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Search: id:A052343
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| A052343 |
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Number of ways to write n as the unordered sum of two triangular numbers (zero allowed). |
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+0
16
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| 1, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 1, 1, 1, 2, 0, 0, 1, 0, 2, 1, 1, 1, 0, 0, 2, 1, 0, 1, 2, 0, 1, 1, 0, 2, 0, 0, 0, 2, 2, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 1, 0, 1, 1, 0, 2, 1, 0, 0, 2, 0, 1, 1, 0, 3, 0, 1, 1, 0, 0, 1, 1, 0, 1, 2, 1, 1, 2, 0, 0, 1, 0, 1, 1, 1
(list; graph; listen)
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OFFSET
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0,7
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COMMENT
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Number of ways of writing n as a sum of a square and twice a triangular number (zeros allowed).
a(A020757(n))=0; a(A020756(n))>0; a(A119345(n))=1; a(A118139(n))>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 15 2006
Or number of ways to write 4n+1 as the unordered sum of two squares of nonnegative integers [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jan 21 2009]
The average value of a(n) for n<=x is pi/4+O(1/sqrt(x)). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Feb 06 2009]
a(n) = ceiling(A008441(n)/2). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 03 2009]
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REFERENCES
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V. Shevelev, Binary additive problems: recursions for numbers of representations, http://www.arxiv.org/abs/0901.3102 [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jan 21 2009]
V. Shevelev, Binary additive problems: theorems of Landau and Hardy-Littlewood type, http://www.arxiv.org/abs/0902.1046 [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Feb 06 2009]
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FORMULA
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G.f.: (Sum_{k>=0} x^(k^2+k))(Sum_{k>=0} x^(k^2)).
Recurrence: a(n)=sum_{1<=k<=r(n)}r(2n-k^2+k)-C(r(n),2)-a(n-1)-a(n-2)-...-a(0), n>=1,a(0)=1, where r(n) is the nearest integer to square root of n+1, or r(n)=A000194(n+1). For example, since r(6)=3, a(6)=r(12)+r(10)+r(6)-C(3,2)-a(5)-...-a(o)=4+3+3-3-0-1-1-1 -1-1=2. [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Feb 06 2009]
a(n) = A025426(8n+2) [From Max Alekseyev (maxale(AT)gmail.com), Mar 09 2009]
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PROGRAM
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(PARI) a(n)=if(n<0, 0, sum(i=0, (sqrtint(4*n+1)-1)\2, issquare(n-i-i^2)))
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CROSSREFS
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Cf. A000217, A052344-A052348, A053587, A056303, A056304.
Sequence in context: A106799 A127499 A121361 this_sequence A073484 A081396 A100544
Adjacent sequences: A052340 A052341 A052342 this_sequence A052344 A052345 A052346
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KEYWORD
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nonn,new
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AUTHOR
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Christian G. Bower (bowerc(AT)usa.net), Jan 23 2000
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