A134158 - OEIS

%I #28 Aug 16 2017 14:27:26

%S 1,2191,24583,109513,324013,759811,1533331,2785693,4682713,7414903,

%T 11197471,16270321,22898053,31369963,42000043,55126981,71114161,

%U 90349663,113246263,140241433,171797341,208400851,250563523,298821613,353736073,415892551,485901391

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

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

%C This sequence is the case when n is congruent to 1 mod 7.

%C A134159 is the case when n is congruent to 2 mod 7.

%C A134160 is the case when n is congruent to 4 mod 7.

%C A134161 is the case when n is congruent to 5 mod 7.

%C A133180 is the union of this sequence, A134159, A134160, and A134161.

%H Colin Barker, <a href="/A134158/b134158.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = (3(7n + 1)^4 + 6(7n + 1)^3 - 3 (7n + 1) + 1)/7.

%F a(n) = (Sum_{k=1..7n+1} k^6) / (Sum_{k=1..7n+1} k^2).

%F G.f.: -(1 + 2186*x + 13638*x^2 + 8498*x^3 + 373*x^4)/(-1+x)^5. - _R. J. Mathar_, Nov 14 2007

%F 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

%t 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_ *)

%o (PARI) Vec((1 + 2186*x + 13638*x^2 + 8498*x^3 + 373*x^4) / (1 - x)^5 + O(x^30)) \\ _Colin Barker_, Aug 12 2017

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

%K nonn,easy

%O 0,2

%A _Artur Jasinski_, Oct 10 2007