A237616 -id:A237616 - cp's OEIS frontend

A237618 a(n) = n(n + 1)(19*n - 16)/6.

0, 1, 22, 82, 200, 395, 686, 1092, 1632, 2325, 3190, 4246, 5512, 7007, 8750, 10760, 13056, 15657, 18582, 21850, 25480, 29491, 33902, 38732, 44000, 49725, 55926, 62622, 69832, 77575, 85870, 94736, 104192, 114257, 124950, 136290, 148296, 160987, 174382

Offset: 0

Bruno Berselli, Feb 11 2014

Also 21-gonal (or icosihenagonal) pyramidal numbers.

			After 0, the sequence is provided by the row sums of the triangle:
   1;
   2,  20;
   3,  40,  39;
   4,  60,  78,  58;
   5,  80, 117, 116, 77;
   6, 100, 156, 174, 154, 96;
   7, 120, 195, 232, 231, 192, 115;
   8, 140, 234, 290, 308, 288, 230, 134;
   9, 160, 273, 348, 385, 384, 345, 268, 153;
  10, 180, 312, 406, 462, 480, 460, 402, 306, 172; etc.,
where (r = row index, c = column index):
T(r,r) = T(c,c) = 19*r-18 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 (nineteenth row of the table).

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).

Cf. A051873, A226490.

Cf. similar sequences listed in A237616.

Magma
```
[n*(n+1)*(19*n-16)/6: n in [0..40]];
```

I:=[0,1,22,82]; [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

Table[n(n+1)(19n-16)/6, {n, 0, 40}]
CoefficientList[Series[x(1+18x)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)

b=binomial; [b(n+2,3) +18*b(n+1,3) for n in (0..50)] # G. C. Greubel, May 27 2022

G.f.: x*(1 + 18*x) / (1 - x)^4.
a(n) = (1/2)*( n*A226490(n) - Sum_{j=0..n-1} A226490(j) ).
a(n) = Sum_{i=0..n-1} (n-i)*(19*i+1), for n>0; see the generalization in A237616 (Formula field).
From G. C. Greubel, May 27 2022: (Start)
a(n) = binomial(n+2, 3) + 18*binomial(n+1, 3).
E.g.f.: (1/6)*x*(6 + 60*x + 19*x^2)*exp(x). (End)

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)

A256718 a(n) = n(n+1)(7*n-6)/2.

Original entry on oeis.org

0, 1, 24, 90, 220, 435, 756, 1204, 1800, 2565, 3520, 4686, 6084, 7735, 9660, 11880, 14416, 17289, 20520, 24130, 28140, 32571, 37444, 42780, 48600, 54925, 61776, 69174, 77140, 85695, 94860, 104656, 115104, 126225, 138040, 150570, 163836, 177859, 192660

Offset: 0

Bruno Berselli, Apr 09 2015

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (21st row of the table).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Partial sums of A051875.

Cf. similar sequences listed in A237616.

Magma
```
[n*(n+1)*(7*n-6)/2: n in [0..40]];
```

Table[n (n + 1) (7 n - 6)/2, {n, 0, 40}]
LinearRecurrence[{4,-6,4,-1},{0,1,24,90},40] (* Harvey P. Dale, Jan 15 2024 *)

PARI
```
vector(40, n, n--; n*(n+1)*(7*n-6)/2)
```
Sage
```
[n*(n+1)*(7*n-6)/2 for n in (0..40)]
```

G.f.: x*(1 + 20*x)/(1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) with n>3, a(0)=0, a(1)=1, a(2)=24, a(3)=90.
a(n) = Sum_{i=0..n-1} (n-i)*(21*i+1) for n>0.

cp's OEIS Frontend

A237618 a(n) = n*(n + 1)*(19*n - 16)/6.

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

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

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Magma

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A256718 a(n) = n*(n+1)*(7*n-6)/2.

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A237618 a(n) = n(n + 1)(19*n - 16)/6.

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.

A256718 a(n) = n(n+1)(7*n-6)/2.