A172117 -id:A172117 - cp's OEIS frontend

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

Bruno Berselli, Feb 10 2014

Also 17-gonal (or heptadecagonal) pyramidal numbers.

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

			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.

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

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).

Cf. A051869, A104728.

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).

List([0..40], n-> n*(n+1)*(5*n-4)/2); # G. C. Greubel, Aug 30 2019

Magma
```
[n*(n+1)*(5*n-4)/2: n in [0..40]];
```

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

seq(n*(n+1)*(5*n-4)/2, n=0..40); # G. C. Greubel, Aug 30 2019

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 *)

a(n)=n*(n+1)*(5*n-4)/2 \\ Charles R Greathouse IV, Sep 24 2015

[n*(n+1)*(5*n-4)/2 for n in (0..40)] # G. C. Greubel, Aug 30 2019

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.
a(n) = A104728(A001844(n-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

A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3n(5*n-4).

Original entry on oeis.org

0, 3, 36, 99, 192, 315, 468, 651, 864, 1107, 1380, 1683, 2016, 2379, 2772, 3195, 3648, 4131, 4644, 5187, 5760, 6363, 6996, 7659, 8352, 9075, 9828, 10611, 11424, 12267, 13140, 14043, 14976, 15939, 16932, 17955, 19008, 20091, 21204

Offset: 0

Omar E. Pol, Dec 26 2008

This sequence is related to A172117 by 3*A172117(n) = n*a(n) - Sum_{i=0..n-1} a(i) and this is the case d=10 in the identity n*(3*n*(d*n - d + 2)/2) - Sum_{k=0..n-1} 3*k*(d*k - d + 2)/2 = n*(n+1)*(2*d*n - 2*d + 3)/2. - Bruno Berselli, Aug 26 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051624, A152965.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153875, A172117.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=30: see Comments lines of A226492.

Table[3n(5n-4),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,3,36},40] (* Harvey P. Dale, Jun 18 2014 *)
3*PolygonalNumber[12,Range[0,60]] (* Harvey P. Dale, May 13 2022 *)

a(n)=3*n*(5*n-4) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 15*n^2 - 12*n = A051624(n)*3.
a(n) = 30*n + a(n-1) - 27 with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: 3*x*(1 + 9*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=3, a(2)=36. - Harvey P. Dale, Jun 18 2014
E.g.f.: 3*x*(1 + 5*x)*exp(x). - G. C. Greubel, Aug 21 2016
a(n) = (4*n-2)^2 - (n-2)^2. In general, if P(k,n) is the k-th n-gonal number, then (2*k+1)*P(8*k+4,n) = ((3*k+1)*n-2*k)^2 - (k*n-2*k)^2. - Charlie Marion, Jul 29 2021

A172118 a(n) = n(n+1)(5*n^2 - n - 3)/2.

Original entry on oeis.org

0, 1, 45, 234, 730, 1755, 3591, 6580, 11124, 17685, 26785, 39006, 54990, 75439, 101115, 132840, 171496, 218025, 273429, 338770, 415170, 503811, 605935, 722844, 855900, 1006525, 1176201, 1366470, 1578934, 1815255, 2077155, 2366416, 2684880

Offset: 0

Vincenzo Librandi, Jan 26 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A172117.

[n*(n+1)*(5*n^2-n-3)/2: n in [0..50]]; // Vincenzo Librandi, Aug 20 2014

Table[n*(n+1)*(5*n^2-n-3)/2, {n, 0, 40}] (* or *) CoefficientList[Series[x(1 +40x +19x^2)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 20 2014 *)
LinearRecurrence[{5,-10,10,-5,1},{0,1,45,234,730},40] (* Harvey P. Dale, Jul 20 2022 *)

[n*(n+1)*(5*n^2-n-3)/2 for n in (0..50)] # G. C. Greubel, Apr 15 2022

From Bruno Berselli, May 07 2010: (Start)
a(n) = n*(n*(n+1)*(20*n-17)/6) - Sum_{i=0..n-1} ( i*(i+1)*(20*i-17)/2 ).
a(n) = n*(n+1)*(5*n^2-n-3)/2.
More generally: n*(n*(n+1)*(2*d*n-2*d+3)/6) - Sum_{i=0..n-1} ( i*(i+1)*(2*d*i-2*d+3)/6, i=0..n-1 ) = n*(n+1)*(3*d*n^2 - d*n + 4*n - 2*d + 2)/12; in this sequence is d=10. (End)
G.f. x*(1+40*x+19*x^2)/(1-x)^5. - R. J. Mathar, Nov 17 2011
From G. C. Greubel, Apr 15 2022: (Start)
a(n) = 12*binomial(n+3,4) - 78*binomial(n+2,3) + 19*binomial(n+1,2).
E.g.f.: (1/2)*x*(2 + 43*x + 34*x^2 + 5*x^3)*exp(x). (End)

Formula simplified and sequence A172117 corrected by Bruno Berselli, May 07 2010

cp's OEIS Frontend

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

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A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3*n*(5*n-4).

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A172118 a(n) = n*(n+1)*(5*n^2 - n - 3)/2.

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

A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3n(5*n-4).

A172118 a(n) = n(n+1)(5*n^2 - n - 3)/2.