A080827 -id:A080827 - cp's OEIS frontend

A001844 Centered square numbers: a(n) = 2n(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.

1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, 2113, 2245, 2381, 2521, 2665, 2813, 2965, 3121, 3281, 3445, 3613, 3785, 3961, 4141, 4325, 4513

Offset: 0

N. J. A. Sloane

These are Hogben's central polygonal numbers denoted by
...2...
....P..
...4.n.
Numbers of the form (k^2+1)/2 for k odd.
(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
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
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
Number of convex polyominoes with a 2 X (n+1) minimal bounding rectangle.
The prime terms are given by A027862. - Lekraj Beedassy, Jul 09 2004
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
Integers of the form 1 + x + x^2/2 (generating polynomial is Schur's polynomial as in A127876). - Artur Jasinski, Feb 04 2007
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
Row sums of triangle A132778. - Gary W. Adamson, Sep 02 2007
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
Narayana transform (A001263) of [1, 4, 0, 0, 0, ...]. Equals A128064 (unsigned) * [1, 2, 3, ...]. - Gary W. Adamson, Dec 29 2007
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
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
a(n) is also the least weight of self-conjugate partitions having n+1 different odd parts. - Augustine O. Munagi, Dec 18 2008
Prefaced with a "1": (1, 1, 5, 13, 25, 41, ...) = A153869 * (1, 2, 3, ...). - Gary W. Adamson, Jan 03 2009
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
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
The running sum of squares taken two at a time. - Al Hakanson (hawkuu(AT)gmail.com), May 18 2009
Equals the odd integers convolved with (1, 2, 2, 2, ...). - Gary W. Adamson, May 25 2009
Equals the triangular numbers convolved with [1, 2, 1, 0, 0, 0, ...]. - Gary W. Adamson & Alexander R. Povolotsky, May 29 2009
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
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
From Keith Tyler, Aug 10 2010: (Start)
Running sum of A008574.
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)
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
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
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
From Ant King, Jun 15 2012: (Start)
a(n) is congruent to 1 (mod 4) for all n.
The digital roots of the a(n) form a purely periodic palindromic 9-cycle 1, 5, 4, 7, 5, 7, 4, 5, 1.
The units' digits of the a(n) form a purely periodic palindromic 5-cycle 1, 5, 3, 5, 1.
(End)
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
(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]
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
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
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
Subsequence of A004431, for n >= 1. - Bob Selcoe, Mar 23 2016
Numbers k such that 2k - 1 is a perfect square. - Juri-Stepan Gerasimov, Apr 06 2016
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
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
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
Intersection of A000982 and A080827. - David James Sycamore, Aug 07 2018
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
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
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
If k is a centered square number, its index in this sequence is n = (sqrt(2k-1)-1)/2. - Rémi Guillaume, Mar 30 2025.
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

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

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.
A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, p. 125, 1964.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 50.
Pertti Lounesto, Clifford Algebras and Spinors, second edition, Cambridge University Press, 2001.
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.
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.
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).
Travers et al., The Mysterious Lost Proof, Using Advanced Algebra, (1976), pp. 27.

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Ahmed, J. De Loera and R. Hemmecke, Polyhedral Cones of Magic Cubes and Squares, arXiv:math/0201108 [math.CO], 2002.
U. Alfred, n and n+1 consecutive integers with equal sums of squares, Math. Mag., 35 (1962), 155-164.
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Section 2.3.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Matthias Beck, Moshe Cohen, Jessica Cuomo, and Paul Gribelyuk, The number of "magic" squares and hypercubes, arXiv:math/0201013 [math.CO], 2002-2005.
Arthur T. Benjamin and Doron Zeilberger, Pythagorean Primes and Palindromic Continued Fractions, Electronic Journal of Combinatorial Number Theory, 5(1) 2005, #A30.
J. A. De Loera, D. C. Haws and M. Koppe, Ehrhart Polynomials of Matroid Polytopes and Polymatroids, arXiv:0710.4346 [math.CO], 2007; Discrete Comput. Geom., 42 (2009), 670-702.
FiveThirtyEight, "Riddler Express" paper cutting problem and solution, Jan 28 2022.
D. C. Haws, Matroids [Broken link, Oct 30 2017]
D. C. Haws, Matroids [Copy on website of Matthias Koeppe]
D. C. Haws, Matroids [Cached copy, pdf file only]
L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, pp. 22 and 36.
Milan Janjic, Two Enumerative Functions. [Broken link; WayBackMachine archive.]
Milan Janjic, On a class of polynomials with integer coefficients, JIS 11 (2008) 08.5.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
Ron Knott, Pythagorean Triples and Online Calculators.
G. Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501.
Mitchell Paukner, Lucy Pepin, Manda Riehl, and Jarred Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015-2016.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
John A. Jr. Rochowicz, Harmonic Numbers: Insights, Approximations and Applications, Spreadsheets in Education (eJSiE), 2015, Vol. 8: Iss. 2, Article 4.
Amelia Carolina Sparavigna, Groupoid of OEIS A001844 Numbers (centered square numbers), Politecnico di Torino, Italy (2019).
R. G. Stanton and D. D. Cowan, Note on a "square" functional equation, SIAM Rev., 12 (1970), 277-279.
David James Sycamore, Triangular array
Leo Tavares, Illustration: Diamond Rows
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
Eric Weisstein's World of Mathematics, Centered Polygonal Number, Centered Square Number, Diamond, Pythagorean Triple, and von Neumann Neighborhood.
Index entries for sequences related to centered polygonal numbers
Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Nicolay Avilov, Graphical representation of the sequence members

