%I A001844 M3826 N1567
%S A001844 1,5,13,25,41,61,85,113,145,181,221,265,313,365,421,481,545,613,685,761,
%T A001844 841,925,1013,1105,1201,1301,1405,1513,1625,1741,1861,1985,2113,2245,
%U A001844 2381,2521,2665,2813,2965,3121,3281,3445,3613,3785,3961,4141,4325
%N 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.
%C A001844 These are Hogben's central polygonal numbers denoted by
%C A001844 ...2...
%C A001844 ....P..
%C A001844 ...4.n.
%C A001844 a(n) = 1 + 3 + 5 + ... + 2n-1 + 2n+1 + 2n-1 + ... + 3 + 1. - Amarnath
Murthy (amarnath_murthy(AT)yahoo.com), May 28 2001
%C A001844 Numbers of the form (k^2+1)/2 for k odd.
%C A001844 a(n) is also the number of 3 X 3 magic squares with sum 3n . - Sharon
Sela (sharonsela(AT)hotmail.com), May 11 2002
%C A001844 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
%C A001844 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
%C A001844 Number of convex polyominoes with a 2 X (n+1) minimal bounding rectangle.
%C A001844 The subsequence of a(n) with only prime terms is given by A027862. -
Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 09 2004
%C A001844 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
%C A001844 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
%C A001844 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
%C A001844 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]
%C A001844 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]
%C A001844 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]
%C A001844 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]
%C A001844 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]
%C A001844 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]
%C A001844 Contribution from Doug Bell (bell.doug(AT)gmail.com), Feb 27 2009: (Start)
%C A001844 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
%C A001844 ((m*(a(n)-2n))^2, (m*a(n))^2, (m*(a(n)+2n-2)))^2)
%C A001844 where m is a whole number. (End)
%C A001844 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]
%C A001844 The running sum of squares taken two at a time. [From Al Hakanson (hawkuu(AT)gmail.com),
May 18 2009]
%C A001844 Equals the odd integers convolved with (1, 2, 2, 2,...). [From Gary W.
Adamson (qntmpkt(AT)yahoo.com), May 25 2009]
%C A001844 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]
%C A001844 Contribution from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org),
Jul 07 2009: (Start)
%C A001844 When the positive integers are written in a square array by diagonals,
a(n) gives the numbers appearing on the main diagonal. That is,
%C A001844 ...1..2..4..7.11.16
%C A001844 ...3..5..8.12.17
%C A001844 ...6..9.13.18
%C A001844 ..10.14.19
%C A001844 ..15.20
%C A001844 and 1, 5, 13, ... can be read off the main diagonal. (End)
%D A001844 U. Alfred, n and n+1 consecutive integers with equal sums of squares,
Math. Mag., 35 (1962), 155-164.
%D A001844 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag,
1976, page 3.
%D A001844 A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover,
p. 125, 1964.
%D A001844 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
%D A001844 L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer
Press, NY, 1950, pp. 22 and 36.
%D A001844 Clark Kimberling, Complementary Equations, Journal of Integer Sequences,
Vol. 10 (2007), Article 07.1.4.
%D A001844 S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003;
see p. 483.
%D A001844 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]
%D A001844 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A001844 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A001844 R. G. Stanton and D. D. Cowan, Note on a "square" functional equation,
SIAM Rev., 12 (1970), 277-279.
%D A001844 B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral
clusters, Inorgan. Chem. 24 (1985), 4545-4558.
%D A001844 Travers et al., The Mysterious Lost Proof, Using Advanced Algebra, (1976),
pp. 27.
%H A001844 T. D. Noe, <a href="b001844.txt">Table of n, a(n) for n=0..1000</a>
%H A001844 M. Ahmed, J. De Loera and R. Hemmecke, <a href="http://front.math.ucdavis.edu/
math.CO/0201108">Polyhedral Cones of Magic Cubes and Squares</a>
%H A001844 Matthias Beck, <a href="http://arXiv.org/abs/math.CO/0201013">The number
of "magic" squares and hypercubes</a>
%H A001844 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative
Functions</a>
%H A001844 Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/
journals/JIS/index.html">Partial Complements and Transposable Dispersions</
a>, J. Integer Seqs., Vol. 7, 2004.
