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Search: id:A072670
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| A072670 |
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Number of ways to write n as i*j+i+j, 0<i<=j. |
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+0
7
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| 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 2, 0, 3, 0, 2, 1, 1, 1, 4, 0, 1, 1, 3, 0, 3, 0, 2, 2, 1, 0, 4, 1, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 5, 0, 1, 2, 3, 1, 3, 0, 2, 1, 3, 0, 5, 0, 1, 2, 2, 1, 3, 0, 4, 2, 1, 0, 5, 1, 1, 1, 3, 0, 5, 1, 2, 1, 1, 1, 5, 0, 2, 2, 4, 0, 3, 0, 3, 3
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OFFSET
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0,12
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COMMENT
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a(n) = A038548(n+1) - 1.
a(n) is the number of partitions of n+1 with summands in arithmetic progression having common difference 2. For example a(29)=3 because there are 3 partitions of 30 that are in arithmetic progressions : 2+4+6+8+10, 8+10+12 and 14+16. - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Feb 01 2008
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REFERENCES
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J. W. Andrushkiw, R. I. Andrushkiw and C. E. Corzatt, Representations of Positive Integers as Sums of Arithmetic Progressions, Mathematics Magazine, Vol. 49, No. 5 (Nov., 1976), pp. 245-248.
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LINKS
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M. A. Nyblom and C. Evans, On the enumeration of partitions with summands in arithmetic progression, Australian Journal of Combinatorics, Vol. 28 (2003), pp. 149-159.
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FORMULA
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a(n)= p2(n+1), where p2(n)= 1/2(d(n)-2+((-1)^{d(n)+1}+1)/2); d(n) is the number of divisors of n: A000005. G.F.: Sum_{n >= 1} a(n) x^n = 1/x Sum_{k>=2} x^(k^2)/(1-x^k). - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Feb 01 2008
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EXAMPLE
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a(11)=2: 11 = 1*5+1+5 = 2*3+2+3.
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MATHEMATICA
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p2[n_]:= 1/2 (Length[Divisors[n]]-2+((-1)^(Length[Divisors[n]]+1)+1)/2); Table[p2[n+1], {n, 0, 104}] - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Feb 01 2008
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CROSSREFS
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Cf. A067432, A066938, A072668, A006093, A072671.
Sequence in context: A116949 A114708 A084927 this_sequence A087624 A085122 A083715
Adjacent sequences: A072667 A072668 A072669 this_sequence A072671 A072672 A072673
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KEYWORD
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nonn
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AUTHOR
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Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 30 2002
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