X values are A005408; Y values are A046092.
Cf. A008586 (first differences), A005900 (partial sums), A254373 (digital roots).
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.
Subsequence of A004431.
Right edge of A055096; main diagonal of A069480, A078475, A129312.
Row n=2 (or column k=2) of A008288.
Cf. A016754.

a001844 n = 2 * n * (n + 1) + 1
a001844_list = zipWith (+) a000290_list $ tail a000290_list
-- Reinhard Zumkeller, Dec 04 2012

[2*n^2 + 2*n + 1: n in [0..50]]; // Vincenzo Librandi, Jan 19 2013

[n: n in [0..4400] | IsSquare(2*n-1)]; // Juri-Stepan Gerasimov, Apr 06 2016

A001844:=-(z+1)**2/(z-1)**3; # Simon Plouffe in his 1992 dissertation

Table[2n(n + 1) + 1, {n, 0, 50}]
FoldList[#1 + #2 &, 1, 4 Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)
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 *)
CoefficientList[ Series[-(x^2 + 2x + 1)/(x - 1)^3, {x, 0, 48}], x] (* or *)
LinearRecurrence[{3, -3, 1}, {1, 5, 13}, 48] (* Robert G. Wilson v, Aug 01 2018 *)
Total/@Partition[Range[0,50]^2,2,1] (* Harvey P. Dale, Dec 05 2020 *)
Table[ j! Coefficient[Series[Exp[x]*(1 + 4*x + 2*x^2), {x, 0, 20}], x,
j], {j, 0, 20}] (* Nikolaos Pantelidis, Feb 07 2023 *)

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

x='x+O('x^200); Vec((1+x)^2/(1-x)^3) \\ Altug Alkan, Mar 23 2016

print([2*n*(n+1)+1 for n in range(48)]) # Michael S. Branicky, Jan 05 2021

[i**2 + (i + 1)**2 for i in range(46)] # Zerinvary Lajos, Jun 27 2008

a(n) = 2*n^2 + 2*n + 1 = n^2 + (n+1)^2.
a(n) = 1 + 3 + 5 + ... + 2*n-1 + 2*n+1 + 2*n-1 + ... + 3 + 1. - Amarnath Murthy, May 28 2001
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
Nearest integer to 1/Sum_{k>n} 1/k^3. - Benoit Cloitre, Jun 12 2003
G.f.: (1+x)^2/(1-x)^3.
E.g.f.: exp(x)*(1+4x+2x^2).
a(n) = a(n-1) + 4n.
a(-n) = a(n-1).
a(n) = A064094(n+3, n) (fourth diagonal).
a(n) = 1 + Sum_{j=0..n} 4*j. - Xavier Acloque, Oct 08 2003
a(n) = A046092(n)+1 = (A016754(n)+1)/2. - Lekraj Beedassy, May 25 2004
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
a(n) = ceiling((2n+1)^2/2). - Paul Barry, Jul 16 2006
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
a(n)*a(n-1) = 4*n^4 + 1 for n > 0. - Reinhard Zumkeller, Feb 12 2009
Prefaced with a "1" (1, 1, 5, 13, 25, 41, ...): a(n) = 2*n*(n-1)+1. - Doug Bell, Feb 27 2009
a(n) = sqrt((A056220(n)^2 + A056220(n+1)^2) / 2). - Doug Bell, Mar 08 2009
a(n) = floor(2*(n+1)^3/(n+2)). - Gary Detlefs, May 20 2010
a(n) = A000330(n) - A000330(n-2). - Keith Tyler, Aug 10 2010
a(n) = A069894(n)/2. - J. M. Bergot, Jun 11 2012
a(n) = 2*a(n-1) - a(n-2) + 4. - Ant King, Jun 12 2012
Sum_{n>=0} 1/a(n) = (Pi/2)*tanh(Pi/2) = 1.4406595199775... = A228048. - Ant King, Jun 15 2012
a(n) = A209297(2*n+1,n+1). - Reinhard Zumkeller, Jan 19 2013
a(n)^3 = A048395(n)^2 + A048395(-n-1)^2. - Vincenzo Librandi, Jan 19 2013
a(n) = A000217(2n+1) - n. - Ivan N. Ianakiev, Nov 08 2013
a(n) = A251599(3*n+1). - Reinhard Zumkeller, Dec 13 2014
a(n) = A101321(4,n). - R. J. Mathar, Jul 28 2016
From Ilya Gutkovskiy, Jul 30 2016: (Start)
a(n) = Sum_{k=0..n} A008574(k).
Sum_{n>=0} (-1)^(n+1)*a(n)/n! = exp(-1) = A068985. (End)
a(n) = 4 * A000217(n) + 1. - Bruce J. Nicholson, Jul 10 2017
a(n) = A002522(n) + A005563(n) = A002522(n+1) + A005563(n-1). - Bruce J. Nicholson, Aug 05 2017
Sum_{n>=0} a(n)/n! = 7*e. Sum_{n>=0} 1/a(n) = A228048. - Amiram Eldar, Jun 20 2020
a(n) = A000326(n+1) + A000217(n-1). - Charlie Marion, Nov 16 2020
a(n) = Integral_{x=0..2n+2} |1-x| dx. - Pedro Caceres, Dec 29 2020
From Amiram Eldar, Feb 17 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = cosh(sqrt(3)*Pi/2)*sech(Pi/2).
Product_{n>=1} (1 - 1/a(n)) = Pi*csch(Pi)*sinh(Pi/2). (End)
a(n) = A001651(n+1) + 1 - A028242(n). - Charlie Marion, Apr 05 2022
a(n) = A016754(n) - A046092(n). - Leo Tavares, Sep 16 2022
For n>0, a(n) = A101096(n+2) / 30. - Andy Nicol, Feb 06 2025
From Rémi Guillaume, Apr 21 2025: (Start)
a(n) = (2*A003215(n)+1)/3.
a(n) = (4*A005448(n+1)-1)/3.
a(n) + a(n-1) = A001845(n) - A001845(n-1), for n >= 1.
a(n) = (A005917(n+1))/(2n+1). (End)

