A256645 -id:A256645 - 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

A254474 30-gonal numbers: a(n) = n(14n-13).

Original entry on oeis.org

0, 1, 30, 87, 172, 285, 426, 595, 792, 1017, 1270, 1551, 1860, 2197, 2562, 2955, 3376, 3825, 4302, 4807, 5340, 5901, 6490, 7107, 7752, 8425, 9126, 9855, 10612, 11397, 12210, 13051, 13920, 14817, 15742, 16695, 17676, 18685, 19722, 20787, 21880

Offset: 0

Luciano Ancora, Apr 04 2015

See comments in A255184.

Also star 15-gonal numbers.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6 (28th row of the table).

Luciano Ancora, Table of n, a(n) for n = 0..1000
Luciano Ancora, Polygonal and Pyramidal numbers, Section 1.
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. similar sequences listed in A255184.

Mathematica
```
Table[n (14 n - 13), {n, 40}]
```

a(n)=n*(14*n-13) \\ Charles R Greathouse IV, Jun 17 2017

G.f.: x*(-1 - 27*x)/(-1 + x)^3.
a(n) = A000217(n) + 27*A000217(n-1).
a(n) = A051867(n) + 15*A000217(n-1).
Product_{n>=2} (1 - 1/a(n)) = 14/15. - Amiram Eldar, Jan 22 2021
E.g.f.: exp(x)*(x + 14*x^2). - Nikolaos Pantelidis, Feb 05 2023

A255185 26-gonal numbers: a(n) = n(12n-11).

Original entry on oeis.org

0, 1, 26, 75, 148, 245, 366, 511, 680, 873, 1090, 1331, 1596, 1885, 2198, 2535, 2896, 3281, 3690, 4123, 4580, 5061, 5566, 6095, 6648, 7225, 7826, 8451, 9100, 9773, 10470, 11191, 11936, 12705, 13498, 14315, 15156, 16021, 16910, 17823, 18760

Offset: 0

Luciano Ancora, Apr 04 2015

See comments in A255184.

Also star 13-gonal number: a(n) = A051865(n) + 13*A000217(n-1).

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6 (24th row of the table).

Luciano Ancora, Table of n, a(n) for n = 0..1000
Luciano Ancora, Polygonal and Pyramidal numbers, Section 1.
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. similar sequences listed in A255184.

[n*(12*n-11): n in [0..50]]; // G. C. Greubel, Jul 12 2024

Table[n (12 n - 11), {n, 50}]
PolygonalNumber[26,Range[0,50]] (* Requires Mathematica version 10 or later *) (* or *) LinearRecurrence[{3,-3,1},{0,1,26},50] (* Harvey P. Dale, Feb 02 2017 *)

a(n)=n*(12*n-11) \\ Charles R Greathouse IV, Jun 17 2017

[n*(12*n-11) for n in range(51)] # G. C. Greubel, Jul 12 2024

G.f.: x*(1 + 23*x)/(1 - x)^3.
a(n) = A000217(n) + 23*A000217(n-1).
Product_{n>=2} (1 - 1/a(n)) = 12/13. - Amiram Eldar, Jan 22 2021
E.g.f.: exp(x)*(x + 12*x^2). - Nikolaos Pantelidis, Feb 05 2023

A256646 26-gonal pyramidal numbers: a(n) = n(n+1)(8*n-7)/2.

Original entry on oeis.org

0, 1, 27, 102, 250, 495, 861, 1372, 2052, 2925, 4015, 5346, 6942, 8827, 11025, 13560, 16456, 19737, 23427, 27550, 32130, 37191, 42757, 48852, 55500, 62725, 70551, 79002, 88102, 97875, 108345, 119536, 131472, 144177, 157675, 171990, 187146, 203167, 220077

Offset: 0

Luciano Ancora, Apr 07 2015

See comments in A256645.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (24th row of the table).

Luciano Ancora, Table of n, a(n) for n = 0..1000
Luciano Ancora, Polygonal and Pyramidal numbers, Section 3.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Partial sums of A255185.
Cf. similar sequences listed in A237616.
Cf. A000292, A051866, A256645.

