a(n) form a purely periodic palindromic 9-cycle 1, 5, 4, 7, 5, 7, 4, 5, 1.

%C The units' digits of the a(n) form a purely periodic palindromic 5-cycle 1, 5, 3, 5, 1.

%C (End)

%C Number of integer solutions (x,y) of |x| + |y| <= n. Geometrically: number of lattice points inside a square with vertex (n,0), (0,-n), (-n,0), (0,n). - _César Eliud Lozada_, Sep 18 2012

%C (a(n)-1)/a(n) = 2*x / (1+x^2) where x = (n-1)/n. Note that in this form, this is the velocity-addition formula according to the special theory of relativity (two objects traveling at 1/n 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

%C 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 Draw n ellipses in the plane (n>0), any 2 meeting in 4 points; sequence gives number of internal regions into which the plane is divided (cf. A051890, A046092); a(n-1) = A051890(n) - 1 = A046092(n-1) - 2. - _Jaroslav Krizek_, Dec 27 2013

%C 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 Subsequence of A004431, for n >= 1. - _Bob Selcoe_, Mar 23 2016

%C Numbers n such that 2n - 1 is a perfect square. - _Juri-Stepan Gerasimov_, Apr 06 2016

%C 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 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 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 Intersection of A000982 and A080827. - _David James Sycamore_, Aug 07 2018

%C 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 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 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

%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.

%D A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, p. 125, 1964.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.

%D Pertti Lounesto, Clifford Algebras and Spinors, second edition, Cambridge University Press, 2001.

%D S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.

%D 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 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).

%D Travers et al., The Mysterious Lost Proof, Using Advanced Algebra, (1976), pp. 27.

%H T. D. Noe, <a href="/A001844/b001844.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 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 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 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 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 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 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 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 D. C. Haws, <a href="http://www.math.ucdavis.edu/~haws/Matroids/">Matroids</a> [Broken link, Oct 30 2017]

%H D. C. Haws, <a href="https://www.math.ucdavis.edu/~mkoeppe/art/Matroids/">Matroids</a> [Copy on website of Matthias Koeppe]

%H D. C. Haws, <a href="/A160747/a160747.pdf">Matroids</a> [Cached copy, pdf file only]

%H 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 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 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 Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.

%H 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 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 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>.

%H 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 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 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 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 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 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 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 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 David James Sycamore, <a href="/A001844/a001844.jpg">Triangular array</a>

%H Leo Tavares, <a href="/A001844/a001844_1.jpg">Illustration: Diamond Rows</a>

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Number</a>, <a href="http://mathworld.wolfram.com/CenteredSquareNumber.html">Centered Square Number</a>, <a href="http://mathworld.wolfram.com/Diamond.html">Diamond</a>, <a href="http://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>, and <a href="http://mathworld.wolfram.com/vonNeumannNeighborhood.html">von Neumann Neighborhood</a>.

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

%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>.

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

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

%F 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 Nearest integer to 1/Sum_{k>n} 1/k^3. - _Benoit Cloitre_, Jun 12 2003

%F G.f.: (1+x)^2/(1-x)^3.

%F E.g.f.: exp(x)*(1+4x+2x^2).

%F a(n) = a(n-1) + 4n.

%F a(-n) = a(n-1).

%F a(n) = A064094(n+3, n) (fourth diagonal).

%F a(n) = 1 + Sum_{j=0..n} 4*j. - Xavier Acloque, Oct 08 2003

%F a(n) = A046092(n)+1 = (A016754(n)+1)/2. - _Lekraj Beedassy_, May 25 2004

%F 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 a(n) = ceiling((2n+1)^2/2). - _Paul Barry_, Jul 16 2006

%F 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 a(n)*a(n-1) = 4*n^4 + 1 for n > 0. - _Reinhard Zumkeller_, Feb 12 2009

%F Prefaced with a "1" (1, 1, 5, 13, 25, 41, ...): a(n) = 2*n*(n-1)+1. - _Doug Bell_, Feb 27 2009

%F a(n) = sqrt((A056220(n)^2 + A056220(n+1)^2) / 2). - _Doug Bell_, Mar 08 2009

%F a(n) = floor(2*(n+1)^3/(n+2)). - _Gary Detlefs_, May 20 2010

%F a(n) = A000330(n) - A000330(n-2). - _Keith Tyler_, Aug 10 2010

%F a(n) = A069894(n)/2. - _J. M. Bergot_, Jun 11 2012

%F a(n) = 2*a(n-1) - a(n-2) + 4. - _Ant King_, Jun 12 2012

%F Sum_{n>=0} 1/a(n) = Pi/2*tanh(Pi/2) = 1.4406595199775... = A228048. - _Ant King_, Jun 15 2012

%F a(n) = A209297(2*n+1,n+1). - _Reinhard Zumkeller_, Jan 19 2013

%F a(n)^3 = A048395(n)^2 + A048395(-n-1)^2. - _Vincenzo Librandi_, Jan 19 2013

%F a(n) = A000217(2n+1) - n. - _Ivan N. Ianakiev_, Nov 08 2013

%F a(n) = A251599(3*n+1). - _Reinhard Zumkeller_, Dec 13 2014

%F a(n) = A101321(4,n). - _R. J. Mathar_, Jul 28 2016

%F From _Ilya Gutkovskiy_, Jul 30 2016: (Start)

%F a(n) = Sum_{k=0..n} A008574(k).

