A180232 -id:A180232 - 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

A237617 a(n) = n(n + 1)(17*n - 14)/6.

Original entry on oeis.org

0, 1, 20, 74, 180, 355, 616, 980, 1464, 2085, 2860, 3806, 4940, 6279, 7840, 9640, 11696, 14025, 16644, 19570, 22820, 26411, 30360, 34684, 39400, 44525, 50076, 56070, 62524, 69455, 76880, 84816, 93280, 102289, 111860, 122010, 132756, 144115, 156104, 168740

Offset: 0

Bruno Berselli, Feb 11 2014

Also 19-gonal (or nonadecagonal) pyramidal numbers.

This sequence is related to A180232 by 2*a(n) = n*A180232(n) - Sum_{i=0..n-1} A180232(i).

			After 0, the sequence is provided by the row sums of the triangle:
   1;
   2,  18;
   3,  36,  35;
   4,  54,  70,  52;
   5,  72, 105, 104,  69;
   6,  90, 140, 156, 138,  86;
   7, 108, 175, 208, 207, 172, 103;
   8, 126, 210, 260, 276, 258, 206, 120;
   9, 144, 245, 312, 345, 344, 309, 240, 137;
  10, 162, 280, 364, 414, 430, 412, 360, 274, 154; etc.,
where (r = row index, c = column index):
T(r,r) = T(c,c) = 17*r - 16 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 (seventeenth 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. A051871, A180232.

Cf. similar sequences listed in A237616.

List([0..40], n-> n*(n+1)*(17*n-14)/6); # G. C. Greubel, Aug 30 2019

Magma
```
[n*(n+1)*(17*n-14)/6: n in [0..40]];
```

I:=[0,1,20,74]; [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..40]]; // Vincenzo Librandi, Feb 12 2014

seq(n*(n+1)*(17*n-14)/6, n=0..40); # G. C. Greubel, Aug 30 2019

Table[n(n+1)(17*n-14)/6, {n, 0, 40}]
CoefficientList[Series[x(1+16x)/(1-x)^4, {x,0,40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
LinearRecurrence[{4,-6,4,-1},{0,1,20,74},40] (* Harvey P. Dale, Aug 04 2021 *)

vector(40, n, n*(n-1)*(17*n-31)/6) \\ G. C. Greubel, Aug 30 2019

[n*(n+1)*(17*n-14)/6 for n in (0..40)] # G. C. Greubel, Aug 30 2019

G.f.: x*(1 + 16*x)/(1 - x)^4.
For n>0, a(n) = Sum_{i=0..n-1} (n-i)*(17*i+1); see the generalization in A237616 (Formula field).
E.g.f.: x*(6 + 54*x + 17*x^2)*exp(x)/6. - G. C. Greubel, Aug 30 2019

cp's OEIS Frontend

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

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A237617 a(n) = n(n + 1)(17*n - 14)/6.

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Magma

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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

A237617 a(n) = n*(n + 1)*(17*n - 14)/6.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Magma

Maple

Mathematica

PARI

Sage

Formula

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

A237617 a(n) = n(n + 1)(17*n - 14)/6.