

A016754


Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.


224



1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625, 5929, 6241, 6561, 6889, 7225, 7569
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OFFSET

0,2


COMMENTS

The brown rat (rattus norwegicus) breeds very quickly. It can give birth to other rats 7 times a year, starting at the age of three months. The average number of pups is 8. The present sequence gives the total number of rats, when the intervals are 12/7 of a year and a young rat starts having offspring at 24/7 of a year.  Hans Isdahl, Jan 26 2008
Numbers n such that tau(n) is odd where tau(x) denotes the Ramanujan tau function (A000594).  Benoit Cloitre, May 01 2003
If Y is a fixed 2subset of a (2n+1)set X then a(n1) is the number of 3subsets of X intersecting Y.  Milan Janjic, Oct 21 2007
Binomial transform of [1, 8, 8, 0, 0, 0, ...]; Narayana transform (A001263) of [1, 8, 0, 0, 0, ...].  Gary W. Adamson, Dec 29 2007
All terms of this sequence are of the form 8k+1. For numbers 8k+1 which aren't squares see A138393. Numbers 8k+1 are squares iff k is a triangular number from A000217. And squares have form 4n(n+1)+1.  Artur Jasinski, Mar 27 2008
Sequence arises from reading the line from 1, in the direction 1, 25, ... and the line from 9, in the direction 9, 49, ..., in the square spiral whose vertices are the squares A000290.  Omar E. Pol, May 24 2008
First quadrisection of A061038: A061038(4n).  Paul Curtz, Oct 26 2008
Equals the triangular numbers convolved with [1, 6, 1, 0, 0, 0, ...].  Gary W. Adamson & Alexander R. Povolotsky, May 29 2009
First differences: A008590(n) = a(n)  a(n1) for n>0.  Reinhard Zumkeller, Nov 08 2009
Central terms of the triangle in A176271; cf. A000466, A053755.  Reinhard Zumkeller, Apr 13 2010
Odd numbers with odd abundance. Odd numbers with even abundance are in A088828. Even numbers with odd abundance are in A088827. Even numbers with even abundance are in A088829.  Jaroslav Krizek, May 07 2011
Appear as numerators in the nonsimple continued fraction expansion of Pi3: Pi3 = K_(k=1)^infinity (12*k)^2/6 = 1/(6+9/(6+25/(6+49/(6+...)))), see also the comment in A007509.  Alexander R. Povolotsky, Oct 12 2011
Ulam's spiral (SE spoke).  Robert G. Wilson v, Oct 31 2011
All terms end in 1, 5 or 9. Modulo 100, all terms are among { 1, 9, 21, 25, 29, 41, 49, 61, 69, 81, 89 }.  M. F. Hasler, Mar 19 2012
Right edge of both triangles A214604 and A214661: a(n) = A214604(n+1,n+1) = A214661(n+1,n+1).  Reinhard Zumkeller, Jul 25 2012
Also: Odd numbers which have an odd sum of divisors (= sigma = A000203).  M. F. Hasler, Feb 23 2013
Consider primitive Pythagorean triangles (a^2 + b^2 = c^2, gcd(a, b) = 1) with hypotenuse c (A020882) and respective even leg b (A231100); sequence gives values cb, sorted with duplicates removed.  K. G. Stier, Nov 04 2013
For n>1 a(n) is twice the area of the irregular quadrilateral created by the points ((n2)*(n1),(n1)*n/2), ((n1)*n/2,n*(n+1)/2), ((n+1)*(n+2)/2,n*(n+1)/2), and ((n+2)*(n+3)/2,(n+1)*(n+2)/2).  J. M. Bergot, May 27 2014
Number of pairs (x, y) of Z^2, such that max(abs(x), abs(y)) <= n.  Michel Marcus, Nov 28 2014
Except for a(1)=4, the number of active (ON,black) cells in nth stage of growth of twodimensional cellular automaton defined by "Rule 737", based on the 5celled von Neumann neighborhood.  Robert Price, May 23 2016
a(n) is the sum of 2n+1 consecutive numbers, the first of which is n+1.  Ivan N. Ianakiev, Dec 21 2016
a(n) is the number of 2 X 2 matrices with all elements in {0..n} with determinant = 2*permanent.  Indranil Ghosh, Dec 25 2016
Engel expansion of Pi*StruveL_0(1)/2 where StruveL_0(1) is A197037.  Benedict W. J. Irwin, Jun 21 2018
Consider all Pythagorean triples (X,Y,Z=Y+1) ordered by increasing Z; the segments on the hypotenuse {p = a(n)/A001844(n), q = A060300(n)/A001844(n) = A001844(n)  p} and their ratio p/q = a(n)/A060300(n) are irreducible fractions in Q\Z. X values are A005408, Y values are A046092, Z values are A001844.  Ralf Steiner, Feb 25 2020


LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000
Jeremiah Bartz, Bruce Dearden and Joel Iiams, Classes of Gap Balancing Numbers, arXiv:1810.07895 [math.NT], 2018.
Bruce C. Berndt and Ken Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary, SÃ©minaire Lotharingien de Combinatoire, B42c (1999), 63 pp.
Milan Janjic, Two Enumerative Functions; also on Semantic Scholar.
Amelia Carolina Sparavigna, Groupoids of OEIS A002378 and A016754 Numbers (oblong and odd square numbers), Politecnico di Torino (Italy, 2019).
Eric Weisstein's World of Mathematics, Moore Neighborhood.
Robert G. Wilson v, Cover of the March 1964 issue of Scientific American.
Index entries for sequences related to centered polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,3,1).


FORMULA

a(n) = 1 + Sum_{i=1..n} 8*i = 1 + 8*A000217(n).  Xavier Acloque, Jan 21 2003; Zak Seidov, May 07 2006; Robert G. Wilson v, Dec 29 2010
O.g.f.: (1+6*x+x^2)/(1x)^3.  R. J. Mathar, Jan 11 2008
a(n) = 4*n*(n + 1) + 1 = 4*n^2 + 4*n + 1.  Artur Jasinski, Mar 27 2008
Sum_{n>=0} 1/a(n) = Pi^2/8.  Jaume Oliver Lafont, Mar 07 2009
a(n) = A000290(A005408(n)).  Reinhard Zumkeller, Nov 08 2009
a(n) = a(n1) + 8*n with n>0, a(0)=1.  Vincenzo Librandi, Aug 01 2010
a(n) = A033951(n) + n.  Reinhard Zumkeller, May 17 2009
a(n) = A033996(n) + 1.  Omar E. Pol, Oct 03 2011
a(n) = (A005408(n))^2.  Zak Seidov, Nov 29 2011
From George F. Johnson, Sep 05 2012: (Start)
a(n+1) = a(n) + 4 + 4*sqrt(a(n)).
a(n1) = a(n) + 4  4*sqrt(a(n)).
a(n+1) = 2*a(n)  a(n1) + 8.
a(n+1) = 3*a(n)  3*a(n1) + a(n2).
(a(n+1)  a(n1))/8 = sqrt(a(n)).
a(n+1)*a(n1) = (a(n)4)^2.
a(n) = 2*A046092(n) + 1 = 2*A001844(n)  1 = A046092(n) + A001844(n).
Limit as n > infinity of a(n)/a(n1) = 1.
(End)
a(n) = binomial(2*n+2,2) + binomial(2*n+1,2).  John Molokach, Jul 12 2013
E.g.f.: (1 + 8*x + 4*x^2)*exp(x).  Ilya Gutkovskiy, May 23 2016
a(n) = A101321(8,n).  R. J. Mathar, Jul 28 2016
Product_{n>=1} A033996(n)/a(n) = Pi/4.  Daniel Suteu, Dec 25 2016
a(n) = A014105(n) + A000384(n+1).  Bruce J. Nicholson, Nov 11 2017
a(n) = A003215(n) + A002378(n).  Klaus Purath, Jun 09 2020
From Amiram Eldar, Jun 20 2020: (Start)
Sum_{n>=0} a(n)/n! = 13*e.
Sum_{n>=0} (1)^(n+1)*a(n)/n! = 3/e. (End)
Sum_{n>=0} (1)^n/a(n) = A006752.  Amiram Eldar, Oct 10 2020
From Amiram Eldar, Jan 28 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = cosh(Pi/2).
Product_{n>=1} (1  1/a(n)) = Pi/4 (A003881). (End)


PROG

(PARI) (n+n+1)^2 \\ Charles R Greathouse IV, Jun 16 2011
(Haskell)
a016754 n = a016754_list !! n
a016754_list = scanl (+) 1 $ tail a008590_list
 Reinhard Zumkeller, Apr 02 2012
(Maxima) A016754(n):=(n+n+1)^2$
makelist(A016754(n), n, 0, 20); /* Martin Ettl, Nov 12 2012 */
(MAGMA) [n^2: n in [1..100 by 2]]; // Vincenzo Librandi, Jan 03 2017


CROSSREFS

Cf. A000290, A000384, A001263, A001539, A001844, A003881, A005408, A006752, A014105, A016742, A016802, A016814, A016826, A016838, A033996, A046092, A060300, A138393, A167661, A167700.
Cf. A000447 (partial sums).
Partial sums of A022144.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Sequence in context: A113659 A325701 A113745 * A110487 A259417 A030156
Adjacent sequences: A016751 A016752 A016753 * A016755 A016756 A016757


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

Additional description from Terrel Trotter, Jr., Apr 06 2002


STATUS

approved



