Search: id:A016754
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%I A016754
%S A016754 1,9,25,49,81,121,169,225,289,361,441,529,625,729,841,961,1089,1225,
%T A016754 1369,1521,1681,1849,2025,2209,2401,2601,2809,3025,3249,3481,3721,3969,
%U A016754 4225,4489,4761,5041,5329,5625,5929,6241,6561,6889,7225,7569
%N A016754 Odd squares: (2n+1)^2. Also centered octagonal numbers.
%C A016754 Comment from Hans Isdahl (hansi(AT)nordtroms.net), Jan 26 2008: 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.
%C A016754 Numbers n such that tau(n) is odd where tau(x) denotes the Ramanujan 
               tau function (A000594). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               May 01 2003
%C A016754 If Y is a fixed 2-subset of a (2n+1)-set X then a(n-1) is the number 
               of 3-subsets of X intersecting Y. - Milan R. Janjic (agnus(AT)blic.net), 
               Oct 21 2007
%C A016754 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 
               (grafix(AT)csl.pl), Mar 27 2008
%C A016754 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 (info(AT)polprimos.com), 
               May 24 2008
%C A016754 First quadrisection of A061038: A061038(4n). From Balmer spectrum of 
               hydrogen. [From Paul Curtz (bpcrtz(AT)free.fr), Oct 26 2008]
%C A016754 Sum_{n>=0} 1/a(n) = Pi^2/8 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), 
               Mar 07 2009]
%C A016754 Equals the triangular numbers convolved with [1, 6, 1, 0, 0, 0,...] [From 
               Gary W. Adamson & Alexander Povolotsky (qntmpkt(AT)yahoo.com), May 
               29 2009]
%C A016754 Except for the first term, a(n)=8*n+a(n-1), (with a(1)=9) [From Vincenzo 
               Librandi (vincenzo.librandi(AT)tin.it), Oct 24 2009]
%C A016754 First differences: A008590(n) = a(n) - a(n-1) for n>0. [From Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 08 2009]
%H A016754 T. D. Noe, <a href="b016754.txt">Table of n, a(n) for n=0..1000</a>
%H A016754 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A016754 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A016754 B. C. Berndt & K. Ono, <a href="http://www.mat.univie.ac.at/~slc/wpapers/
               s42berndt.pdf">Ramanujan's unpublished manuscript on the partition 
               and tau functions with proofs and commentary</a>
%H A016754 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               MooreNeighborhood.html">Moore Neighborhood</a>
%H A016754 <a href="Sindx_Ce.html#CENTRALCUBE">Index entries for sequences related 
               to centered polygonal numbers</a>
%F A016754 a(n) = 1 + Sum [(8*i),{i,1,n}] =(2n+1)^2 - Zak Seidov, May 07 2006
%F A016754 Binomial transform of [1, 8, 8, 0, 0, 0,...]; Narayana transform (A001263) 
               of [1, 8, 0, 0, 0,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Dec 29 2007
%F A016754 O.g.f.: (1+6*x+x^2)/(1-x)^3 = 1/(1-x)-8/(1-x)^2+8/(1-x)^3 . - R. J. Mathar 
               (mathar(AT)strw.leidenuniv.nl), Jan 11 2008
%F A016754 a(n) = 8n(n + 1))/2 + 1 = 4n (n + 1) + 1 = 4n^2 + 4n + 1 - Artur Jasinski 
               (grafix(AT)csl.pl), Mar 27 2008
%F A016754 a(n) = A000290(A005408(n)). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 08 2009]
%F A016754 a(n)=8*n+a(n-1)-8 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 09 2009]
%e A016754 For n=2, a(2)=8*2+1-8=9; n=3, a(3(=8*3+9-8=25; n=4, a(4)=8*4+25-8=49 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 09 2009]
%t A016754 a = {}; Do[If[Sqrt[8k + 1] == Floor[Sqrt[8k + 1]], AppendTo[a, 8k + 1]], 
               {k, 0, 1000}]; a or Table[4n(n + 1) + 1, {n, 0, 500}] - Artur Jasinski 
               (grafix(AT)csl.pl), Mar 27 2008
%o A016754 (Other) sage: [crt(2, n, 4, 5)^2/2^2 for n in xrange(3, 47)] # [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 07 2009]
%Y A016754 Cf. A016742, A033996.
%Y A016754 Cf. A001263.
%Y A016754 Cf. A000217, A138393.
%Y A016754 Cf. A000290, A001539, A016742, A016802, A016814, A016826, A016838.
%Y A016754 a(n) = A033951(n) + n. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               May 17 2009]
%Y A016754 Cf. A167661, A167700. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 09 2009]
%Y A016754 Sequence in context: A075026 A113659 A113745 this_sequence A110487 A030156 
               A141768
%Y A016754 Adjacent sequences: A016751 A016752 A016753 this_sequence A016755 A016756 
               A016757
%K A016754 nonn,easy,new
%O A016754 0,2
%A A016754 N. J. A. Sloane (njas(AT)research.att.com).
%E A016754 Additional description from Terry Trotter, Apr 06 2002.
%E A016754 More terms from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 30 
               2006

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