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A237616 a(n) = n*(n + 1)*(5*n - 4)/2. 28

%I

%S 0,1,18,66,160,315,546,868,1296,1845,2530,3366,4368,5551,6930,8520,

%T 10336,12393,14706,17290,20160,23331,26818,30636,34800,39325,44226,

%U 49518,55216,61335,67890,74896,82368,90321,98770,107730,117216,127243,137826,148980,160720

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

%C Also 17-gonal (or heptadecagonal) pyramidal numbers.

%C This sequence is related to A226489 by 2*a(n) = n*A226489(n) - Sum_{i=0..n-1} A226489(i).

%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (fifteenth row of the table).

%H Bruno Berselli, <a href="/A237616/b237616.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PyramidalNumber.html">Pyramidal Number</a>.

%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</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*(1 + 14*x) / (1 - x)^4.

%F 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 a(n) = A104728(A001844(n-1)) for n>0.

%F 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

%e After 0, the sequence is provided by the row sums of the triangle:

%e 1;

%e 2, 16;

%e 3, 32, 31;

%e 4, 48, 62, 46;

%e 5, 64, 93, 92, 61;

%e 6, 80, 124, 138, 122, 76;

%e 7, 96, 155, 184, 183, 152, 91;

%e 8, 112, 186, 230, 244, 228, 182, 106;

%e 9, 128, 217, 276, 305, 304, 273, 212, 121;

%e 10, 144, 248, 322, 366, 380, 364, 318, 242, 136, etc.,

%e where (r = row index, c = column index):

%e 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.

%t Table[n (n + 1) (5 n - 4)/2, {n, 0, 40}]

%t CoefficientList[Series[x (1 + 14 x)/(1 - x)^4, {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 12 2014 *)

%t LinearRecurrence[{4,-6,4,-1},{0,1,18,66},50] (* _Harvey P. Dale_, Jan 11 2015 *)

%o (MAGMA) [n*(n+1)*(5*n-4)/2: n in [0..40]];

%o (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 (PARI) a(n)=n*(n+1)*(5*n-4)/2 \\ _Charles R Greathouse IV_, Sep 24 2015

%Y Cf. A051869, A104728.

%Y 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 nonn,easy

%O 0,3

%A _Bruno Berselli_, Feb 10 2014

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Last modified February 19 16:02 EST 2019. Contains 320311 sequences. (Running on oeis4.)