Partially edited by Joerg Arndt, Mar 11 2010

A047838 a(n) = floor(n^2/2) - 1.

Original entry on oeis.org

1, 3, 7, 11, 17, 23, 31, 39, 49, 59, 71, 83, 97, 111, 127, 143, 161, 179, 199, 219, 241, 263, 287, 311, 337, 363, 391, 419, 449, 479, 511, 543, 577, 611, 647, 683, 721, 759, 799, 839, 881, 923, 967, 1011, 1057, 1103, 1151, 1199, 1249, 1299, 1351, 1403

Offset: 2

Michael Somos, May 07 1999

Define the organization number of a permutation pi_1, pi_2, ..., pi_n to be the following. Start at 1, count the steps to reach 2, then the steps to reach 3, etc. Add them up. Then the maximal value of the organization number of any permutation of [1..n] for n = 0, 1, 2, 3, ... is given by 0, 1, 3, 7, 11, 17, 23, ... (this sequence). This was established by Graham Cormode (graham(AT)research.att.com), Aug 17 2006, see link below, answering a question raised by Tom Young (mcgreg265(AT)msn.com) and Barry Cipra, Aug 15 2006
From Dmitry Kamenetsky, Nov 29 2006: (Start)
This is the length of the longest non-self-intersecting spiral drawn on an n X n grid. E.g., for n=5 the spiral has length 17:
  1 0 1 1 1
  1 0 1 0 1
  1 0 1 0 1
  1 0 0 0 1
  1 1 1 1 1 (End)
It appears that a(n+1) is the maximum number of consecutive integers (beginning with 1) that can be placed, one after another, on an n-peg Towers of Hanoi, such that the sum of any two consecutive integers on any peg is a square. See the problem: http://online-judge.uva.es/p/v102/10276.html. - Ashutosh Mehra, Dec 06 2008
a(n) = number of (w,x,y) with all terms in {0,...,n} and w = |x+y-w|. - Clark Kimberling, Jun 11 2012
The same sequence also represents the solution to the "pigeons problem": maximal value of the sum of the lengths of n-1 line segments (connected at their end-points) required to pass through n trail dots, with unit distance between adjacent points, visiting all of them without overlaping two or more segments. In this case, a(0)=0, a(1)=1, a(2)=3, and so on. - Marco Ripà, Jan 28 2014
Also the longest path length in the n X n white bishop graph. - Eric W. Weisstein, Mar 27 2018
a(n) is the number of right triangles with sides n*(h-floor(h)), floor(h) and h, where h is the hypotenuse. - Andrzej Kukla, Apr 14 2021

			x^2 + 3*x^3 + 7*x^4 + 11*x^5 + 17*x^6 + 23*x^7 + 31*x^8 + 39*x^9 + 49*x^10 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000
Laurent Bulteau, Samuele Giraudo and Stéphane Vialette, Disorders and permutations , 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Article No. 18; pp. 18:1-18:14.
Graham Cormode, Notes on the organization number of a permutation.
Eric Weisstein's World of Mathematics, Longest Path Problem.
Eric Weisstein's World of Mathematics, White Bishop Graph.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Complement of A047839. First difference is A052928.
Cf. A000217, A007590, A080827, A134151, A135151, A135152, A188653.
Partial sums: A213759(n-1) for n > 1. - Guenther Schrack, May 12 2018

