cp's OEIS Frontend

A237616 a(n) = n*(n + 1)*(5*n - 4)/2.

%I A237616 #75 Aug 04 2025 14:21:17
%S A237616 0,1,18,66,160,315,546,868,1296,1845,2530,3366,4368,5551,6930,8520,
%T A237616 10336,12393,14706,17290,20160,23331,26818,30636,34800,39325,44226,
%U A237616 49518,55216,61335,67890,74896,82368,90321,98770,107730,117216,127243,137826,148980,160720
%N A237616 a(n) = n*(n + 1)*(5*n - 4)/2.
%C A237616 Also 17-gonal (or heptadecagonal) pyramidal numbers.
%C A237616 This sequence is related to A226489 by 2*a(n) = n*A226489(n) - Sum_{i=0..n-1} A226489(i).
%D A237616 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (fifteenth row of the table).
%H A237616 Bruno Berselli, <a href="/A237616/b237616.txt">Table of n, a(n) for n = 0..1000</a>
%H A237616 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PyramidalNumber.html">Pyramidal Number</a>.
%H A237616 <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>.
%H A237616 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A237616 G.f.: x*(1 + 14*x)/(1 - x)^4.
%F A237616 For n>0, a(n) = Sum_{i=0..n-1} (n-i)*(15*i+1). More generally, the sequence with the closed form n*(n+1)*(k*n-k+3)/6 is also given by Sum_{i=0..n-1} (n-i)*(k*i+1) for n>0.
%F A237616 a(n) = A104728(A001844(n-1)) for n>0.
%F A237616 Sum_{n>=1} 1/a(n) = (2*sqrt(5*(5 + 2*sqrt(5)))*Pi + 10*sqrt(5)*arccoth(sqrt(5)) + 25*log(5) - 16)/72 = 1.086617842136293176... . - _Vaclav Kotesovec_, Dec 07 2016
%F A237616 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. - _Wesley Ivan Hurt_, Dec 18 2020
%F A237616 E.g.f.: exp(x)*x*(2 + 16*x + 5*x^2)/2. - _Elmo R. Oliveira_, Aug 04 2025
%e A237616 After 0, the sequence is provided by the row sums of the triangle:
%e A237616    1;
%e A237616    2,  16;
%e A237616    3,  32,  31;
%e A237616    4,  48,  62,  46;
%e A237616    5,  64,  93,  92,  61;
%e A237616    6,  80, 124, 138, 122,  76;
%e A237616    7,  96, 155, 184, 183, 152,  91;
%e A237616    8, 112, 186, 230, 244, 228, 182, 106;
%e A237616    9, 128, 217, 276, 305, 304, 273, 212, 121;
%e A237616   10, 144, 248, 322, 366, 380, 364, 318, 242, 136; etc.,
%e A237616 where (r = row index, c = column index):
%e A237616 T(r,r) = T(c,c) = 15*r-14 and T(r,c) = T(r-1,c)+T(r,r) = (r-c+1)*T(r,r), with r>=c>0.
%p A237616 seq(n*(n+1)*(5*n-4)/2, n=0..40); # _G. C. Greubel_, Aug 30 2019
%t A237616 Table[n(n+1)(5n-4)/2, {n, 0, 40}]
%t A237616 CoefficientList[Series[x (1+14x)/(1-x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 12 2014 *)
%t A237616 LinearRecurrence[{4,-6,4,-1},{0,1,18,66},50] (* _Harvey P. Dale_, Jan 11 2015 *)
%o A237616 (Magma) [n*(n+1)*(5*n-4)/2: n in [0..40]];
%o A237616 (Magma) I:=[0,1,18,66]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Feb 12 2014
%o A237616 (PARI) a(n)=n*(n+1)*(5*n-4)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%o A237616 (Sage) [n*(n+1)*(5*n-4)/2 for n in (0..40)] # _G. C. Greubel_, Aug 30 2019
%o A237616 (GAP) List([0..40], n-> n*(n+1)*(5*n-4)/2); # _G. C. Greubel_, Aug 30 2019
%Y A237616 Cf. A051869, A104728.
%Y A237616 Cf. sequences with formula n*(n+1)*(k*n-k+3)/6: A000217 (k=0), A000292 (k=1), A000330 (k=2), A002411 (k=3), A002412 (k=4), A002413 (k=5), A002414 (k=6), A007584 (k=7), A007585 (k=8), A007586 (k=9), A007587 (k=10), A050441 (k=11), A172073 (k=12), A177890 (k=13), A172076 (k=14), this sequence (k=15), A172078(k=16), A237617 (k=17), A172082 (k=18), A237618 (k=19), A172117(k=20), A256718 (k=21), A256716 (k=22), A256645 (k=23), A256646(k=24), A256647 (k=25), A256648 (k=26), A256649 (k=27), A256650(k=28).
%K A237616 nonn,easy
%O A237616 0,3
%A A237616 _Bruno Berselli_, Feb 10 2014