[n*(n+1)*(8*n-7)/2: n in [0..50]]; // Vincenzo Librandi, Apr 08 2015

Table[n (n + 1) (8 n - 7)/2, {n, 0, 40}]
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 27, 102}, 40] (* Vincenzo Librandi, Apr 08 2015 *)

[(8*n-7)*binomial(n+1,2) for n in range(51)] # G. C. Greubel, Jul 12 2024

G.f.: x*(1 + 23*x)/(1 - x)^4.
a(n) = A000292(n) + 23*A000292(n-1).
a(n) = n*A051866(n) - Sum_{i=0..n-1} A051866(i). - Bruno Berselli, Apr 09 2015
Sum_{n>=1} 1/a(n) = 2*(4*(sqrt(2)+1)*Pi - 4*(sqrt(2)-8)*log(2) + 8*sqrt(2)*log(sqrt(2)+2) - 7)/105. - Amiram Eldar, Jan 10 2022
E.g.f.: (1/2)*x*(2 + 25*x + 8*x^2)*exp(x). - G. C. Greubel, Jul 12 2024

A256647 27-gonal pyramidal numbers: a(n) = n(n+1)(25*n-22)/6.

Original entry on oeis.org

0, 1, 28, 106, 260, 515, 896, 1428, 2136, 3045, 4180, 5566, 7228, 9191, 11480, 14120, 17136, 20553, 24396, 28690, 33460, 38731, 44528, 50876, 57800, 65325, 73476, 82278, 91756, 101935, 112840, 124496, 136928, 150161, 164220, 179130, 194916, 211603, 229216

Offset: 0

Luciano Ancora, Apr 07 2015

See comments in A256645.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (25th row of the table).

Luciano Ancora, Table of n, a(n) for n = 0..1000
Luciano Ancora, Polygonal and Pyramidal numbers, Section 3.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Partial sums of A255186.
Cf. similar sequences listed in A237616.
Cf. A000292, A256645.

[n*(n+1)*(25*n-22)/6: n in [0..50]]; // Vincenzo Librandi, Apr 08 2015

Table[n (n + 1) (25 n - 22)/6, {n, 0, 40}]
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 28, 106}, 40] (* Vincenzo Librandi, Apr 08 2015 *)

G.f.: x*(1 + 24*x)/(1 - x)^4.
a(n) = A000292(n) + 24*A000292(n-1).
From Elmo R. Oliveira, Aug 04 2025: (Start)
E.g.f.: exp(x)*x*(6 + 78*x + 25*x^2)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

A256648 28-gonal pyramidal numbers: a(n) = n(n+1)(26*n-23)/6.

Original entry on oeis.org

0, 1, 29, 110, 270, 535, 931, 1484, 2220, 3165, 4345, 5786, 7514, 9555, 11935, 14680, 17816, 21369, 25365, 29830, 34790, 40271, 46299, 52900, 60100, 67925, 76401, 85554, 95410, 105995, 117335, 129456, 142384, 156145, 170765, 186270, 202686, 220039, 238355

Offset: 0

Luciano Ancora, Apr 07 2015

See comments in A256645.

This sequence is related to A051867 by a(n) = n*A051867(n) - Sum_{i=0..n-1} A051867(i). - Bruno Berselli, Apr 09 2015

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (26th row of the table).

Luciano Ancora, Table of n, a(n) for n = 0..1000
Luciano Ancora, Polygonal and Pyramidal numbers, Section 3.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Partial sums of A161935.
Cf. similar sequences listed in A237616.
Cf. A000292, A051867, A256645.

[n*(n+1)*(26*n-23)/6: n in [0..50]]; // Vincenzo Librandi, Apr 08 2015

Table[n (n + 1)(26 n - 23)/6, {n, 0, 40}]
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 29, 110}, 40] (* Vincenzo Librandi, Apr 08 2015 *)

G.f.: x*(1 + 25*x)/(1 - x)^4.
a(n) = A000292(n) + 25*A000292(n-1).
From Elmo R. Oliveira, Aug 04 2025: (Start)
E.g.f.: exp(x)*x*(6 + 81*x + 26*x^2)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

A256649 29-gonal pyramidal numbers: a(n) = n(n+1)(9*n-8)/2.

