A226491 -id:A226491 - 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

A152759 3 times 9-gonal (or nonagonal) numbers: a(n) = 3n(7*n-5)/2.

Original entry on oeis.org

0, 3, 27, 72, 138, 225, 333, 462, 612, 783, 975, 1188, 1422, 1677, 1953, 2250, 2568, 2907, 3267, 3648, 4050, 4473, 4917, 5382, 5868, 6375, 6903, 7452, 8022, 8613, 9225, 9858, 10512, 11187, 11883, 12600, 13338, 14097, 14877, 15678, 16500, 17343, 18207, 19092, 19998

Offset: 0

Omar E. Pol, Dec 14 2008

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

Cf. A001106, A139268, A226491.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152767, A153783, A153448, A153875.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=21: see Comments lines of A226492.

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,3,6!,21}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
CoefficientList[Series[3 x (1 + 6 x) / (1 - x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 05 2013 *)
LinearRecurrence[{3,-3,1},{0,3,27},40] (* Harvey P. Dale, May 26 2015 *)

a(n)=3*n*(7*n-5)/2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = (21*n^2 - 15*n)/2 = 3*A001106(n).
a(n) = a(n-1) + 21*n - 18 with n > 0, a(0)=0. - Vincenzo Librandi, Nov 26 2010
G.f.: 3*x*(1+6*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
a(n) = n + A226491(n). - Bruno Berselli, Jun 11 2013
From Elmo R. Oliveira, Dec 15 2024: (Start)
E.g.f.: 3*exp(x)*x*(2 + 7*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A051875 23-gonal numbers: a(n) = n(21n-19)/2.

Original entry on oeis.org

0, 1, 23, 66, 130, 215, 321, 448, 596, 765, 955, 1166, 1398, 1651, 1925, 2220, 2536, 2873, 3231, 3610, 4010, 4431, 4873, 5336, 5820, 6325, 6851, 7398, 7966, 8555, 9165, 9796, 10448, 11121, 11815, 12530, 13266, 14023, 14801, 15600

Offset: 0

N. J. A. Sloane, Dec 15 1999

Sequence found by reading the line from 0, in the direction 0, 23, ..., and the parallel line from 1, in the direction 1, 66, ..., in the square spiral whose vertices are the generalized 23-gonal numbers. - Omar E. Pol, Jul 18 2012

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).

CoefficientList[Series[x (1 + 20 x) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 19 2013 *)
Table[(21n^2 - 19n)/2, {n, 0, 39}] (* Alonso del Arte, Jan 23 2015 *)
PolygonalNumber[23,Range[0,40]] (* Harvey P. Dale, Aug 01 2022 *)

a(n)=n*(21*n-19)/2 \\ Charles R Greathouse IV, Jan 24 2014

G.f.: x*(1+20*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011
a(n) = 21*n + a(n-1) - 20 with n > 0, a(0) = 0. - Vincenzo Librandi, Aug 06 2010
a(n) = A226491(n) - n. - Bruno Berselli, Jun 11 2013
a(21*a(n)+211*n+1) = a(21*a(n)+211*n) + a(21*n+1). - Vladimir Shevelev, Jan 24 2014
Product_{n>=2} (1 - 1/a(n)) = 21/23. - Amiram Eldar, Jan 22 2021
E.g.f.: exp(x)*(x + 21*x^2/2). - Nikolaos Pantelidis, Feb 06 2023

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Magma

Maple

Mathematica

PARI

Sage

Formula

A152759 3 times 9-gonal (or nonagonal) numbers: a(n) = 3n(7*n-5)/2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A051875 23-gonal numbers: a(n) = n(21n-19)/2.

Views

Author

Keywords

Comments

References

Links

Programs

Mathematica

PARI

Formula

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Magma

Maple

Mathematica

PARI

Sage

Formula

A152759 3 times 9-gonal (or nonagonal) numbers: a(n) = 3*n*(7*n-5)/2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A051875 23-gonal numbers: a(n) = n(21n-19)/2.

Views

Author

Keywords

Comments

References

Links

Programs

Mathematica

PARI

Formula

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

A152759 3 times 9-gonal (or nonagonal) numbers: a(n) = 3n(7*n-5)/2.