Re: [math-fun] 4-5-6
Man, you guys work fast. On 2018-07-19 09:17, Neil Sloane wrote:
Steve, Thank you, that was very helpful, and I updated A316833.
And ...834 and ...835. But Oy, that Maple code! In[886]:= ev@q_:=Evaluate@Echo@Sum[q^(2n)^2,{n,0,\[Infinity]}] 1/2 (1+EllipticTheta[3,0,q^4]) In[888]:= od@q_:=Evaluate@Echo@PowerExpand@Sum[q^(2n-1)^2,{n,\[Infinity]}] 1/2 EllipticTheta[2,0,q^4] In[893]:= #@od\[Subset]#@ev&[(#[q]^4-6#[q^2]#[q]^2+8#[q^3]#@q+3#[q^2]^2-6#[q^4])/24&] See 1998 mail below Out[893]= 1/24 (1/16 EllipticTheta[2,0,q^4]^4-3/4 EllipticTheta[2,0,q^4]^2 EllipticTheta[2,0,q^8]+3/4 EllipticTheta[2,0,q^8]^2+2 EllipticTheta[2,0,q^4] EllipticTheta[2,0,q^12]-3 EllipticTheta[2,0,q^16]) ⊂ 1/24 (1/16 (1+EllipticTheta[3,0,q^4])^4-3/4 (1+EllipticTheta[3,0,q^4])^2 (1+EllipticTheta[3,0,q^8])+3/4 (1+EllipticTheta[3,0,q^8])^2+2 (1+EllipticTheta[3,0,q^4]) (1+EllipticTheta[3,0,q^12])-3 (1+EllipticTheta[3,0,q^16])) In[897]:= PowerExpand[Series[#,{q,0,420}]&/@%893] Out[897]= q^84+q^116+q^140+2 q^156+q^164+q^180+q^196+2 q^204+q^212+2 q^228+q^236+q^244+2 q^252+3 q^260+2 q^276+2 q^284+3 q^300+2 q^308+2 q^316+3 q^324+q^332+2 q^340+2 q^348+2 q^356 +3 q^364+3 q^372+4 q^380+4 q^396+3 q^404+6 q^420+O[q]^421 ⊂ q^56+q^84+q^104+q^116+2 q^120+q^140+q^152+q^156+q^164+q^168+q^180+2 q^184+q^196+2 q^200+q^204+q^212+2 q^216+q^224+q^228+q^236+q^244+3 q^248+q^252+2 q^260+2 q^264+2 q^276 +2 q^280+q^284+3 q^296+2 q^300+2 q^308+4 q^312+q^316+2 q^324+q^332+2 q^336+q^340+3 q^344+q^348+2 q^356+4 q^360+2 q^364+2 q^372+4 q^376+2 q^380+3 q^392+2 q^396+3 q^404+3 q^408+q^416 +4 q^420+O[q]^421 E.g., 420 is the sum of four distinct odd squares six different ways and the sum of four distinct even squares four different ways --rwg
Michael Hirschhorn's book looks very interesting and judging by the excerpts that your link showed (look at page 295!) could lead to many other new sequences. If anyone
would
like to help entering them into the OEIS, please do!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Thu, Jul 19, 2018 at 11:26 AM, Lucas, Stephen K - lucassk < lucassk@jmu.edu> wrote:
I don’t know about formally published, but the proof is discussed on the Mathworld site on square numbers, about halfway down, and references Michael Hirschhorn. It gives the identity (4a+1)^2+(4b+1)^2+(4c+1)^2+(4d+1)^2 = 4[ (a+b+c+d+1)^2 + (a-b-c+d)^2 + (a-b+c-d)^2 + (a+b-c-d)^2 ].
You can find this mentioned in his own book at https://books.google.com/books?id=60QwDwAAQBAJ&pg= PA295&lpg=PA295&dq=hirschhorn+sum+four+distinct+odd+squares& source=bl&ots=P0bLaMkf63&sig=B1tCdmh6jDWeYBfza_arOJwvtgE&hl=en&sa=X&ved= 2ahUKEwiUrtmOuKvcAhWhct8KHTzfCe8Q6AEwB3oECAEQAQ#v=onepage&q= hirschhorn%20sum%20four%20distinct%20odd%20squares&f=false
Steve
On Jul 19, 2018, at 10:57 AM, Neil Sloane <njasloane@gmail.com<mailto:nj asloane@gmail.com>> wrote:
Bill, "Sums of four distinct odd squares" seemed to be missing from OEIS so I created A316833 - could you check? Was Mike H.'s proof ever published?
