OFFSET
0,3
COMMENTS
REFERENCES
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (fifteenth row of the table).
LINKS
Bruno Berselli, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pyramidal Number.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
G.f.: x*(1 + 14*x) / (1 - x)^4.
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.
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
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
EXAMPLE
After 0, the sequence is provided by the row sums of the triangle:
1;
2, 16;
3, 32, 31;
4, 48, 62, 46;
5, 64, 93, 92, 61;
6, 80, 124, 138, 122, 76;
7, 96, 155, 184, 183, 152, 91;
8, 112, 186, 230, 244, 228, 182, 106;
9, 128, 217, 276, 305, 304, 273, 212, 121;
10, 144, 248, 322, 366, 380, 364, 318, 242, 136; etc.,
where (r = row index, c = column index):
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.
MAPLE
seq(n*(n+1)*(5*n-4)/2, n=0..40); # G. C. Greubel, Aug 30 2019
MATHEMATICA
Table[n(n+1)(5n-4)/2, {n, 0, 40}]
CoefficientList[Series[x (1+14x)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 18, 66}, 50] (* Harvey P. Dale, Jan 11 2015 *)
PROG
(Magma) [n*(n+1)*(5*n-4)/2: n in [0..40]];
(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
(PARI) a(n)=n*(n+1)*(5*n-4)/2 \\ Charles R Greathouse IV, Sep 24 2015
(Sage) [n*(n+1)*(5*n-4)/2 for n in (0..40)] # G. C. Greubel, Aug 30 2019
(GAP) List([0..40], n-> n*(n+1)*(5*n-4)/2); # G. C. Greubel, Aug 30 2019
CROSSREFS
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).
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Feb 10 2014
STATUS
approved