[Floor(n^2/2)-1 : n in [2..100]]; // Wesley Ivan Hurt, Aug 06 2015

seq(floor((n^2+4*n+2)/2), n=0..20) # Gary Detlefs, Feb 10 2010

Table[Floor[n^2/2] - 1, {n, 2, 60}] (* Robert G. Wilson v, Aug 31 2006 *)
LinearRecurrence[{2, 0, -2, 1}, {1, 3, 7, 11}, 60] (* Harvey P. Dale, Jan 16 2015 *)
Floor[Range[2, 20]^2/2] - 1 (* Eric W. Weisstein, Mar 27 2018 *)
Table[((-1)^n + 2 n^2 - 5)/4, {n, 2, 20}] (* Eric W. Weisstein, Mar 27 2018 *)
CoefficientList[Series[(-1 - x - x^2 + x^3)/((-1 + x)^3 (1 + x)), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 27 2018 *)

PARI
```
a(n) = n^2\2 - 1
```

a(2)=1; for n > 2, a(n) = a(n-1) + n - 1 + (n-1 mod 2). - Benoit Cloitre, Jan 12 2003
a(n) = T(n-1) + floor(n/2) - 1 = T(n) - floor((n+3)/2), where T(n) is the n-th triangular number (A000217). - Robert G. Wilson v, Aug 31 2006
Equals (n-1)-th row sums of triangles A134151 and A135152. Also, = binomial transform of [1, 2, 2, -2, 4, -8, 16, -32, ...]. - Gary W. Adamson, Nov 21 2007
G.f.: x^2*(1+x+x^2-x^3)/((1-x)^3*(1+x)). - R. J. Mathar, Sep 09 2008
a(n) = floor((n^2 + 4*n + 2)/2). - Gary Detlefs, Feb 10 2010
a(n) = abs(A188653(n)). - Reinhard Zumkeller, Apr 13 2011
a(n) = (2*n^2 + (-1)^n - 5)/4. - Bruno Berselli, Sep 14 2011
a(n) = a(-n) = A007590(n) - 1.
a(n) = A080827(n) - 2. - Kevin Ryde, Aug 24 2013
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n > 4. - Wesley Ivan Hurt, Aug 06 2015
a(n) = A000217(n-1) + A004526(n-2), for n > 1. - J. Stauduhar, Oct 20 2017
From Guenther Schrack, May 12 2018: (Start)
Set a(0) = a(1) = -1, a(n) = a(n-2) + 2*n - 2 for n > 1.
a(n) = A000982(n-1) + n - 2 for n > 1.
a(n) = 2*A033683(n) - 3 for n > 1.
a(n) = A061925(n-1) + n - 3 for n > 1.
a(n) = A074148(n) - n - 1 for n > 1.
a(n) = A105343(n-1) + n - 4 for n > 1.
a(n) = A116940(n-1) - n for n > 1.
a(n) = A179207(n) - n + 1 for n > 1.
a(n) = A183575(n-2) + 1 for n > 2.
a(n) = A265284(n-1) - 2*n + 1 for n > 1.
a(n) = 2*A290743(n) - 5 for n > 1. (End)
E.g.f.: 1 + x + ((x^2 + x - 2)*cosh(x) + (x^2 + x - 3)*sinh(x))/2. - Stefano Spezia, May 06 2021
Sum_{n>=2} 1/a(n) = 3/2 + tan(sqrt(3)*Pi/2)*Pi/(2*sqrt(3)) - cot(Pi/sqrt(2))*Pi/(2*sqrt(2)). - Amiram Eldar, Sep 15 2022

Edited by Charles R Greathouse IV, Apr 23 2010

A099392 a(n) = floor((n^2 - 2*n + 3)/2).

Original entry on oeis.org

1, 1, 3, 5, 9, 13, 19, 25, 33, 41, 51, 61, 73, 85, 99, 113, 129, 145, 163, 181, 201, 221, 243, 265, 289, 313, 339, 365, 393, 421, 451, 481, 513, 545, 579, 613, 649, 685, 723, 761, 801, 841, 883, 925, 969, 1013, 1059, 1105, 1153, 1201, 1251, 1301, 1353, 1405

Offset: 1

Ralf Stephan following a suggestion from Luke Pebody, Oct 20 2004

Guenther Schrack, Table of n, a(n) for n = 1..10010
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Differs from A085913 at n = 61. Apart from leading term, identical to A080827.
Cf. A000217, A001844, A002522, A007494, A007590, A058331 (bisections).
From Guenther Schrack, Apr 17 2018: (Start)
First differences: A052928.
Partial sums: A212964(n) + n for n > 0.
Also A058331 and A001844 interleaved. (End)
Cf. A014601, A033683, A047838, A074148, A109613, A183575, A290743.

Array[Floor[(#^2 - 2 # + 3)/2] &, 54] (* or *)
Rest@ CoefficientList[Series[x (-1 + x - x^2 - x^3)/((1 + x) (x - 1)^3), {x, 0, 54}], x] (* Michael De Vlieger, Apr 21 2018 *)

