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A001844
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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)
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122
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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; internal format)
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OFFSET
| 0,2
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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 (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]
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]
Contribution from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), 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
...3..5..8.12.17
...6..9.13.18
..10.14.19
..15.20
and 1, 5, 13, ... can be read off the main diagonal. (End)
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 15 2010: (Start)
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. (End)
Contribution from Keith D. Tyler (oeis(AT)keithtyler.com), Aug 10 2010: (Start)
A000330(n)-A000330(n-2)
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. [From James Buddenhagen (jbuddenh(AT)gmail.com), 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].
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REFERENCES
| 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.
J. A. De Loera, D. C. Haws and M. Koppe, Ehrhart Polynomials of Matroid Polytopes and Polymatroids, Discrete Comput. Geom., 42 (2009), 670-702. [From N. J. A. Sloane, Nov 09 2009]
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.
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]
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).
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.
Arthur T. Benjamin and Doron Zeilberger, "Pythagorean Primes and Palindromic Continued Fractions", Electronic Journal of Combinatorial Number Theory, 5(1) 2005, #A30 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 15 2010]
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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
Matthias Beck, The number of "magic" squares and hypercubes
D. C. Haws, Matroids [From N. J. A. Sloane, Nov 09 2009]
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, 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).
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FORMULA
| 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_{j=0..n} (4*j). - 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]
[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) = floor(2*(n+1)^3/(n+2)) [From Gary Detlefs (gdetlefs(AT)aol.com), May 20 2010]
a(n) = a(n-1)+4*n (with a(0)=1). [From Vincenzo Librandi, Nov 18 2010]
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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. [From Augustine O. Munagi (amunagi(AT)yahoo.com), Dec 18 2008]
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MAPLE
| A001844:=-(z+1)**2/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| Table[2n(n + 1) + 1, {n, 0, 50}]
FoldList[#1 + #2 &, 1, 4 Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)
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PROG
| (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
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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.
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]
Main diagonal of the matrices described in A078475 [From Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Jul 07 2009]
Sequence in context: A098972 A081961 A096891 * A099776 A133322 A146590
Adjacent sequences: A001841 A001842 A001843 * A001845 A001846 A001847
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Partially edited by Joerg Arndt (arndt(AT)jjj.de), Mar 11 2010
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