see also A033954 - Charlie Marion (charliemath@optonline.net), Dec 8 2007 - Charlie Marion (charliemath@optonline.net), Dec 8 2007 If anyone could supply me with a past reference to these sequences, I Thanks very much. Content-type: text/html; charset=us-ascii Content-transfer-encoding: 7BIT name="place" downloadurl="http://www.5iantlavalamp.com/"/> name="City"/> name="State"/> name="PostalCode"/> st1\:*{behavior:url(#default#ieooui) } /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {margin:0in; margin-bottom:.0001pt; font-size:12.0pt; font-family:"Times New Roman";} {color:blue; text-decoration:underline;} {color:purple; text-decoration:underline;} {font-family:"Courier New";} span.EmailStyle17 {mso-style-type:personal-compose; font-family:Arial; color:windowtext;} {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in;} {page:Section1;} style='width:100.0%'>

Rhombic matchstick numbers: n*(3*n+2).

+20
7

0, 5, 16, 33, 56, 85, 120, 161, 208, 261, 320, 385, 456, 533, 616, 705, 800, 901, 1008,

stage for considering whether or not it is also true that 5^3 + 6^3 = 7^3 and style='width:100.0%'>

A033954

n*(4*n+3). Also, second 10-gonal (or decagonal) numbers.

+20
16

0, 7, 22, 45, 76, 115, 162, 217, 280, 351, 430, 517, 612, 715, 826, 945, 1072, 1207,

stage for considering whether or not it is also true that 5^3 + 6^3 = 7^3 and see also 045944

