A134159 -id:A134159 - cp's OEIS frontend

A134158 a(n) = 1 + 27n + 252n^2 + 882n^3 + 1029n^4.

1, 2191, 24583, 109513, 324013, 759811, 1533331, 2785693, 4682713, 7414903, 11197471, 16270321, 22898053, 31369963, 42000043, 55126981, 71114161, 90349663, 113246263, 140241433, 171797341, 208400851, 250563523, 298821613, 353736073, 415892551, 485901391

Offset: 0

Artur Jasinski, Oct 10 2007

A000540(n) is divisible by A000330(n) if and only if n is congruent to {1,2,4,5} mod 7 (see A047380).
This sequence is the case when n is congruent to 1 mod 7.
A134159 is the case when n is congruent to 2 mod 7.
A134160 is the case when n is congruent to 4 mod 7.
A134161 is the case when n is congruent to 5 mod 7.
A133180 is the union of this sequence, A134159, A134160, and A134161.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000330, A000540, A119617, A134153, A134154, A133180, A134155, A134159, A134160, A134161.

Table[(3(7n + 1)^4 + 6(7n + 1)^3 - 3 (7n + 1) + 1)/7, {n, 0, 100}] (* or *) Table[Sum[k^6, {k, 1, 7n + 1}]/Sum[k^2, {k, 1, 7n + 1}], {n, 0, 100}] (* Artur Jasinski *)

Vec((1 + 2186*x + 13638*x^2 + 8498*x^3 + 373*x^4) / (1 - x)^5 + O(x^30)) \\ Colin Barker, Aug 12 2017

a(n) = (3(7n + 1)^4 + 6(7n + 1)^3 - 3 (7n + 1) + 1)/7.
a(n) = (Sum_{k=1..7n+1} k^6) / (Sum_{k=1..7n+1} k^2).
G.f.: -(1 + 2186*x + 13638*x^2 + 8498*x^3 + 373*x^4)/(-1+x)^5. - R. J. Mathar, Nov 14 2007
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4. - Colin Barker, Aug 12 2017

A134160 a(n) = 163 + 1053n + 2520n^2 + 2646n^3 + 1029n^4.

Original entry on oeis.org

163, 7411, 49981, 180793, 477463, 1042303, 2002321, 3509221, 5739403, 8893963, 13198693, 18904081, 26285311, 35642263, 47299513, 61606333, 78936691, 99689251, 124287373, 153179113, 186837223, 225759151, 270467041, 321507733

Offset: 0

Artur Jasinski, Oct 10 2007

A000540(n) is divisible by A000330(n) if and only n is congruent to {1,2,4,5} mod 7 (see A047380) A134158 is case when n is congruent to 1 mod 7 A134159 is case when n is congruent to 2 mod 7 A134160 is case when n is congruent to 4 mod 7 A134161 is case when n is congruent to 5 mod 7 A133180 is union of A134158 and A134159 and A134160 and A134161

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000330, A000540, A119617, A134153, A134154, A133180, A134158, A134159, A134161.

Table[(3(7n + 4)^4 + 6(7n + 4)^3 - 3 (7n + 4) + 1)/7, {n, 0, 100}] (*Artur Jasinski*)
Table[Sum[k^6, {k, 1, 7n + 4}]/Sum[k^2, {k, 1, 7n + 4}], {n, 0, 100}] (*Artur Jasinski*)
LinearRecurrence[{5,-10,10,-5,1},{163,7411,49981,180793,477463},30] (* Harvey P. Dale, Jul 20 2024 *)

a(n)=163+1053*n+2520*n^2+2646*n^3+1029*n^4 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (3*(7*n + 4)^4 + 6*(7*n + 4)^3 - 3*(7*n + 4) + 1)/7.
a(n) = sum(k=1..7*n+4, k^6) / sum(k=1..7*n+4, k^2).
G.f.: (163+6596*x+14556*x^2+3368*x^3+13*x^4)/(1-x)^5. - Colin Barker, May 25 2012

A133180 a(n) = (Sum_{k=1..A047380(n)} k^6) / (Sum_{k=1..A047380(n)} k^2).

Original entry on oeis.org

1, 13, 163, 373, 2191, 3433, 7411, 10363, 24583, 31591, 49981, 61723, 109513, 130351, 180793, 210901, 324013, 370273, 477463, 539041, 759811, 846613, 1042303, 1151983, 1533331, 1679323, 2002321, 2180263, 2785693, 3013051, 3509221

Offset: 1

Artur Jasinski, Oct 10 2007

A000540(n) is divisible by A000330(n) if and only if n is congruent to {1,2,4,5} mod 7 (see A047380).

This sequence is the union of A134158 and A134159 and A134160 and A134161.

Cf. A000330, A000540, A047380, A119617, A134153, A134154, A134158, A134159, A134160, A134161.

a = {}; Do[j = Sum[k^6, {k, 1, n}]/Sum[k^2, {k, 1, n}]; If[IntegerQ[j], AppendTo[a, j]], {n, 1, 100}] ; a (*Artur Jasinski*)
Select[Table[Sum[k^6,{k,n}]/Sum[k^2,{k,n}],{n,100}],IntegerQ] (* Harvey P. Dale, Nov 26 2019 *)

a(n) = A000540(A047380(n)) / A000330(A047380(n)). - Jason Yuen, Sep 23 2024

Offset corrected by Jason Yuen, Sep 23 2024

A134161 a(n) = 373 + 1947n + 3780n^2 + 3234n^3 + 1029n^4.