a(n)=(n^2+3)\2-n \\ Charles R Greathouse IV, Aug 01 2013

a(n) = ceiling(n^2/2)-n+1. - Paul Barry, Jul 16 2006; index shifted by R. J. Mathar, Jul 29 2007
a(n) = ceiling(A002522(n-1)/2). - Branko Curgus, Sep 02 2007
From R. J. Mathar, Feb 20 2011: (Start)
G.f.: x *( -1+x-x^2-x^3 ) / ( (1+x)*(x-1)^3 ).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
a(n+1) = (3 + 2*n^2 + (-1)^n)/4. (End)
a(n) = A007590(n-1) + 1 for n >= 2. - Richard R. Forberg, Aug 01 2013
a(n) = A000217(n) - A007494(n-1). - Bui Quang Tuan, Mar 27 2015
From Guenther Schrack, Apr 17 2018: (Start)
a(n) = (2*n^2 - 4*n + 5 -(-1)^n)/4.
a(n+2) = a(n) + 2*n for n > 0.
a(n) = 2*A033683(n-1) - 1 for n > 0.
a(n) = A047838(n-1) + 2 for n > 2.
a(n) = A074148(n-1) - n + 2 for n > 1.
a(n) = A183575(n-3) + 3 for n > 3.
a(n) = 2*A290743(n-1) - 3 for n > 0.
a(n) = 2*A290743(n-2) + A109613(n-5) for n > 4.
a(n) = A074148(n) - A014601(n-1) for n > 0. (End)
Sum_{n>=1} 1/a(n) = tanh(Pi/2)*Pi/2 + coth(Pi/sqrt(2))*Pi/(2*sqrt(2)) + 1/2. - Amiram Eldar, Sep 16 2022
E.g.f.: ((2 - x + x^2)*cosh(x) + (3 - x + x^2)*sinh(x) - 2)/2. - Stefano Spezia, Jan 28 2024

A238010 Number A(n,k) of partitions of k^n into parts that are at most n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 5, 10, 1, 1, 0, 1, 9, 75, 64, 1, 1, 0, 1, 13, 374, 4410, 831, 1, 1, 0, 1, 19, 1365, 123464, 1366617, 26207, 1, 1, 0, 1, 25, 3997, 1736385, 393073019, 2559274110, 2239706, 1, 1

Offset: 0

Alois P. Heinz, Feb 16 2014

In general, column k>=2 is asymptotic to k^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

			A(3,2) = 10: 332, 2222, 3221, 3311, 22211, 32111, 221111, 311111, 2111111, 11111111.
A(2,3) = 5: 22221, 222111, 2211111, 21111111, 111111111.
A(2,4) = 9: 22222222, 222222211, 2222221111, 22222111111, 222211111111, 2221111111111, 22111111111111, 211111111111111, 1111111111111111.
Square array A(n,k) begins:
  0, 0,   0,       0,         0,           0, ...
  1, 1,   1,       1,         1,           1, ...
  1, 1,   3,       5,         9,          13, ...
  1, 1,  10,      75,       374,        1365, ...
  1, 1,  64,    4410,    123464,     1736385, ...
  1, 1, 831, 1366617, 393073019, 33432635477, ...

Alois P. Heinz, Antidiagonals n = 0..43, flattened
A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz) (arXiv:1108.4391 [math.CO])

Columns k=0+1,2-10 give: A057427, A237998, A238560, A238561, A238562, A238563, A238564, A238565, A238566, A238567.
Rows n=0-2 give: A000004, A000012, A080827.
Main diagonal gives A238000.
Cf. A238012, A238016.

A[n_, k_] := SeriesCoefficient[Product[1/(1-x^j), {j, 1, n}], {x, 0, k^n}]; A[0, 0] = 0; Table[A[n-k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Feb 17 2017 *)

A(n,k) = [x^(k^n)] Product_{j=1..n} 1/(1-x^j).

A058919 a(n) = n^4/2 - n^3 + 3n^2/2 - n + 1 = (n^2 + 1)(n^2 - 2*n + 2)/2.

Original entry on oeis.org

1, 1, 5, 25, 85, 221, 481, 925, 1625, 2665, 4141, 6161, 8845, 12325, 16745, 22261, 29041, 37265, 47125, 58825, 72581, 88621, 107185, 128525, 152905, 180601, 211901, 247105, 286525, 330485, 379321, 433381, 493025, 558625, 630565, 709241, 795061, 888445, 989825, 1099645

Offset: 0

Henry Bottomley, Jan 11 2001

On an n X (n - 1)(n - 2)/2 X n(n - 1)/2 cuboid with n >= 5, the two points at greatest surface distance from a corner are the opposite corner and the point 1 in from each of the two edges on a smallest face which meet at the opposite corner; this greatest surface distance is sqrt(a(n)).

Subsequence of A080827, see formula. - David James Sycamore, Jul 31 2018

Harry J. Smith, Table of n, a(n) for n = 0..500
Henry Bottomley, Source
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

For n >= 4 the sequence is a subsequence of A007692.

Cf. A002522, A080827.