Original entry on oeis.org

0, 1, 30, 114, 280, 555, 966, 1540, 2304, 3285, 4510, 6006, 7800, 9919, 12390, 15240, 18496, 22185, 26334, 30970, 36120, 41811, 48070, 54924, 62400, 70525, 79326, 88830, 99064, 110055, 121830, 134416, 147840, 162129, 177310, 193410, 210456, 228475, 247494

Offset: 0

Luciano Ancora, Apr 07 2015

See comments in A256645.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (27th row of the table).

Luciano Ancora, Table of n, a(n) for n = 0..1000
Luciano Ancora, Polygonal and Pyramidal numbers, Section 3.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Partial sums of A255187.
Cf. similar sequences listed in A237616.
Cf. A000292, A256645.

[n*(n+1)*(9*n-8)/2: n in [0..50]]; // Vincenzo Librandi, Apr 08 2015

Table[n (n + 1)(9 n - 8)/2, {n, 0, 40}]
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 30, 114}, 40] (* Vincenzo Librandi, Apr 08 2015 *)

G.f.: x*(1 + 26*x)/(1 - x)^4.
a(n) = A000292(n) + 26*A000292(n-1).
From Elmo R. Oliveira, Aug 04 2025: (Start)
E.g.f.: exp(x)*x*(2 + 28*x + 9*x^2)/2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

A256650 30-gonal pyramidal numbers: a(n) = n(n+1)(28*n-25)/6.

Original entry on oeis.org

0, 1, 31, 118, 290, 575, 1001, 1596, 2388, 3405, 4675, 6226, 8086, 10283, 12845, 15800, 19176, 23001, 27303, 32110, 37450, 43351, 49841, 56948, 64700, 73125, 82251, 92106, 102718, 114115, 126325, 139376, 153296, 168113, 183855, 200550, 218226, 236911, 256633

Offset: 0

Luciano Ancora, Apr 07 2015

See comments in A256645.

This sequence is related to A051868 by a(n) = n*A051868(n) - Sum_{i=0..n-1} A051868(i). [Bruno Berselli, Apr 09 2015]

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (28th row of the table).

Luciano Ancora, Table of n, a(n) for n = 0..1000
Luciano Ancora, Polygonal and Pyramidal numbers, Section 3.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Partial sums of A254474.
Cf. similar sequences listed in A237616.
Cf. A000292, A051868, A256645.

[n*(n+1)*(28*n-25)/6: n in [0..50]]; // Vincenzo Librandi, Apr 08 2015

Table[n (n + 1) (28 n - 25)/6, {n, 0, 40}]
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 31, 118}, 40] (* Vincenzo Librandi, Apr 08 2015 *)

G.f.: x*(1 + 27*x)/(1 - x)^4.
a(n) = A000292(n) + 27*A000292(n-1).
From Elmo R. Oliveira, Aug 04 2025: (Start)
E.g.f.: exp(x)*x*(6 + 87*x + 28*x^2)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

cp's OEIS Frontend

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

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A254474 30-gonal numbers: a(n) = n*(14*n-13).

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A255185 26-gonal numbers: a(n) = n*(12*n-11).

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A256646 26-gonal pyramidal numbers: a(n) = n*(n+1)*(8*n-7)/2.

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A256647 27-gonal pyramidal numbers: a(n) = n*(n+1)*(25*n-22)/6.

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A256648 28-gonal pyramidal numbers: a(n) = n*(n+1)*(26*n-23)/6.

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Magma

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

A254474 30-gonal numbers: a(n) = n(14n-13).

A255185 26-gonal numbers: a(n) = n(12n-11).

A256646 26-gonal pyramidal numbers: a(n) = n(n+1)(8*n-7)/2.

A256647 27-gonal pyramidal numbers: a(n) = n(n+1)(25*n-22)/6.

A256648 28-gonal pyramidal numbers: a(n) = n(n+1)(26*n-23)/6.

A256649 29-gonal pyramidal numbers: a(n) = n(n+1)(9*n-8)/2.

A256650 30-gonal pyramidal numbers: a(n) = n(n+1)(28*n-25)/6.