

A001844


Centered square numbers: 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.
(Formerly M3826 N1567)


233



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

These are Hogben's central polygonal numbers denoted by
...2...
....P..
...4.n.
a(n) = 1 + 3 + 5 + ... + 2n1 + 2n+1 + 2n1 + ... + 3 + 1.  Amarnath Murthy, May 28 2001
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 3n.  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
Let z(1)=I, (I^2=1), z(k+1) = 1/(z(k)+2I); then a(n)=1/real(z(n+1)).  Benoit Cloitre, Aug 06 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 nset and Y and Z disjoint 2subsets of X then a(n4) is equal to the number of 4subsets 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
n such that the Diophantine equation x^3  y^3 = x*y + n has a solution with y = x1. 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
Numbers n such that (n, n, 2*n2) are the sides of an isosceles triangle with integer area. Also, n such that 2*n1 is a square.  James R. Buddenhagen, Oct 17 2008
a(n) is also the least weight of selfconjugate 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*(n1)+1, all tuples of square numbers (XY, X, X+Y) are produced by ((m*(a(n)2n))^2, (m*a(n))^2, (m*(a(n)+2n2)))^2) where m is a whole number.  Doug Bell, Feb 27 2009
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
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) / (4*n^4 + 8*n^3 + 6*n^2 + 2*n + 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 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 NivenZuckermanMontgomery 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)
The limiting value of the partial sums of the reciprocals of the a(n) is Pi/2*tanh(Pi/2) = 1.4406595199775...
a(n) is congruent to 1 (mod 4) for all n.
The digital roots of the a(n) form a purely periodic palindromic 9cycle 1, 5, 4, 7, 5, 7, 4, 5, 1.
The units' digits of the a(n) form a purely periodic palindromic 5cycle 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 vertex (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 = (n1)/n. Note that in this form, this is the velocityaddition formula according to the special theory of relativity (two objects traveling at 1/nth slower than c relative to each other appear to travel at 1/a(n)th less than c to a stationary observer).  Christian N. K. Anderson, May 20 2013
A geometric curiosity: the envelope of the circles x^2 + (ya(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.  JeanFrançois Alcover, Dec 02 2013
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(n1) = A051890(n)  1 = A046092(n1)  2.  Jaroslav Krizek, Dec 27 2013
a(n) is also, of course, the scalar product of the 2vector (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 2space 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. 714. See also the Oct 15 2014 comment in A147973.  Wolfdieter Lang, Nov 06 2014
a(n) = A251599(3*n+1).  Reinhard Zumkeller, Dec 13 2014
Subsequence of A004431, for n >= 1.  Bob Selcoe, Mar 23 2016
Numbers n such that 2n  1 is a perfect square.  JuriStepan Gerasimov, Apr 06 2016
The number of active (ON,black) cells in nth stage of growth of twodimensional cellular automaton defined by "Rule 574", based on the 5celled 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


REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, 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.
L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer Press, NY, 1950, pp. 22 and 36.
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.


LINKS

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), 155164.
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Section 2.3.
Matthias Beck, The number of "magic" squares and hypercubes, arXiv:math/0201013 [math.CO], 20022005.
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, Discrete Comput. Geom., 42 (2009), 670702.
D. C. Haws, Matroids.
Milan Janjic, Two Enumerative Functions. [Broken link; WayBackMachine archive.]
M. Janjic, On a class of polynomials with integer coefficients, JIS 11 (2008) 08.5.
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.
A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 24922501.
Mitchell Paukner, Lucy Pepin, Manda Riehl, Jarred Wieser, Pattern Avoidance in TaskPrecedence Posets, arXiv:1511.00080 [math.CO], 20152016.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
John A. Jr. Rochowicz, Harmonic Numbers: Insights, Approximations and Applications, Spreadsheets in Education (eJSiE), 2015, Vol. 8: Iss. 2, Article 4.
R. G. Stanton and D. D. Cowan, Note on a "square" functional equation, SIAM Rev., 12 (1970), 277279.
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 45454558.
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 twoway infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (3,3,1).


FORMULA

a(n) = 2*n^2 + 2*n + 1.
Nearest integer to 1/Sum_{k>n} 1/k^3.  Benoit Cloitre, Jun 12 2003
G.f.: (1+x)^2/(1x)^3.
E.g.f.: exp(x)*(1+4x+2x^2).
a(n) = a(n1) + 4n.
a(n) = a(n1).
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..nk+1} binomial(nk+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(n1)  3*a(n2) + a(n3), a(0)=1, a(1)=5, a(2)=13.  Jaume Oliver Lafont, Dec 02 2008
a(n)*a(n1) = 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*(n1)+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(n2).  Keith Tyler, Aug 10 2010
a(n) = A069894(n)/2.  J. M. Bergot, Jun 11 2012
a(n) = 2*a(n1)  a(n2) + 4.  Ant King, Jun 12 2012
a(n) = A209297(2*n+1,n+1).  Reinhard Zumkeller, Jan 19 2013
a(n)^3 = A048395(n)^2 + A048395(n1)^2.  Vincenzo Librandi, Jan 19 2013
a(n) = A000217(2n+1)  n.  Ivan N. Ianakiev, Nov 08 2013
a(n) = sqrt(A046092(n)^2 + A005408(n)^2).  Wassan Letourneur, Apr 09 2016
Let f(m,n) = (2*m*n^2 + 2*m*n + m)^2 = (2*m*n^2 + 2*m*n)^2 + (2*m*n + m)^2. If m = 1 or m = 1, then f(1,n) = f(1,n) = a(n) = (2*n^2 + 2*n + 1)^2 = (2*n^2 + 2*n)^2 + (2*n + 1)^2.  Wassan Letourneur, Apr 09 2016
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(n1).  Bruce J. Nicholson, Aug 05 2017


EXAMPLE

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 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


MAPLE

A001844:=(z+1)**2/(z1)**3; # Simon Plouffe in his 1992 dissertation


MATHEMATICA

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 *)


PROG

(PARI) {a(n) = 2*n*(n+1) + 1};
(PARI) x='x+O('x^200); Vec((1+x)^2/(1x)^3) \\ Altug Alkan, Mar 23 2016
(Sage) [i^2+(i+1)^2 for i in xrange(0, 46)] # Zerinvary Lajos, Jun 27 2008
(Haskell)
a001844 n = 2 * n * (n + 1) + 1
a001844_list = zipWith (+) a000290_list $ tail a000290_list
 Reinhard Zumkeller, Dec 04 2012
(MAGMA) [2*n^2 + 2*n + 1: n in [0..50]]; // Vincenzo Librandi, Jan 19 2013
(MAGMA) [n: n in [0..4400]  IsSquare(2*n1)]; // JuriStepan Gerasimov, Apr 06 2016


CROSSREFS

X values are A005408; Y values are A046092.
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.
Right edge of A055096; main diagonal of A069480, A078475, A129312.
First difference gives A008586; first differences of A005900.
Row n=2 (or column k=2) of A008288.
Sequence in context: A081961 A096891 * A099776 A133322 A268525 A146590
Adjacent sequences: A001841 A001842 A001843 * A001845 A001846 A001847


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

Partially edited by Joerg Arndt, Mar 11 2010


STATUS

approved



