A372751 - OEIS

%I #20 Jun 19 2024 07:34:19

%S 1,21,139,554,1645,4031,8631,16724,30009,50665,81411,125566,187109,

%T 270739,381935,527016,713201,948669,1242619,1605330,2048221,2583911,

%U 3226279,3990524,4893225,5952401,7187571,8619814,10271829,12167995,14334431,16799056,19591649

%N a(n) = (3*n^5 + 4*n^3 - n)/6.

%C Sums of hexagonal numbers (A000384) in successive groups of length 1,2,3,etc, so 1, 6+15, 28+45+66, 91+120+153+190, etc.

%H Paolo Xausa, <a href="/A372751/b372751.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F From _Stefano Spezia_, May 12 2024: (Start)

%F G.f.: x*(1 + 15*x + 28*x^2 + 15*x^3 + x^4)/(1 - x)^6.

%F E.g.f.: exp(x)*x*(6 + 57*x + 79*x^2 + 30*x^3 + 3*x^4)/6. (End)

%e The hexagonal numbers and their groups summed begin

%e   1, 6, 15, 28, 45, 66, 91, 120, 153, 190

%e   \/ \---/  \--------/  \---------------/

%e   1,   21,     139,            554

%t A372751[n_] := (3*n^5 + 4*n^3 - n)/6; Array[A372751, 50] (* _Paolo Xausa_, Jun 19 2024 *)

%Y Cf. A000384 (hexagonal numbers), A002412 (their partial sums).

%Y Cf. A260513 (for triangular numbers), A072474 (for squares), A372583 (for pentagonal numbers), A075664 (cubes).

%K nonn,easy

%O 1,2

%A _Kelvin Voskuijl_, May 12 2024