A177890 15-gonal (or pentadecagonal) pyramidal numbers: a(n) = n*(n+1)*(13*n-10)/6.

0, 1, 16, 58, 140, 275, 476, 756, 1128, 1605, 2200, 2926, 3796, 4823, 6020, 7400, 8976, 10761, 12768, 15010, 17500, 20251, 23276, 26588, 30200, 34125, 38376, 42966, 47908, 53215, 58900, 64976, 71456, 78353, 85680, 93450, 101676, 110371, 119548, 129220

Offset: 0

Author

Bruno Berselli, Dec 14 2010

Keywords

• nonn
• easy

Comments

Also a(n) = (15-m)*A000292(n-1) + n*(n+1)*((m-2)*n - (m-5))/6 being n*(n+1)*((m-2)*n - (m-5))/6 a m-gonal pyramidal number (1 < m < 15). For m=6, a(n) = 9*A000292(n-1) + A002412(n).

Inverse binomial transform of this sequence: 0, 1, 14, 13, 0, 0 (0 continued).

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index to sequences related to pyramidal numbers
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. similar sequences listed in A237616.

Cf. A000292, A002412, A051867.

Programs

GAP

List([0..40], n-> n*(n+1)*(13*n-10)/6); # G. C. Greubel, Aug 30 2019

Magma

I:=[0,1,16,58]; [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, Jul 04 2012

Magma

[n*(n+1)*(13*n-10)/6: n in [0..40]]; // G. C. Greubel, Aug 30 2019

Maple

seq(n*(n+1)*(13*n-10)/6, n=0..40); # G. C. Greubel, Aug 30 2019

Mathematica

CoefficientList[Series[x*(1+12*x)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jul 04 2012 *)
Table[n*(n-1)*(13*n-23)/6, {n,40}] (* G. C. Greubel, Aug 30 2019 *)
LinearRecurrence[{4,-6,4,-1},{0,1,16,58},40] (* Harvey P. Dale, Dec 21 2022 *)

PARI

vector(40, n, n*(n-1)*(13*n-23)/6) \\ G. C. Greubel, Aug 30 2019

Sage

[n*(n+1)*(13*n-10)/6 for n in (0..40)] # G. C. Greubel, Aug 30 2019

Formula

G.f.: x*(1+12*x)/(1-x)^4.

a(n) = Sum_{i=0..n} A051867(i).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jul 04 2012

a(n) = Sum_{i=0..n-1} (n-i)*(13*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014

E.g.f.: x*(6 + 42*x + 13*x^2)*exp(x)/6. - G. C. Greubel, Aug 30 2019