[n^4/2 - n^3 + 3*n^2/2 - n + 1: n in [0..30]]; // Wesley Ivan Hurt, May 10 2014

A058919:=n->n^4/2 - n^3 + 3*n^2/2 - n + 1; seq(A058919(n), n=0..30); # Wesley Ivan Hurt, May 10 2014

Table[n^4/2 - n^3 + 3 n^2/2 - n + 1, {n, 0, 30}] (* Wesley Ivan Hurt, May 10 2014 *)

a(n) = { (n^4 + 3*n^2)/2 - n^3 - n + 1 } \\ Harry J. Smith, Jun 23 2009

G.f.: (1 - 4*x + 10*x^2 + 5*x^4)/(1 - x)^5. - Colin Barker, Jan 01 2012
a(n) = A002522(n)*A002522(n-1)/2, with A002522(-1)=2. - Bruno Berselli, Nov 11 2014
a(n) = A080827(n^2-n+1). - David James Sycamore, Jul 31 2018
E.g.f.: exp(x)*(2 + 4*x^2 + 4*x^3 + x^4)/2. - Stefano Spezia, Oct 08 2022
For n>1, a(n) = A000217(n-1)^2 + (A000217(n-1)+1)^2 = (A000217(n)-1)^2 + (A000217(n-2)-1)^2. - Charlie Marion, Feb 08 2024

A251599 Centers of rows of the triangular array formed by the natural numbers.

Original entry on oeis.org

1, 2, 3, 5, 8, 9, 13, 18, 19, 25, 32, 33, 41, 50, 51, 61, 72, 73, 85, 98, 99, 113, 128, 129, 145, 162, 163, 181, 200, 201, 221, 242, 243, 265, 288, 289, 313, 338, 339, 365, 392, 393, 421, 450, 451, 481, 512, 513, 545, 578, 579, 613, 648, 649, 685, 722, 723

Offset: 1

Dave Durgin, Dec 05 2014

nonn

Forms a cascade of 3-number triangles down the center of the triangle array. Related to A000124 (left/west bank of same triangular array), A000217 (right/east bank) and A001844 (center column).
Sums of the mentioned cascading triangles: a(3*n-2) + a(3*n-1) + a(3*n) = A058331(n) + A001105(n) + A001844(n-1) = 2*A056106(n) = 2*(3*n^2-n+1). - Reinhard Zumkeller, Dec 13 2014
Union of A080827 and A000982. - David James Sycamore, Aug 09 2018

			First ten terms (1,2,3,5,8,9,13,18,19,25) may be read down the center of the triangular formation:
               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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David James Sycamore, A080827 & A000982
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A000124, A000217, A001844.
Cf. A092942 (first differences).
Cf. A001105, A056106, A058331.
Cf. A080827, A000982.

a251599 n = a251599_list !! (n-1)
a251599_list = f 0 $ g 1 [1..] where
   f i (us:vs:wss) = [head $ drop i us] ++ (take 2 $ drop i vs) ++
                     f (i + 1) wss
   g k zs = ys : g (k + 1) xs where (ys,xs) = splitAt k zs
-- Reinhard Zumkeller, Dec 12 2014

a:= n-> (m-> 2*(m+1)^2-[2*m+1, 0, -1][1+r])(iquo(n-1, 3, 'r')):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 10 2014

LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {1, 2, 3, 5, 8, 9, 13}, 60] (* Jean-François Alcover, Jan 09 2016 *)

Vec(-x*(x^2+1)*(x^4-x^3+x+1)/((x^2+x+1)^2*(x-1)^3) + O(x^80)) \\ Michel Marcus, Jan 09 2016

Terms for n=1 (mod 3): 2m^2+2m+1, for n=2 (mod 3): 2m^2+4m+2, for n=0 (mod 3): 2m^2+4m+3, where m = floor((n-1)/3).

G.f.: -x*(x^2+1)*(x^4-x^3+x+1)/((x^2+x+1)^2*(x-1)^3). - Alois P. Heinz, Dec 10 2014

A124350 a(n) = 4n(floor(n^2/2)+1). For n >= 3, this is the number of directed Hamiltonian paths on the n-prism graph.

Original entry on oeis.org

0, 4, 24, 60, 144, 260, 456, 700, 1056, 1476, 2040, 2684, 3504, 4420, 5544, 6780, 8256, 9860, 11736, 13756, 16080, 18564, 21384, 24380, 27744, 31300, 35256, 39420, 44016, 48836, 54120, 59644, 65664, 71940, 78744, 85820, 93456, 101380, 109896, 118716

Offset: 0

Eric W. Weisstein, Oct 26 2006

Stefano Spezia, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Hamiltonian Path.
Eric Weisstein's World of Mathematics, Prism Graph.
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A080827, A124349.

LinearRecurrence[{2, 1, -4, 1, 2, -1}, {0, 4, 24, 60, 144, 260}, 60] (* Vincenzo Librandi, Jan 26 2016 *)

Vec(4*x*(x^2+1)*(x^2+4*x+1)/((x-1)^4*(x+1)^2) + O(x^100)) \\ Colin Barker, Sep 06 2013

From Colin Barker, Sep 06 2013: (Start)
a(n) = n*(3 + (-1)^n + 2*n^2).
G.f.: 4*x*(x^2+1)*(x^2+4*x+1) / ((x-1)^4*(x+1)^2). (End)
a(n) = 4*n*A080827(n). - R. J. Mathar, Jan 25 2016
E.g.f.: 2*x*((2 + 3*x + x^2)*cosh(x) + (3 + 3*x + x^2)*sinh(x)). - Stefano Spezia, Jan 27 2024

Formula and further terms from Max Alekseyev, Feb 07 2008

A248800 a(n) = (2*n^2 + 3 + (-1)^n)/2.