%H A001844 Ron Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/
pythag.html">Pythagorean Triples and Online Calculators</a>
%H A001844 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A001844 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001844 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
CenteredPolygonalNumber.html">Link to a section of The World of Mathematics.</
a>
%H A001844 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
CenteredSquareNumber.html">Link to a section of The World of Mathematics.</
a>
%H A001844 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
PythagoreanTriple.html">Pythagorean Triple</a>
%H A001844 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
vonNeumannNeighborhood.html">von Neumann Neighborhood</a>
%H A001844 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Diamond.html">Diamond</a>
%H A001844 <a href="Sindx_Ce.html#CENTRALCUBE">Index entries for sequences related
to centered polygonal numbers</a>
%H A001844 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
a>
%H A001844 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%F A001844 Nearest integer to 1/sum(k>n, 1/k^3) - Benoit Cloitre (benoit7848c(AT)orange.fr),
Jun 12 2003
%F A001844 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).
%F A001844 a(n)= 1+ sum (4*n) - Xavier Acloque Oct 08 2003
%F A001844 a(n)=A046092(n)+1=(A016754(n)+1)/2. - Lekraj Beedassy (blekraj(AT)yahoo.com),
May 25 2004
%F A001844 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
%F A001844 a(n)=ceiling((2n+1)^2/2); - Paul Barry (pbarry(AT)wit.ie), Jul 16 2006
%F A001844 Row sums of triangle A132778. - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Sep 02 2007
%F A001844 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
%F A001844 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
%F A001844 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]
%F A001844 a(n)*a(n-1) = 4*n^4 + 1 for n > 0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Feb 12 2009]
%F A001844 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]
%F A001844 a(n) = sqrt((A056220(n)^2 + A056220(n+1)^2) / 2) [From Doug Bell (bell.doug(AT)gmail.com),
Mar 08 2009]
%F A001844 a(n)= 2*n^2+10*n+13. Offset -1. a(3)=61. [From Al Hakanson (hawkuu(AT)gmail.com),
May 18 2009]
%F A001844 a(n)=4*n+a(n-1)-4 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it),
Nov 09 2009]
%F A001844 [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
%e A001844 The first few triples are (1,0,1), (3,4,5), (5,12,13), (7,24,25),...
%e A001844 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]
%e A001844 For n=2, a(2)=4*2+1-4=5; n=3, a(3)=4*3+5-4=13; n=4, a(4)=4*4+13-4=25
[From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 09 2009]
%p A001844 A001844:=-(z+1)**2/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
%t A001844 Table[2n(n + 1) + 1, {n, 0, 50}]
%t A001844 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
%t A001844 1. Table[2*n^2 + 2*n + 1, {n, 0, 46}] (.) 2. lst = {}; Do[a = 2*n^2 +
2*n + 1; AppendTo[lst, a], {n, 0, 46}]; lst (.) [From Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), Jul 10 2009]
%t A001844 s = 1; lst = {s}; Do[s += n + 3; AppendTo[lst, s], {n, 1, 200, 4}]; lst
[From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 11 2009]
%o A001844 (PARI) a(n)=2*n*(n+1)+1
%o A001844 sage: [i^2+(i+1)^2 for i in xrange(0,46)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jun 27 2008
%Y A001844 X values are 1, 3, 5, 7, 9, ... (A005408), Y values are A046092. Cf.
A005448, A005891, A002061, A051890.
%Y A001844 Right edge of A055096. First difference gives A008586. The first differences
of A005900.
%Y A001844 a(n)= A064094(n+3, n) (fourth diagonal).
%Y A001844 Main diagonal of A069480, A078475.
%Y A001844 Cf. A001788, A132778, A001263, A128064, A127876.
%Y A001844 Cf. A046092 [From Augustine O. Munagi (amunagi(AT)yahoo.com), Dec 18
2008]
%Y A001844 A153869 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 03 2009]
%Y A001844 Cf. A000217 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan
11 2009]
%Y A001844 Cf. A048395, A059722 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it),
Feb 01 2009]
%Y A001844 Main diagonal of the matrices described in A078475 [From Joshua Zucker
(joshua.zucker(AT)stanfordalumni.org), Jul 07 2009]
%Y A001844 Sequence in context: A098972 A081961 A096891 this_sequence A099776 A133322
A146590
%Y A001844 Adjacent sequences: A001841 A001842 A001843 this_sequence A001845 A001846
A001847
%K A001844 nonn,easy,nice
%O A001844 0,2
%A A001844 N. J. A. Sloane (njas(AT)research.att.com).
|
