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A031366 Expansion of Dirichlet series related to icosians. 0
1, 0, 0, 25, 36, 0, 0, 0, 100, 0, 288, 0, 0, 0, 0, 440, 0, 0, 800, 900, 0, 0, 0, 0, 960, 0, 0, 0, 1800, 0, 2048, 0, 0, 0, 0, 2500, 0, 0, 0, 0, 3528, 0, 0, 7200, 3600, 0, 0, 0, 2550, 0, 0, 0, 0, 0, 10368, 0, 0, 0, 7200, 0, 7688, 0, 0, 7330, 0, 0, 0, 0, 0, 0, 10368, 0, 0, 0, 0, 20000, 0, 0, 12800, 15840, 8362, 0, 0, 0, 0, 0, 0, 0, 16200, 0, 0, 0, 0, 0, 28800, 0, 0, 0, 28800, 23899 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

This sequence may be multiplicative. - Mitch Harris, Apr 19 2005

LINKS

Table of n, a(n) for n=1..100.

M. Baake, Solution of the coincidence problem in dimensions d <= 4, in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.; arXiv:math/0605222 [math.MG], 2006.

MAPLE

read("transforms") :

# expansion of 1/(1-5^(-s)) in (5.10)

L1 := [1, seq(0, i=2..200)] :

for k from 1 do

    if 5^k <= nops(L1) then

        L1 := subsop(5^k=1, L1) ;

    else

        break ;

    end if;

end do:

# multiplication with 1/(1-p^(-2s)) in (5.10)

for i from 1 do

    p := ithprime(i) ;

    if modp(p, 5) = 2 or modp(p, 5)=3 then

        Laux := [1, seq(0, i=2..200)] :

        for k from 1 do

            if p^(2*k) <= nops(Laux) then

                Laux := subsop(p^(2*k)=1, Laux) ;

            else

                break ;

            end if;

        end do:

        L1 := DIRICHLET(L1, Laux) ;

    end if;

    if p > nops(L1) then

        break;

    end if;

end do:

# multiplication with 1/(1-p^(-s))^2 in (5.10)

for i from 1 do

    p := ithprime(i) ;

    if modp(p, 5) = 1 or modp(p, 5)=4 then

        Laux := [1, seq(0, i=2..200)] :

        for k from 1 do

            if p^k <= nops(Laux) then

                Laux := subsop(p^k=k+1, Laux) ;

            else

                break ;

            end if;

        end do:

        L1 := DIRICHLET(L1, Laux) ;

    end if;

    if p > nops(L1) then

        break;

    end if;

end do:

# this is now Zeta_L(s), seems to be A035187

# print(L1) ;

# generate Zeta_L(s-1)

L1shft := [seq(op(i, L1)*i, i=1..nops(L1))] ;

# generate 1/Zeta_L(s)

L1x := add(op(i, L1)*x^(i-1), i=1..nops(L1)) :

taylor(1/L1x, x=0, nops(L1)) :

L1i := gfun[seriestolist](%) ;

# generate 1/Zeta_L(2s)

L1i2 := [1, seq(0, i=2..nops(L1))] ;

for k from 2 to nops(L1i) do

    if k^2 < nops(L1i2) then

        L1i2 := subsop(k^2=op(k, L1i), L1i2) ;

    else

        break ;

    end if;

end do:

# generate Zeta_L(s)*Zeta_L(s-1)

DIRICHLET(L1, L1shft) ;

# generate Zeta_L(s)*Zeta_L(s-1)/Zeta_L(2s) = Phi(s)

Phis := DIRICHLET(%, L1i2) ;

# generate Phis(s-1)

Phishif := [seq(op(i, Phis)*i, i=1..nops(Phis))] ;

DIRICHLET(Phis, Phishif) ;

CROSSREFS

Sequence in context: A153446 A039457 A243749 * A068165 A086935 A077689

Adjacent sequences:  A031363 A031364 A031365 * A031367 A031368 A031369

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Terms beyond a(36) from R. J. Mathar, Mar 04 2018

STATUS

approved

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Last modified July 16 04:26 EDT 2019. Contains 325064 sequences. (Running on oeis4.)