Original entry on oeis.org

2, 2, 6, 10, 18, 26, 38, 50, 66, 82, 102, 122, 146, 170, 198, 226, 258, 290, 326, 362, 402, 442, 486, 530, 578, 626, 678, 730, 786, 842, 902, 962, 1026, 1090, 1158, 1226, 1298, 1370, 1446, 1522, 1602, 1682, 1766, 1850, 1938, 2026, 2118

Offset: 0

Paul Curtz, Oct 14 2014

Numbers belonging to A016825: a(n) = A016825( A002620(n) ). - Bruno Berselli, Oct 15 2014

For n>1, a(n) is the number of row vectors of length 2n with entries in [1,n], first entry 1, with maximum inner distance. That is, vectors where the modular distance between adjacent entries is at least (n-2)/2. Modular distance is the minimum of remainders of (x - y) and (y - x) when dividing by n. Geometrically, this metric counts how far the integers mod n are from each other if 1 and n are adjacent as on a circle. - Omar Aceval Garcia, Jan 30 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Omar Aceval Garcia, On the Number of Maximum Inner Distance Latin Squares, arXiv:2112.13912 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000034, A000290, A002620, A016825, A042964, A080827, A168273, A005899, A069894.

[n^2+3/2+(-1)^n/2: n in [0..50]]; // Vincenzo Librandi, Oct 15 2014

Table[n^2 + 3/2 + (-1)^n/2, {n, 0, 50}] (* Bruno Berselli, Oct 15 2014 *)
CoefficientList[Series[2(x^3+x^2-x+1)/((1-x)^3 (x+1)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 15 2014 *)
LinearRecurrence[{2,0,-2,1},{2,2,6,10},60] (* Harvey P. Dale, Apr 08 2019 *)

Vec(-2*(x^3+x^2-x+1)/((x-1)^3*(x+1)) + O(x^100)) \\ Colin Barker, Oct 15 2014

[(2*n^2 +3 +(-1)^n)/2 for n in (0..50)] # G. C. Greubel, Dec 14 2021

a(n) = A000290(n) + A000034(n+1) = 4*A002620(n) + 2.
a(n+1) = 2*A080827(n+1) = (n+2)^2 - A042964(n+1) = a(n) + 2*n + 1 -(-1)^n.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Colin Barker, Oct 15 2014
G.f.: 2*(1-x+x^2+x^3) / ((1-x)^3*(x+1)). - Colin Barker, Oct 15 2014
E.g.f.: cosh(x) + (1 + x + x^2)*exp(x). - G. C. Greubel, Dec 14 2021
a(2n) = A005899(n) for n > 0, a(2n+1) = A069894(n). - Omar Aceval Garcia, Dec 30 2021

Typo in data fixed by Colin Barker, Oct 15 2014

A132892 Square array T(m,n) read by antidiagonals; T(m,n) is the number of equivalence classes in the set of sequences of n nonnegative integers that sum to m, generated by the equivalence relation defined in the following manner: we write a sequence in the form a[1]0a[2]0...0a[p], where each a[i] is a (possibly empty) sequence of positive integers; two sequences in this form, a[1]0a[2]0...0a[p] and b[1]0b[2]0...0b[q] are said to be equivalent if p=q and b[1],b[2],...,b[q] is a cyclic permutation of a[1],a[2],...a[p].

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 3, 1, 1, 5, 9, 7, 4, 1, 1, 6, 13, 14, 10, 4, 1, 1, 7, 19, 25, 22, 12, 5, 1, 1, 8, 25, 41, 42, 30, 15, 5, 1, 1, 9, 33, 63, 79, 66, 43, 19, 6, 1, 1, 10, 41, 92, 131, 132, 99, 55, 22, 6, 1, 1, 11, 51, 129, 213, 245, 217, 143, 73, 26, 7, 1, 1, 12, 61, 175, 325, 428, 429, 335, 201, 91, 31, 7, 1

Offset: 1

Emeric Deutsch and Ira M. Gessel, Oct 02 2007

T(n,n) = A000108(n) (the Catalan numbers; see R. P. Stanley, Catalan addendum, problem starting "Equivalence classes of the equivalence relation ..."). T(m,m+1) = A007595(m+1); T(m,m+2) = A003441(m+1); T(m,m+3) = A003444(m+3); T(n+2,n) = A001453(n+1) (Catalan numbers - 1); T(m,1)=1; T(m,2)=m; T(m,3) = A080827(m) = A099392(m+1); T(m,4) = A004006(m).

			T(2,4) = 3 because we have {2000, 0200, 0020, 0002}, {1100, 0110, 0011} and {1010, 0101, 1001}.
T(4,2) = 4 because we have {40, 04}, {31}, {13} and {22}.
The square array starts:
  1....1.....1.....1......1.....1.....1...
  1....2.....3.....3......4.....4.....5...
  1....3.....5.....7.....10....12....15...
  1....4.....9....14.....22....30....43...
  1....5....13....25.....42....66....99...

