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%I #24 Sep 02 2023 18:51:45
%S 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,
%T 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,
%U 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
%N Number of ways 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.
%H Reinhard Zumkeller, <a href="/A099751/b099751.txt">Table of n, a(n) for n = 1..10000</a>
%F Multiplicative with a(2^e) = e-1 if e>0, a(3^e) = 1, a(p^e) = e+1 if p>3.
%F Moebius transform is period 12 sequence [ 1, -1, 0, 1, 1, 0, 1, 1, 0, -1, 1, 0, ...].
%F G.f.: Sum_{k>0} (x^k - x^(2*k) + x^(4*k) + x^(5*k) + x^(7*k) + x^(8*k) - x^(10*k) + x^(11*k)) / (1 - x^(12*k)). - _Michael Somos_, Sep 20 2005
%F a(3*n) = a(n). a(4*n + 2) = 0. - _Michael Somos_, Nov 16 2011
%F a(4*n) = A035191(n). - _Michael Somos_, Mar 19 2015
%F From _Amiram Eldar_, Nov 30 2022: (Start)
%F Dirichlet g.f.: zeta(s)^2*(1 + 2^(1-2*s) - 2^(1-s))*(1 - 1/3^s).
%F Sum_{k=1..n} a(k) ~ n*log(n)/3 + (2*gamma - 1 + log(3)/2)*n/3, where gamma is Euler's constant (A001620). (End)
%e G.f. = x + x^3 + x^4 + 2*x^5 + 2*x^7 + 2*x^8 + x^9 + 2*x^11 + x^12 + 2*x^13 + ...
%e a(5)=2 because there are two ways of differences: First pe(3)-pe(-2)=(-15)-(-20)=5 and second pe(1)-pe(2)=(1)-(-4)=5, for e=-4.
%p 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;
%t a[ n_] := If[ n < 1, 0, If[ Divisible[n, 4], -1, 1] Sum[ KroneckerSymbol[ -3, d] (-1)^Quotient[ d, 3], {d, Divisors@n}]]; (* _Michael Somos_, Mar 19 2015 *)
%o (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 */
%o (PARI) {a(n) = if( n<1, 0, (-1)^(n%4 == 0) * sumdiv( n, d, (-1)^(d\3) * kronecker( -3, d)))}; /* _Michael Somos_, Nov 16 2011 */
%o (Haskell)
%o a099751 n = product $ zipWith f (a027748_row n) (a124010_row n)
%o where f 2 e = e - 1; f 3 e = 1; f _ e = e + 1
%o -- _Reinhard Zumkeller_, Mar 20 2015
%Y Cf. A001227 for e in {3, -2, 6}, A048272 for e in {0, 1, 4, 8} and A035218 for e=-1.
%Y Cf. A001620, A027748, A124010, A256232, A035191.
%K mult,easy,nonn
%O 1,5
%A Volker Schmitt (clamsi(AT)gmx.net), Nov 10 2004
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