A226490 -id:A226490 - cp's OEIS frontend

A226488 a(n) = n(13n - 9)/2.

0, 2, 17, 45, 86, 140, 207, 287, 380, 486, 605, 737, 882, 1040, 1211, 1395, 1592, 1802, 2025, 2261, 2510, 2772, 3047, 3335, 3636, 3950, 4277, 4617, 4970, 5336, 5715, 6107, 6512, 6930, 7361, 7805, 8262, 8732, 9215, 9711, 10220, 10742, 11277, 11825, 12386, 12960

Offset: 0

Bruno Berselli, Jun 09 2013

Sum of n-th octagonal number and n-th 9-gonal (nonagonal) number.

Sum of reciprocals of a(n), for n>0: 0.629618994194109711163742089971688...

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000567, A001106, A153080 (first differences).

Cf. numbers of the form n*(n*k-k+4)/2 listed in A005843 (k=0), A000096 (k=1), A002378 (k=2), A005449 (k=3), A001105 (k=4), A005476 (k=5), A049450 (k=6), A218471 (k=7), A002939 (k=8), A062708 (k=9), A135706 (k=10), A180223 (k=11), A139267 (n=12), this sequence (k=13), A139268 (k=14), A226489 (k=15), A139271 (k=16), A180232 (k=17), A152995 (k=18), A226490 (k=19), A152965 (k=20), A226491 (k=21), A152997 (k=22).

List([0..50], n-> n*(13*n-9)/2); # G. C. Greubel, Aug 30 2019

Magma
```
[n*(13*n-9)/2: n in [0..50]];
```

I:=[0,2,17]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2) +Self(n-3): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

A226488:=n->n*(13*n - 9)/2; seq(A226488(n), n=0..50); # Wesley Ivan Hurt, Feb 25 2014

Table[n(13n-9)/2, {n, 0, 50}]
LinearRecurrence[{3, -3, 1}, {0, 2, 17}, 50] (* Harvey P. Dale, Jun 19 2013 *)
CoefficientList[Series[x(2+11x)/(1-x)^3, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)

a(n)=n*(13*n-9)/2 \\ Charles R Greathouse IV, Sep 24 2015

[n*(13*n-9)/2 for n in (0..50)] # G. C. Greubel, Aug 30 2019

G.f.: x*(2+11*x)/(1-x)^3.
a(n) + a(-n) = A152742(n).
a(0)=0, a(1)=2, a(2)=17; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jun 19 2013
E.g.f.: x*(4 + 13*x)*exp(x)/2. - G. C. Greubel, Aug 30 2019
a(n) = A000567(n) + A001106(n). - Michel Marcus, Aug 31 2019

A051873 21-gonal numbers: a(n) = n*(19n - 17)/2.

Original entry on oeis.org

0, 1, 21, 60, 118, 195, 291, 406, 540, 693, 865, 1056, 1266, 1495, 1743, 2010, 2296, 2601, 2925, 3268, 3630, 4011, 4411, 4830, 5268, 5725, 6201, 6696, 7210, 7743, 8295, 8866, 9456, 10065, 10693, 11340, 12006, 12691, 13395, 14118

Offset: 0

N. J. A. Sloane, Dec 15 1999

Sequence found by reading the line from 0, in the direction 0, 21, ... and the parallel line from 1, in the direction 1, 60, ..., in the square spiral whose vertices are the generalized 21-gonal numbers. - Omar E. Pol, Jul 18 2012

Partial sums of A215144. - Leo Tavares, Mar 17 2023

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A215144, A226490.

A051873 := proc(n) n*(19*n-17)/2 ;end proc: seq(A051873(n),n=0..30) ; # R. J. Mathar, Feb 05 2011

PolygonalNumber[21,Range[0,40]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 22 2016 *)
Table[n*(19*n - 17)/2, {n, 0, 100}] (* Robert Price, Oct 11 2018 *)

n*(19*n-17)/2 \\ Charles R Greathouse IV, Jan 24 2014

G.f.: x*(1+18*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011
a(n) = 19*n+a(n-1)-18 with n>0, a(0)=0. - Vincenzo Librandi, Aug 06 2010
a(n) = A226490(n) - n. - Bruno Berselli, Jun 11 2013
a(19*a(n)+172*n+1) = a(19*a(n)+172*n) + a(19*n+1). - Vladimir Shevelev, Jan 24 2014
Product_{n>=2} (1 - 1/a(n)) = 19/21. - Amiram Eldar, Jan 22 2021
E.g.f.: exp(x)*(x + 19*x^2/2). - Nikolaos Pantelidis, Feb 06 2023

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

Original entry on oeis.org

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)

cp's OEIS Frontend

A226488 a(n) = n*(13*n - 9)/2.

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A051873 21-gonal numbers: a(n) = n*(19n - 17)/2.

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Maple

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

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A226488 a(n) = n(13n - 9)/2.

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