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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)
168
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 (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 + ... + 2n-1 + 2n+1 + 2n-1 + ... + 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 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

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

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

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 (bell.doug(AT)gmail.com), 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

From Joshua Zucker, Jul 07 2009: (Start)

When the positive integers are written in a square array by diagonals, a(n) gives the numbers appearing on the main diagonal. That is,

...1....2....4....7...11...16...22...29...37...46.

...3....5....8...12...17...23...30...38...47...57.

...6....9...13...18...24...31...39...48...58...69.

..10...14...19...25...32...40...49...59...70...82.

..15...20...26...33...41...50...60...71...83...96.

..21...27...34...42...51...61...72...84...97..111.

..28...35...43...52...62...73...85...98..112..127.

..36...44...53...63...74...86...99..113..128..144.

..45...54...64...75...87..100..114..129..145..162.

..55...65...76...88..101..115..130..146..163..181.

and 1, 5, 13, ... can be read off the main diagonal. (End)

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 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)

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 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 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 = (n-1)/n. Note that in this form, this is the velocity-addition formula according to the special theory of relativity (two objects travelling at 1/n-th 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 + (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; 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

REFERENCES

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.

L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer Press, NY, 1950, pp. 22 and 36.

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]

U. Alfred, n and n+1 consecutive integers with equal sums of squares, Math. Mag., 35 (1962), 155-164.

Matthias Beck, The number of "magic" squares and hypercubes

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), 670-702.

D. C. Haws, Matroids

Milan Janjic, Two Enumerative Functions

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

A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501.

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.

R. G. Stanton and D. D. Cowan, Note on a "square" functional equation, SIAM Rev., 12 (1970), 277-279.

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

Eric Weisstein's World of Mathematics, Centered Square Number

Eric Weisstein's World of Mathematics, Pythagorean Triple

Eric Weisstein's World of Mathematics, von Neumann Neighborhood

Eric Weisstein's World of Mathematics, Diamond

Index entries for sequences related to centered polygonal numbers

Index entries for two-way infinite sequences

Index entries for sequences related to 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/(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) = 1+ sum(4*j, j=0..n). - 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)^kC(n, k)*sum{j=0..n-k+1, C(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 (bell.doug(AT)gmail.com), Feb 27 2009

a(n) = sqrt((A056220(n)^2 + A056220(n+1)^2) / 2). - Doug Bell (bell.doug(AT)gmail.com), 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

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 8 2013

EXAMPLE

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

G.f. = 1 + 5*x + 13*x^2 + 25*x^3 + 41*x^4 + 61*x^5 + 85*x^6 + 113*x^7 + ...

MAPLE

A001844:=-(z+1)**2/(z-1)**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};

(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

CROSSREFS

X values are 1, 3, 5, 7, 9, ... (A005408), Y values are A046092. Cf. A005448, A005891, A002061, A051890.

Right edge of A055096. First difference gives A008586. The first differences of A005900.

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

Main diagonal of A069480, A078475.

Cf. A001788, A132778, A001263, A128064, A127876, A046092, A153869.

Main diagonal of the matrices described in A078475.

Cf. A000290, A048395.

Cf. A240876.

Main diagonal of A129312.

Sequence in context: A098972 A081961 A096891 * A099776 A133322 A146590

Adjacent sequences:  A001841 A001842 A001843 * A001845 A001846 A001847

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

Partially edited by Joerg Arndt, Mar 11 2010

STATUS

approved

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Last modified October 2 05:40 EDT 2014. Contains 247537 sequences.