A172076 a(n) = n*(n+1)*(14*n-11)/6.

0, 1, 17, 62, 150, 295, 511, 812, 1212, 1725, 2365, 3146, 4082, 5187, 6475, 7960, 9656, 11577, 13737, 16150, 18830, 21791, 25047, 28612, 32500, 36725, 41301, 46242, 51562, 57275, 63395, 69936, 76912, 84337, 92225, 100590, 109446, 118807, 128687

Offset: 0

Author

Vincenzo Librandi, Jan 25 2010

Keywords

• nonn
• easy

Comments

Generated by the formula n*(n+1)*(2*d*n-(2*d-3))/6 for d=7.

From Bruno Berselli, Dec 14 2010: (Start)

In fact, the sequence is related to A001106 by a(n) = n*A001106(n) - Sum_{k=0..n-1} A001106(k) and this is the case d=7 in the identity n*(n*(d*n-d+2)/2) - Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6.

Also 16-gonal (or hexadecagonal) pyramidal numbers.

Inverse binomial transform of this sequence: 0, 1, 15, 14, 0, 0 (0 continued). (End)

References

• E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93. [From Bruno Berselli, Feb 13 2014]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian), 2008.
• Index to sequences related to pyramidal numbers.
• 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.

Programs

GAP

List([0..40], n-> n*(n+1)*(14*n-11)/6); # G. C. Greubel, Aug 30 2019

Magma

[n*(n+1)*(14*n-11)/6: n in [0..40]] // G. C. Greubel, Aug 30 2019

Maple

A172076:=n->n*(n+1)*(14*n-11)/6; seq(A172076(n), n=0..50); # Wesley Ivan Hurt, Feb 26 2014

Mathematica

LinearRecurrence[{4,-6,4,-1}, {0, 1, 17, 62}, 50] (* Vincenzo Librandi, Mar 01 2012 *)

PARI

vector(40, n, n*(n-1)*(14*n-25)/6) \\ G. C. Greubel, Aug 30 2019

Sage

[n*(n+1)*(14*n-11)/6 for n in (0..40)] # G. C. Greubel, Aug 30 2019

Formula

G.f.: x*(1+13*x)/(1-x)^4. - Bruno Berselli, Dec 15 2010

a(n) = Sum_{i=0..n} A051868(i). - Bruno Berselli, Dec 15 2010

a(n) = Sum_{i=0..n-1} (n-i)*(14*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014

E.g.f.: x*(6 + 45*x + 14*x^2)*exp(x)/6. - G. C. Greubel, Aug 30 2019