%F Sum_{n>=0} (-1)^(n+1)*a(n)/n! = exp(-1) = A068985. (End)

%F a(n) = 4 * A000217(n) + 1. - _Bruce J. Nicholson_, Jul 10 2017

%F a(n) = A002522(n) + A005563(n) = A002522(n+1) + A005563(n-1). - _Bruce J. Nicholson_, Aug 05 2017

%F Sum_{n>=0} a(n)/n! = 7*e. Sum_{n>=0} 1/a(n) = A228048. - _Amiram Eldar_, Jun 20 2020

%F a(n) = A000326(n+1) + A000217(n-1). - _Charlie Marion_, Nov 16 2020

%F a(n) = Integral_{x=0..2n+2} |1-x| dx. - _Pedro Caceres_, Dec 29 2020

%F From _Amiram Eldar_, Feb 17 2021: (Start)

%F Product_{n>=0} (1 + 1/a(n)) = cosh(sqrt(3)*Pi/2)*sech(Pi/2).

%F Product_{n>=1} (1 - 1/a(n)) = Pi*csch(Pi)*sinh(Pi/2). (End)

%F a(n) = A001651(n+1) + 1 - A028242(n). - _Charlie Marion_, Apr 05 2022

%F a(n) = A016754(n) - A046092(n). - _Leo Tavares_, Sep 16 2022

%e 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 The first few triples are (1,0,1), (3,4,5), (5,12,13), (7,24,25), ...

%e The first four such partitions, corresponding to a(n) = 0,1,2,3, 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:=-(z+1)**2/(z-1)**3; # _Simon Plouffe_ in his 1992 dissertation

%t Table[2n(n + 1) + 1, {n, 0, 50}]

%t FoldList[#1 + #2 &, 1, 4 Range@ 50] (* _Robert G. Wilson v_, Feb 02 2011 *)

%t 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 CoefficientList[ Series[-(x^2 + 2x + 1)/(x - 1)^3, {x, 0, 48}], x] (* or *)

%t LinearRecurrence[{3, -3, 1}, {1, 5, 13}, 48] (* _Robert G. Wilson v_, Aug 01 2018 *)

%t Total/@Partition[Range[0,50]^2,2,1] (* _Harvey P. Dale_, Dec 05 2020 *)

%t Table[ j! Coefficient[Series[Exp[x]*(1 + 4*x + 2*x^2), {x, 0, 20}], x,

%t j], {j, 0, 20}] (* _Nikolaos Pantelidis_, Feb 07 2023 *)

%o (PARI) {a(n) = 2*n*(n+1) + 1};

%o (PARI) x='x+O('x^200); Vec((1+x)^2/(1-x)^3) \\ _Altug Alkan_, Mar 23 2016

%o (Sage) [i**2 + (i + 1)**2 for i in range(46)] # _Zerinvary Lajos_, Jun 27 2008

%o (Haskell)

%o a001844 n = 2 * n * (n + 1) + 1

%o a001844_list = zipWith (+) a000290_list $ tail a000290_list

%o -- _Reinhard Zumkeller_, Dec 04 2012

%o (Magma) [2*n^2 + 2*n + 1: n in [0..50]]; // _Vincenzo Librandi_, Jan 19 2013

%o (Magma) [n: n in [0..4400] | IsSquare(2*n-1)]; // _Juri-Stepan Gerasimov_, Apr 06 2016

%o (Python) print([2*n*(n+1)+1 for n in range(48)]) # _Michael S. Branicky_, Jan 05 2021

%Y X values are A005408; Y values are A046092.

%Y Cf. A008586 (first differences), A005900 (partial sums), A254373 (digital root).

%Y Cf. A000217, A000290, A001263, A001788, A002061, A004431 (numbers that are the sum of 2 distinct nonzero squares), A005448, A005891, A008844 (terms which are perfect squares), A048395, A051890, A056106, A127876, A128064, A132778, A147973, A153869, A240876, A251599 A000982, A080827, A008288.

%Y Right edge of A055096; main diagonal of A069480, A078475, A129312.

%Y Row n=2 (or column k=2) of A008288.

%Y Cf. A016754.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_

%E Partially edited by _Joerg Arndt_, Mar 11 2010