face="Times New Roman">Yorktown Heights NY 10598

Return-Path: X-Ids: 168 DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=gamma; h=domainkey-signature:received:received:message-id:date:from:to:subject:cc:in-reply-to:mime-version:content-type:content-transfer-encoding:content-disposition:references; bh=3YVcn5XBGwXFzmW2hnPSIdcii1eODnAgOjYvB4jB7B8=; b=seww3vFiKkgVwflBYiGwDuYQ7Ek3bMQXgOvIv+GSW2U4PMYx/pTT3g6SuNZuFgqae909gXd9TplU2MVv5XJml/PUniKzgtcdzFPv5pdfk1a+bMsDdBq7N+RtoHER+n4p5U3sAc5mzw6wmziKJD24sSkt6svuCFxEIuFsqv6OhcA= DomainKey-Signature: a=rsa-sha1; c=nofws; d=gmail.com; s=gamma; h=message-id:date:from:to:subject:cc:in-reply-to:mime-version:content-type:content-transfer-encoding:content-disposition:references; b=SuONu8yt9Ma3Z4reslY7EU9RsOQHJO/LqQiYVjltzurIFGGEqrTAnr5mKPe6PfZlozs3rih24zuZuSYvxjjPHl51oa5DyDjJOposZLdatQ9/V9VfOBGkC+tdbbWSH+CHWWwHYblF2mFowwXm7kc4q9TQnOUM5rdz4l3od2utPYY= Message-ID: <144987c90712151911l7e3c2d8awce67aafa8482dfd3@mail.gmail.com> Date: Sat, 15 Dec 2007 22:11:41 -0500 From: "Alexander Povolotsky" To: "Charles Marion" Subject: Re: Help find reference Cc: seqfan@ext.jussieu.fr In-Reply-To: <0JT300296U307FO0@mta2.srv.hcvlny.cv.net> MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Content-Disposition: inline References: <0JT300296U307FO0@mta2.srv.hcvlny.cv.net> X-Greylist: IP, sender and recipient auto-whitelisted, not delayed by milter-greylist-3.0 (shiva.jussieu.fr [134.157.0.168]); Sun, 16 Dec 2007 04:11:43 +0100 (CET) X-Virus-Scanned: ClamAV 0.88.7/5140/Sat Dec 15 22:09:37 2007 on shiva.jussieu.fr X-Virus-Status: Clean X-j-chkmail-Score: MSGID : 4764976E.002 on shiva.jussieu.fr : j-chkmail score : X : 0/50 1 0.388 -> 1 X-Miltered: at shiva.jussieu.fr with ID 4764976E.002 by Joe's j-chkmail (http://j-chkmail.ensmp.fr)! Can not help with references but if you are interested here is the PARI formulation, confirming your identity a(n)= sum(k=0,n,(n*(3*n+2)+k)^3)+ 2*((-1)^n*sum(k=1,n,(-1)^k*k^2))^2- sum(k=(n+1),2*n,(n*(3*n+2)+k)^3) (22:09) gp > a(0) %9 = 0 (22:09) gp > a(1) %10 = 0 (22:09) gp > a(2) %11 = 0 (22:09) gp > a(3) %12 = 0 (22:09) gp > a(4) %13 = 0 (22:10) gp > a(5) %14 = 0 (22:15) gp > a(33) %15 = 0 On Dec 15, 2007 2:00 PM, Charles Marion wrote: > > > > > Dear Seqfans, > > > > I have submitted the following two comments to OEIS and am planning an > article about the sequences of equations mentioned therein: > > > > > A045944 > > > > Rhombic matchstick numbers: n*(3*n+2). > > > > +20 > 7 > > > > > > > 0, 5, 16, 33, 56, 85, 120, 161, 208, 261, 320, 385, 456, 533, 616, 705, 800, > 901, 1008, > > > > The equations 1 + 2 = 3 and 3^2 + 4^2 =5^2 set the stage for considering > whether or not it is also true that 5^3 + 6^3 = 7^3 and 7^4 + 8^4 = 9^4. > Reflecting on Fermat's Last Theorem or resorting to a calculator dispels any > hope that either of the two equations could be correct. However, 5^3 + 6^3 > + 2 does equal 7^3 and 7^4 + 8^4 + 64 equals 9^4. More significantly, each > of these equations is the first of an infinite sequence of equations > featuring consecutive integers that conform to the spirit of the equations > mentioned in A000384. For n>0, > > a(n)^3+(a(n)+1)^3 +...+(a(n)+n)^3 +2*A000217(n)^2= > (a(n)+n+1)^3+...+(a(n)+2n)^3; > > e.g., 5^3+6^3+2*1^2=7^3; 16^3+17^3+18^3+2*3^2=19^3+20^3; see also A033954 > > see also A033954 - Charlie Marion (charliemath@optonline.net), Dec 8 2007 > > > > > > > A033954 > > > > n*(4*n+3). Also, second 10-gonal (or decagonal) numbers. > > > > +20 > 16 > > > > > > > 0, 7, 22, 45, 76, 115, 162, 217, 280, 351, 430, 517, 612, 715, 826, 945, > 1072, 1207, > > > > The equations 1 + 2 = 3 and 3^2 + 4^2 =5^2 set the stage for considering > whether or not it is also true that 5^3 + 6^3 = 7^3 and 7^4 + 8^4 = 9^4. > Reflecting on Fermat's Last Theorem or resorting to a calculator dispels any > hope that either of the two equations could be correct. However, 5^3 + 6^3 > + 2 does equal 7^3 and 7^4 + 8^4 + 64 equals 9^4. More significantly, each > of these equations is the first of an infinite sequence of equations > featuring consecutive integers that conform to the spirit of the equations > mentioned in A000384. For n>0, > > a(n)^4+(a(n)+1)^4 +...+(a(n)+n)^4 +(4*A000217(n))^3 = > (a(n)+n+1)^4+...+(a(n)+2n)^4; > > e.g., 7^4+8^4+(4*1)^3=9^4; 22^4+23^4+24^4+(4*3)^3=25^4+26^4; see also 045944 > > - Charlie Marion (charliemath@optonline.net), Dec 8 2007 > > > > > > If anyone could supply me with a past reference to these sequences, I > would greatly appreciate it. > > > > Thanks very much. > > > > > > Charlie Marion > > Yorktown Heights NY 10598 > > > >