Original entry on oeis.org

373, 10363, 61723, 210901, 539041, 1151983, 2180263, 3779113, 6128461, 9432931, 13921843, 19849213, 27493753, 37158871, 49172671, 63887953, 81682213, 102957643, 128141131, 157684261, 192063313, 231779263, 277357783, 329349241

Offset: 0

Artur Jasinski, Oct 10 2007

A000540(n) is divisible by A000330(n) if and only n is congruent to {1,2,4,5} mod 7 (see A047380) A134158 is case when n is congruent to 1 mod 7 A134159 is case when n is congruent to 2 mod 7 A134160 is case when n is congruent to 4 mod 7 A134161 is case when n is congruent to 5 mod 7 A133180 is union of A134158 and A134159 and A134160 and A134161.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000330, A000540, A119617, A134153, A134154, A133180, A134158, A134159, A134160.

Table[(3(7n + 5)^4 + 6(7n + 5)^3 - 3 (7n + 5) + 1)/7, {n, 0, 100}]
Table[Sum[k^6, {k, 1, 7n + 5}]/Sum[k^2, {k, 1, 7n + 5}], {n, 0, 100}]
LinearRecurrence[{5,-10,10,-5,1},{373,10363,61723,210901,539041},100] (* Harvey P. Dale, Nov 25 2012 *)

a(n)=373+1947*n+3780*n^2+3234*n^3+1029*n^4 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (3*(7*n + 5)^4 + 6*(7*n + 5)^3 - 3*(7*n + 5) + 1)/7.
a(n) = (Sum_{k=1..7*n+5} k^6) / (Sum_{k=1..7*n+5} k^2).
G.f.: -(373+8498*x+13638*x^2+2186*x^3+x^4)/(-1+x)^5. - R. J. Mathar, Nov 14 2007
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) with a(0)=373, a(1)=10363, a(2)=61723, a(3)=210901, and a(4)=539041. - Harvey P. Dale, Nov 25 2012

A134163 1 + 12n + 81n^3 + n(105n + 81*n^3)/2.

Original entry on oeis.org

1, 187, 1531, 5977, 16441, 36811, 71947, 127681, 210817, 329131, 491371, 707257, 987481, 1343707, 1788571, 2335681, 2999617, 3795931, 4741147, 5852761, 7149241, 8650027, 10375531, 12347137, 14587201, 17119051, 19966987, 23156281, 26713177

Offset: 0

Artur Jasinski, Oct 10 2007

A000541(n) is divisible by A000537(n) if and only n is congruent to 1 mod 3 (see A016777).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000537, A000541, A119617, A134153, A134154, A133180, A134158, A134159, A134160.

[1 + 12*n + 81*n^3 + n*(105*n+ 81*n^3)/2: n in [0..30]]; // Vincenzo Librandi, May 09 2011

A134163:=n->1 + 12*n + 81*n^3 + n*(105*n + 81*n^3)/2: seq(A134163(n), n=0..30); # Wesley Ivan Hurt, Oct 23 2014

Table[(3(3n + 1)^4 + 6(3n + 1)^3 - (3n + 1)^2 - 4 (3n + 1) + 2)/6, {n, 0, 100}] (* or *) Table[Sum[k^7, {k, 1, 3n + 1}]/Sum[k^3, {k, 1, 3n + 1}], {n, 0, 100}]

a(n) = (3(3n + 1)^4 + 6(3n + 1)^3 - (3n + 1)^2 - 4 (3n + 1) + 2)/6.
a(n) = ( sum_{k=1..3n+1} k^7 ) / ( sum_{k=1..3n+1} k^3 ).
G.f.: (1+182*x+606*x^2+182*x^3+x^4)/(1-x)^5. - R. J. Mathar, Nov 14 2007
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Wesley Ivan Hurt, Oct 23 2014

cp's OEIS Frontend

A134158 a(n) = 1 + 27n + 252n^2 + 882n^3 + 1029n^4.

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A134160 a(n) = 163 + 1053*n + 2520*n^2 + 2646*n^3 + 1029*n^4.

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A133180 a(n) = (Sum_{k=1..A047380(n)} k^6) / (Sum_{k=1..A047380(n)} k^2).

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A134161 a(n) = 373 + 1947*n + 3780*n^2 + 3234*n^3 + 1029*n^4.

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A134163 1 + 12*n + 81*n^3 + n*(105*n + 81*n^3)/2.

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Mathematica

Formula

A134160 a(n) = 163 + 1053n + 2520n^2 + 2646n^3 + 1029n^4.

A134161 a(n) = 373 + 1947n + 3780n^2 + 3234n^3 + 1029n^4.

A134163 1 + 12n + 81n^3 + n(105n + 81*n^3)/2.