A035282 -id:A035282 - cp's OEIS frontend

A078472 Partial sums of A035282.

1, 6, 12, 22, 46, 67, 107, 137, 168, 228, 292, 342, 426, 546, 606, 656, 800, 920, 1044, 1129, 1273, 1473, 1633, 1759, 1850, 2030, 2270, 2510, 2665, 2869, 3089, 3389, 3799, 4119, 4275, 4539, 4819, 5029, 5389, 5689, 5993, 6377, 6797, 6967, 7367, 7871, 8231

Offset: 1

Benoit Cloitre, Dec 31 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A078471, A035282.

Distinct terms of A353997.

f[p_, e_] := Which[p == 5, (5^(e + 1) - 1)/4, (m = Mod[p, 5]) == 2 || m == 3, If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), 0], m == 1 || m == 4, Sum[(k + 1)*(e - k + 1)*p^k, {k, 0, e}]]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Select[Array[s, 200], # > 0 &]] (* Amiram Eldar, May 13 2022 *)

a(20) and a(36) corrected by Georg Fischer, Aug 31 2020

A078473 Expansion of zeta function of icosian ring.

Original entry on oeis.org

1, 0, 0, 5, 6, 0, 0, 0, 10, 0, 24, 0, 0, 0, 0, 21, 0, 0, 40, 30, 0, 0, 0, 0, 31, 0, 0, 0, 60, 0, 64, 0, 0, 0, 0, 50, 0, 0, 0, 0, 84, 0, 0, 120, 60, 0, 0, 0, 50, 0, 0, 0, 0, 0, 144, 0, 0, 0, 120, 0, 124, 0, 0, 85, 0, 0, 0, 0, 0, 0, 144, 0, 0, 0, 0, 200, 0, 0, 160, 126, 91, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Benoit Cloitre, Dec 31 2002

Let zetaI(s) be the zeta function of icosian ring: zetaI(s) = zetaQ(tau)(2s)*zetaQ(tau)(2s-1) where zetaQ(tau)(s) is defined in A035187. Then zetaI(s) = Sum_{n>=1} a(n)/n^(2s).

Amiram Eldar, Table of n, a(n) for n = 1..10000
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. 51 (1999), 1258-1276; arXiv preprint, arXiv:math/9904028 [math.MG], 1999.

Cf. A035187, A035282 (nonzero terms of the sequence), A031363 (n for which a(n) is not zero), A078471.

f[p_, e_] := Which[p == 5, (5^(e + 1) - 1)/4, (m = Mod[p, 5]) == 2 || m == 3, If[EvenQ[e], (p^(e + 2) - 1)/(p^2 - 1), 0], m == 1 || m == 4, Sum[(k + 1)*(e - k + 1)*p^k, {k, 0, e}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 13 2022 *)

{a(n)=local(A); if(n<1, 0, A=direuler(p=2,n,1/(1-X)/(1-kronecker(5,p)*X)); sumdiv(n,d,A[d]*d*A[n/d]))} /* Michael Somos, Jun 06 2005 */

pf(p, r) = {if (p==5, (5^(r+1) -1)/4, if (((p % 5) == 2) || ((p % 5) == 3), if (!(r % 2), (p^(r+2) - 1)/(p^2-1), 0), if (((p % 5) == 1) || ((p % 5) == 4), sum(k=0, r, (k+1)*(r-k+1)*p^k))););}
a(n) = {my(f = factor(n)); prod(i=1, #f~, pf(f[i, 1], f[i, 2]));} \\ Michel Marcus, Mar 03 2014

Multiplicative with a(p^e) = (5^(e + 1) - 1)/4 if p = 5, (p^(e + 2) - 1)/(p^2 - 1) or 0 if p == 2 or 3 (mod 5) and e is even or odd, respectively, and Sum_{k=0..e} (k + 1)*(e - k + 1)*p^k if p == 1 or 4 (mod 5). - Amiram Eldar, May 13 2022

A249076 a(n) = (n*(n+1))^6.

Original entry on oeis.org

0, 64, 46656, 2985984, 64000000, 729000000, 5489031744, 30840979456, 139314069504, 531441000000, 1771561000000, 5289852801024, 14412774445056, 36343632130624, 85766121000000, 191102976000000, 404961208827904, 820972403643456, 1600135042849344, 3010936384000000, 5489031744000000

Offset: 0

Jiwoo Lee, Oct 20 2014

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A059978; A002378: n*(n+1); A035282: n^2 *(n+1)^2; A060459: n^3 *(n+1)^3; A248619: n^4 *(n+1)^4;

Magma
```
[(n*(n+1))^6: n in [0..30]];
```
Maple
```
[ seq(n^6*(n+1)^6, n = 0..100) ];
```

Table[(n (n + 1))^6, {n, 0, 70}] (* or *)
CoefficientList[Series[64*x*(x^10 + 716 x^9 + 37257 x^8 + 450048 x^7 + 1822014 x^6 + 2864328 x^5 + 1822014 x^4 + 450048 x^3 + 37257 x^2 + 716 x + 1)/(1 - x)^13, {x, 0, 30}], x]

a(n)=(n*(n+1))^6 \\ Charles R Greathouse IV, Oct 21 2014

a(n) = A002378(n)^6.
a(n) = 64*A059978(n) for n>0.
G.f.: 64*x*(x^10 + 716*x^9 + 37257*x^8 + 450048*x^7 + 1822014*x^6 + 2864328*x^5 + 1822014*x^4 + 450048*x^3 + 37257*x^2 + 716*x + 1)/(1 - x)^13. [corrected by Georg Fischer, May 10 2019]
Sum_{n>=1} 1/a(n) = -462 + 42*Pi^2 + 7*Pi^4/15 + 2*Pi^6/945. - Vaclav Kotesovec, Sep 25 2019

Incorrect term corrected by Colin Barker, Oct 21 2014

Terms a(21) and beyond corrected by Andrew Howroyd, Feb 22 2018

A035111 Numerators in expansion of a certain Dirichlet series.

Original entry on oeis.org

1, 5, 6, 10, 24, 20, 40, 30, 30, 60, 64, 50, 84, 120, 60, 50, 144, 120, 124, 80, 144, 200, 160, 120, 90, 180, 240, 240, 150, 204, 220, 300, 408, 320, 150, 264, 280, 200, 360, 300, 304, 384, 420, 170, 400, 480, 360, 300, 364, 384, 250, 400, 504, 960, 424, 720, 300

Offset: 0

N. J. A. Sloane

This is the SO(3) case, whereas the SO(4) case is reported in A078473 and A035282. [From R. J. Mathar, Jul 16 2010]

M. Baake and R. V. Moody, Similarity submodules and semigroups in Quasicrystals and Discrete Geometry, ed. J. Patera, Fields Institute Monographs, vol. 10 AMS, Providence, RI (1998) pp. 1-13.

More terms from R. J. Mathar, Jul 16 2010

cp's OEIS Frontend

A078472 Partial sums of A035282.

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A078473 Expansion of zeta function of icosian ring.

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PARI

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A249076 a(n) = (n*(n+1))^6.

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PARI

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A035111 Numerators in expansion of a certain Dirichlet series.

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