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A262000 a(n) = n^2*(7*n - 5)/2.

%I A262000 #27 Aug 06 2025 17:09:02
%S A262000 0,1,18,72,184,375,666,1078,1632,2349,3250,4356,5688,7267,9114,11250,
%T A262000 13696,16473,19602,23104,27000,31311,36058,41262,46944,53125,59826,
%U A262000 67068,74872,83259,92250,101866,112128,123057,134674,147000,160056,173863,188442,203814,220000
%N A262000 a(n) = n^2*(7*n - 5)/2.
%C A262000 Also, structured enneagonal prism numbers.
%H A262000 Bruno Berselli, <a href="/A262000/b262000.txt">Table of n, a(n) for n = 0..1000</a>
%H A262000 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A262000 G.f.: x*(1 + 14*x + 6*x^2)/(1 - x)^4.
%F A262000 a(n) = Sum_{i=0..n-1} n*(7*i+1) for n>0, a(0)=0.
%F A262000 a(n+1) + a(-n) = A069125(n+1).
%F A262000 Sum_{i>0} 1/a(i) = 1.082675669875907610300284768825... = (42*(log(14) + 2*(cos(Pi/7)*log(cos(3*Pi/14)) + log(sin(Pi/7))*sin(Pi/14) - log(cos(Pi/14)) * sin(3*Pi/14))) + 21*Pi*tan(3*Pi/14))/75 - Pi^2/15. - _Vaclav Kotesovec_, Oct 04 2016
%F A262000 From _Elmo R. Oliveira_, Aug 06 2025: (Start)
%F A262000 E.g.f.: exp(x)*x*(2 + 16*x + 7*x^2)/2.
%F A262000 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
%e A262000 For n=8, a(8) = 8*(7*0+1)+8*(7*1+1)+8*(7*2+1)+8*(7*3+1)+8*(7*4+1)+8*(7*5+1)+8*(7*6+1)+8*(7*7+1) = 1632.
%t A262000 Table[n^2 (7 n - 5)/2, {n, 0, 40}]
%t A262000 LinearRecurrence[{4,-6,4,-1},{0,1,18,72},50] (* _Harvey P. Dale_, Oct 04 2016 *)
%o A262000 (PARI) vector(40, n, n--; n^2*(7*n-5)/2)
%o A262000 (Sage) [n^2*(7*n-5)/2 for n in (0..40)]
%o A262000 (Magma) [n^2*(7*n-5)/2: n in [0..40]];
%Y A262000 Cf. similar sequences with the formula n^2*(k*n - k + 2)/2: A000290 (k=0), A002411 (k=1), A000578 (k=2), A050509 (k=3), A015237 (k=4), A006597 (k=5), A100176 (k=6), this sequence (k=7), A103532 (k=8).
%Y A262000 Cf. A001106, A069125.
%K A262000 nonn,easy
%O A262000 0,3
%A A262000 _Bruno Berselli_, Sep 08 2015