A237616 a(n) = n*(n + 1)*(5*n - 4)/2.
0, 1, 18, 66, 160, 315, 546, 868, 1296, 1845, 2530, 3366, 4368, 5551, 6930, 8520, 10336, 12393, 14706, 17290, 20160, 23331, 26818, 30636, 34800, 39325, 44226, 49518, 55216, 61335, 67890, 74896, 82368, 90321, 98770, 107730, 117216, 127243, 137826, 148980, 160720
Offset: 0
Examples
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.
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 to sequences related to pyramidal numbers.
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
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).
Programs
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GAP
List([0..40], n-> n*(n+1)*(5*n-4)/2); # G. C. Greubel, Aug 30 2019
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Magma
[n*(n+1)*(5*n-4)/2: n in [0..40]];
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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
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Maple
seq(n*(n+1)*(5*n-4)/2, n=0..40); # G. C. Greubel, Aug 30 2019
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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 *) -
PARI
a(n)=n*(n+1)*(5*n-4)/2 \\ Charles R Greathouse IV, Sep 24 2015
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Sage
[n*(n+1)*(5*n-4)/2 for n in (0..40)] # G. C. Greubel, Aug 30 2019
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
E.g.f.: exp(x)*x*(2 + 16*x + 5*x^2)/2. - Elmo R. Oliveira, Aug 04 2025
Comments