0,2

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 (amarnath_murthy(AT)yahoo.com), May 28 2001

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

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 (benoit7848c(AT)orange.fr), 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 (benoit7848c(AT)orange.fr), Aug 06 2002

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

The subsequence of a(n) with only prime terms is given by A027862. - Lekraj Beedassy (blekraj(AT)yahoo.com), 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 (blekraj(AT)yahoo.com), Jun 04 2006

Integers of the form 1 + x + x^2/2 (generating polynomial is Schur's polynomial as in A127876 - Artur Jasinski (grafix(AT)csl.pl), 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-subests of X intersecting both Y and Z. - Milan R. Janjic (agnus(AT)blic.net), Aug 26 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. [From James Buddenhagen (jbuddenh(AT)gmail.com), 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. [From James Buddenhagen (jbuddenh(AT)gmail.com), Oct 17 2008]

a(n) is also the least weight of self-conjugate partitions having n+1 different odd parts. [From Augustine O. Munagi (amunagi(AT)yahoo.com), Dec 18 2008]

Prefaced with a "1": (1, 1, 5, 13, 25, 41,...) = A153869 * (1, 2, 3,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 03 2009]

a(n)=4n+1 when n is a triangular numbers (1, 3, 6, 10, etc) [A000217] [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 11 2009]

Also, except for the first term of [A059722], then [A001844]^3 = [A048395]^2 + [A059722]^2; [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 01 2009]

Contribution from Doug Bell (bell.doug(AT)gmail.com), Feb 27 2009: (Start)

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. (End)

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). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 01 2009]

The running sum of squares taken two at a time. [From Al Hakanson (hawkuu(AT)gmail.com), May 18 2009]

Equals the odd integers convolved with (1, 2, 2, 2,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 25 2009]

Equals the triangular numbers convolved with [1, 2, 1, 0, 0, 0,...]. [From Gary W. Adamson & Alexander Povolotsky (qntmpkt(AT)yahoo.com), May 29 2009]

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

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.

Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.

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

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.

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

A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501. [From Augustine O. Munagi (amunagi(AT)yahoo.com), Dec 18 2008]

T. D. Noe, Table of n, a(n) for n=0..1000

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients

M. Ahmed, J. De Loera and R. Hemmecke, Polyhedral Cones of Magic Cubes and Squares

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

Milan Janjic, Two Enumerative Functions

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Ron Knott, Pythagorean Triples and Online Calculators

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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

[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

Nearest integer to 1/sum(k>n, 1/k^3) - Benoit Cloitre (benoit7848c(AT)orange.fr), 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*n) - Xavier Acloque Oct 08 2003

a(n)=A046092(n)+1=(A016754(n)+1)/2. - Lekraj Beedassy (blekraj(AT)yahoo.com), 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 (pbarry(AT)wit.ie), Dec 22 2004

a(n)=ceiling((2n+1)^2/2); - Paul Barry (pbarry(AT)wit.ie), Jul 16 2006

Row sums of triangle A132778. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 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 (qntmpkt(AT)yahoo.com), Sep 02 2007

Narayana transform (A001263) of [1, 4, 0, 0, 0,...]. Equals A128064 (unsigned) * [1,2,3,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 29 2007

a(n)=3a(n-1)-3a(n-2)+a(n-3), a(0)=1, a(1)=5, a(2)=13 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 02 2008]

a(n)*a(n-1) = 4*n^4 + 1 for n > 0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 12 2009]

Prefaced with a "1": (1, 1, 5, 13, 25, 41, ...) a(n) = 2n*(n-1)+1 [From Doug Bell (bell.doug(AT)gmail.com), Feb 27 2009]

a(n) = sqrt((A056220(n)^2 + A056220(n+1)^2) / 2) [From Doug Bell (bell.doug(AT)gmail.com), Mar 08 2009]

a(n)= 2*n^2+10*n+13. Offset -1. a(3)=61. [From Al Hakanson (hawkuu(AT)gmail.com), May 18 2009]

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. [From Augustine O. Munagi (amunagi(AT)yahoo.com), Dec 18 2008]

A001844:=-(z+1)**2/(z-1)**3; [S. Plouffe in his 1992 dissertation.]

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

a = {}; Do[If[IntegerQ[1 + x + x^2/2], AppendTo[a, 1 + x + x^2/2]], {x, 0, 100}]; a - Artur Jasinski (grafix(AT)csl.pl), Feb 04 2007

(PARI) a(n)=2*n*(n+1)+1

sage: [i^2+(i+1)^2 for i in xrange(0, 46)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2008

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.

Cf. A046092 [From Augustine O. Munagi (amunagi(AT)yahoo.com), Dec 18 2008]

A153869 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 03 2009]

Cf. A000217 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 11 2009]

Cf. A048395, A059722 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 01 2009]

Adjacent sequences: A001841 A001842 A001843 this_sequence A001845 A001846 A001847

Sequence in context: A098972 A081961 A096891 this_sequence A099776 A133322 A146590

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).