A254407 - OEIS

%I #30 Sep 08 2022 08:46:11

%S 0,7,32,86,180,325,532,812,1176,1635,2200,2882,3692,4641,5740,7000,

%T 8432,10047,11856,13870,16100,18557,21252,24196,27400,30875,34632,

%U 38682,43036,47705,52700,58032,63712,69751,76160,82950,90132,97717,105716,114140,123000

%N a(n) = n*(n+1)*(11*n +10)/6.

%C Similar sequences of the type m*P(s,m) - Sum_{i=1..m} P(s-1,i), where P(s,m) is the m-th s-gonal number:

%C s=3: A027480(n) = (n+1)*A000217(n+1) - Sum_{i=1..n+1} i;

%C s=4: A162148(n) = (n+1)*A000290(n+1) - Sum_{i=1..n+1} A000217(i);

%C s=5: A245301(n) = (n+1)*A000326(n+1) - Sum_{i=1..n+1} A000290(i);

%C s=6: A085788(n) = (n+1)*A000384(n+1) - Sum_{i=1..n+1} A000326(i);

%C s=7:       a(n) = (n+1)*A000566(n+1) - Sum_{i=1..n+1} A000384(i).

%H Vincenzo Librandi, <a href="/A254407/b254407.txt">Table of n, a(n) for n = 0..1000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Polygonal_number#Table_of_values">Polygonal numbers: Table of values</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F G.f.: x*(7 + 4*x)/(1 - x)^4.

%F a(-n) = -A132112(n-1).

%F a(n) = Sum_{k=0..n} A011875(11*k+2).

%F Equivalently, partial sums of A254963.

%F E.g.f.: x*(42 + 54*x + 11*x^2)*exp(x)/6. - _G. C. Greubel_, Mar 31 2021

%e 532 is the 7th term because A000566(7)=112 and Sum_{i=1..7} A000384(i)=252, therefore 7*112-252 = 532.

%p A254407:= n-> n*(n+1)*(11*n+10)/6; seq(A254407(n), n=0..50); # _G. C. Greubel_, Mar 31 2021

%t Table[n (n + 1) (11 n + 10)/6, {n, 0, 40}]

%t Column[CoefficientList[Series[x (7 + 4 x) / (1 - x)^4, {x, 0, 60}], x]] (* _Vincenzo Librandi_, Jan 31 2015 *)

%o (PARI) vector(40, n, n--; n*(n+1)*(11*n+10)/6)

%o (Sage) [n*(n+1)*(11*n+10)/6 for n in (0..40)]

%o (Magma) [n*(n+1)*(11*n+10)/6: n in [0..40]];

%o (Maxima) makelist(n*(n+1)*(11*n+10)/6, n, 0, 40);

%Y Cf. A011875, A027480, A057145, A085788, A132112, A162148, A245301, A254963.

%K nonn,easy

%O 0,2

%A _Bruno Berselli_, Jan 30 2015