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Search: id:A099751
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| A099751 |
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Number of ways to to write n as differences of (-4)-gonal numbers. If pe(n):=1/2*n*((e-2)*n+(4-e)) is the n-th e-gonal number, then a(n) = |{(m,k) of Z X Z; pe(-1)(m+k)-pe(m-1)=n}| for e=-4. cf. A001227 for e in {3, -2, 6}, A048272 for e in {0, 1, 4, 8}, and A035218 for e=-1. |
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+0
1
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| 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 2, 1, 2, 0, 2, 3, 2, 0, 2, 2, 2, 0, 2, 2, 3, 0, 1, 2, 2, 0, 2, 4, 2, 0, 4, 1, 2, 0, 2, 4, 2, 0, 2, 2, 2, 0, 2, 3, 3, 0, 2, 2, 2, 0, 4, 4, 2, 0, 2, 2, 2, 0, 2, 5, 4, 0, 2, 2, 2, 0, 2, 2, 2, 0, 3, 2, 4, 0, 2, 6, 1, 0, 2, 2, 4, 0, 2, 4, 2, 0, 4, 2, 2, 0, 4, 4, 2, 0, 2, 3, 2, 0, 2, 4, 4
(list; graph; listen)
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OFFSET
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1,5
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FORMULA
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Multiplicative with a(2^e)=e-1 if e>0, a(3^e)=1, a(p^e)=e+1 if p>3.
Moebius transform is period 12 sequence [1, -1, 0, 1, 1, 0, 1, 1, 0, -1, 1, 0, ...].
G.f.: Sum_{k>0} (x^k-x^(2k)+x^(4k)+x^(5k)+x^(7k)+x^(8k)-x^(10k)+x^(11k))/(1-x^(12k)). - Michael Somos Sep 20 2005
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EXAMPLE
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a(5)=2 because there two ways of differences: First pe(3)-pe(-2)=(-15)-(-20)=5 and second pe(1)-pe(2)=(1)-(-4)=5, for e=-4.
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MAPLE
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res:=1; ifac:=op(ifactors(i))[2]; for pfac in ifac do; if pfac[1]=2 then res:=res*(pfac[2]-1); else if pfac[1]<>3 then res:=res*(pfac[2]+1); fi; fi; od; a(i):=res;
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PROGRAM
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(PARI) {a(n)=if(n<1, 0, if(n%2==0, (valuation(n, 2)-1)*a(n/2^valuation(n, 2)), if(n%3==0, a(n/3^valuation(n, 3)), numdiv(n)))) } /* Michael Somos Sep 20 2005 */
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CROSSREFS
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Cf. A035218, A048272, A001227.
Adjacent sequences: A099748 A099749 A099750 this_sequence A099752 A099753 A099754
Sequence in context: A138231 A076880 A082115 this_sequence A058728 A143751 A059581
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KEYWORD
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mult,easy,nonn
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AUTHOR
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Volker Schmitt (clamsi(AT)gmx.net), Nov 10 2004
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