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: https://urldefense.proofpoint. com/v2/url?u=http-3A__NeilSloane.com&d=DwIGaQ&c= eLbWYnpnzycBCgmb7vCI4uqNEB9RSjOdn_5nBEmmeq0&r=vge6KOo90zMf7Wx14WFtiQ&m= KlvEAtFe4zKCr3Aql2E0mi4T73AzYJ8kLGVSN9IC010&s=zft- IZ1e4jFVTGQ3CXQOdJ4AHPEdQ1E2Vr39GfbCVSc&e= Email: njasloane@gmail.com<mailto:njasloane@gmail.com>
From rcs Wed Oct 14 11:50:42 1998 Return-Path: <owner-math-fun@baskerville.CS.Arizona.EDU> Received: from optima.cs.arizona.edu (optima.CS.Arizona.EDU [192.12.69.5]) by baskerville.CS.Arizona.EDU (8.9.1a/8.9.1) with ESMTP id LAA04966 for <math-fun@baskerville.cs.arizona.edu>; Wed, 14 Oct 1998 11:50:42 -0700 (MST) Received: from NEWTON.Macsyma.COM ([192.156.175.1]) by optima.cs.arizona.edu (8.9.1a/8.9.1) with SMTP id LAA24261 for <math-fun@OPTIMA.CS.ARIZONA.EDU>; Wed, 14 Oct 1998 11:50:22 -0700 (MST) Received: from SWEATHOUSE.Macsyma.COM by NEWTON.Macsyma.COM via INTERNET with SMTP id 419503; 14 Oct 1998 14:49:02-0400 Date: Wed, 14 Oct 1998 11:50-0700 From: Bill Gosper <rwg@[192.156.175.1]> Reply-To: rwg@NEWTON.Macsyma.COM Subject: integer sequences and partitions, [arcsines -> Mt. Ararat] To: math-fun@optima.CS.Arizona.EDU cc: njas@research.att.com, wilf@math.upenn.edu In-Reply-To: <19980828061701.1.RWG@[192.233.166.105]> Message-ID: <19981014185051.6.RWG@[192.233.166.105]>
Yesterfortnight, I muttered Suppose h(t) has a powerseries with coefficients in {0,1}, and is thus the generating function of the characteristic function of an integer sequence. E.g., to characterize the positive squares, h(t) := (theta_3(t)-1)/2 = sum(t^n^2,n,1,inf). Then the coefficient of s^i t^j in inf ==== i i i \ (- 1) s h(t ) - > --------------- / i ==== i = 1 (d1100) %e =: f(s) is the number of ways to partition j into i distinct elements of the integer sequence. To relieve the distinctness constraint, take all signs positive. Expanding both versions at s=0 gives the generating functions for partitions into 0 members, 1 member, 2 members, ... of whatever set h characterizes (e.g., positive squares.) (c3232) taylor(1/subst(-s,s,d3226),s,0,4); /* allowing repetitions */ Time= 68 msec. 2 2 2 (h(t ) + h (t)) s (d3232)/T/ 1 + h(t) s + ------------------ 2 3 2 3 3 (2 h(t ) + 3 h(t) h(t ) + h (t)) s + ----------------------------------- 6 4 3 2 2 2 2 4 4 (6 h(t ) + 8 h(t) h(t ) + 3 h (t ) + 6 h (t) h(t ) + h (t)) s + -------------------------------------------------------------- 24 + . . . (c3226) taylor(d1100,s,0,4); /* distinct */ 2 2 2 (h (t) - h(t )) s (d3226)/T/ 1 + h(t) s + ------------------ 2 3 2 3 3 (h (t) - 3 h(t ) h(t) + 2 h(t )) s + ----------------------------------- 6 4 2 2 3 2 2 4 4 (h (t) - 6 h(t ) h (t) + 8 h(t ) h(t) + 3 h (t ) - 6 h(t )) s + -------------------------------------------------------------- 24 + . . .
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Bill Gosper