Alois P. Heinz, antidiagonals n = 1..200, flattened
Emeric Deutsch and Ira Gessel, Equivalence Classes and Cyclic Arrangements:Problem 10525, Amer. Math. Monthly, 105, No. 8, 1998, 774-775 (published solution by D. Beckwith).
R. P. Stanley, Catalan addendum. See the interpretation (www, "Vertices of height n-1 of the tree T ...").

Cf. A000108, A007595, A003441, A003444, A001453, A080827, A099392, A004006.

with(numtheory): T:=proc(m,n) local r, div, N: r:=igcd(m,n+1): div:=divisors(r): N:=nops(div): (sum(phi(div[j])*(binomial((m+n+1)/div[j]-1,(n+1)/div[j]-1) -binomial(m/div[j]-1,(n+1)/div[j]-1)),j=1..N))/(n+1) end proc: for m to 12 do seq(T(m, n),n=1..12) end do; # yields the upper left 12 by 12 block of the infinite matrix T(m,n)
# second Maple program:
T:= proc(m, n) uses numtheory; (C-> add(phi(d)*(C((m+n+1)/d-1, (n+1)/d-1)
      -C(m/d-1, (n+1)/d-1))/(n+1), d=divisors(igcd(m, n+1))))(binomial)
    end:
seq(seq(T(1+d-n, n), n=1..d), d=1..14);  # Alois P. Heinz, Jan 28 2025

T[m_, n_] := Module[{r, div, N}, r = GCD[m, n + 1]; div = Divisors[r]; N = Length[div]; (Sum[EulerPhi[div[[j]]]*(Binomial[(m + n + 1)/div[[j]] - 1, (n + 1)/div[[j]] - 1] - Binomial[m/div[[j]] - 1, (n + 1)/div[[j]] - 1]), {j, 1, N}])/(n + 1)];
Table[T[m - n + 1, n], {m, 1, 13}, {n, 1, m}] // Flatten (* Jean-François Alcover, Sep 01 2024, after Maple program *)

T(m,n) = Sum_{d | gcd(m,n+1)} phi(d)*(C((m+n+1)/d-1, (n+1)/d-1) - C(m/d-1, (n+1)/d-1))/(n+1). [corrected by Jason Yuen, Jan 28 2025]

A243813 Table read by antidiagonals: T(n,k) is the curvature (truncated to integer) of a circle in a variation of nested Pappus chains (see Comments for details).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 9, 1, 1, 1, 1, 3, 13, 1, 1, 1, 1, 2, 5, 19, 1, 1, 1, 1, 1, 3, 7, 25, 1, 1, 1, 1, 1, 2, 4, 9, 33, 1, 1, 1, 1, 1, 1, 2, 5, 11, 41, 1, 1, 1, 1, 1, 1, 2, 3, 6, 14, 51, 1, 1, 1, 1, 1, 1, 1, 2, 4, 7, 17, 61, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 9, 21

Offset: 0

Kival Ngaokrajang, Jun 11 2014

Refer to the construction rule used in A243618. For this case, the curvature is defined by (-1/k, 1/(k-1), 1), the circle radius will diverge to infinity (zero curvature). The integral curvatures appearing as periodic, i.e., 2, 6, 6, 10, 30, 42, 28, 12, ..., = A083482(k-1). The integral curvatures seem to align as some sequence, e.g., 3, 7, 13, 21, 31, 43, ..., = A002061(k) and 9, 25, 49, ..., = A016754(k-1). See illustration.

			Table begins:
  n/k  2   3   4   5   6   7  ...
   0   1   1   1   1   1   1  ...
   1   1   1   1   1   1   1  ...
   2   3   1   1   1   1   1  ...
   3   5   2   1   1   1   1  ...
   4   9   3   2   1   1   1  ...
   5  13   5   3   2   1   1  ...
   6  19   7   4   2   2   1  ...
   7  25   9   5   3   2   2  ...
   8  33  11   6   4   3   2  ...
   9  41  14   7   5   3   2  ...
  10  51  17   9   6   4   3  ...
  11  61  21  11   7   5   3  ...
  12  73  25  13   8   5   4  ...
  ...

Kival Ngaokrajang, Illustration for k = 2..7

Cf. A243618, A083482, A002061, A016754.
Cf. Column 1 = A080827(n), column 2 = A056827(n) + 1.
Cf. Integral curvature in column 1..6: [A058331, A227776, A056107, A212656, A158558, A158604].

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Magma

Maple

Mathematica

PARI

PARI

Python

Sage

Formula

Extensions

A047838 a(n) = floor(n^2/2) - 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A099392 a(n) = floor((n^2 - 2*n + 3)/2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A238010 Number A(n,k) of partitions of k^n into parts that are at most n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A058919 a(n) = n^4/2 - n^3 + 3*n^2/2 - n + 1 = (n^2 + 1)*(n^2 - 2*n + 2)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A251599 Centers of rows of the triangular array formed by the natural numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

A001844 Centered square numbers: a(n) = 2n(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.

A058919 a(n) = n^4/2 - n^3 + 3n^2/2 - n + 1 = (n^2 + 1)(n^2 - 2*n + 2)/2.

A124350 a(n) = 4n(floor(n^2/2)+1). For n >= 3, this is the number of directed Hamiltonian paths on the n-prism graph.