A220084 a(n) = (n + 1)*(20*n^2 + 19*n + 6)/6.

1, 15, 62, 162, 335, 601, 980, 1492, 2157, 2995, 4026, 5270, 6747, 8477, 10480, 12776, 15385, 18327, 21622, 25290, 29351, 33825, 38732, 44092, 49925, 56251, 63090, 70462, 78387, 86885, 95976, 105680, 116017, 127007, 138670, 151026, 164095, 177897, 192452

Offset: 0

Author

Bruno Berselli, Dec 11 2012

Keywords

• nonn
• easy

Comments

Sequence related to heptagonal pyramidal numbers (A002413) by a(n) = n*A002413(n) - (n-1)*A002413(n-1).

Other sequences of numbers of the form m*P(k,m)-(m-1)*P(k,m-1), where P(k,m) is the m-th k-gonal pyramidal number:

k=3, A002412(m) = m*A000292(m)-(m-1)*A000292(m-1);

k=4, A051662(m) = (m+1)*A000330(m+1)-m*A000330(m);

k=5, A213772(m) = m*A002411(m)-(m-1)*A002411(m-1);

k=6, A213837(m) = m*A002412(m)-(m-1)*A002412(m-1);

k=7, this sequence;

k=8, A130748(m) = m*A002414(m)-(m-1)*A002414(m-1).

Also, first bisection of A212983.

Binomial transform of (1, 14, 33, 20, 0, 0, 0, ...). - Gary W. Adamson, Aug 26 2015

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000292, A000330, A000566, A002411, A002412, A002413, A002414, A051662, A130748, A212983, A213772, A213837.

Programs

Magma

[(n+1)*(20*n^2+19*n+6)/6: n in [0..40]]; // Bruno Berselli, Jun 28 2016

Magma

/* By first comment: */  A002413:=func; [n*A002413(n)-(n-1)*A002413(n-1): n in [1..40]];

Magma

I:=[1,15,62,162]; [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, Aug 18 2013

Mathematica

Table[(n + 1) (20 n^2 + 19 n + 6)/6, {n, 0, 40}]
LinearRecurrence[{4,-6,4,-1},{1,15,62,162},40] (* Harvey P. Dale, Dec 23 2012 *)
CoefficientList[Series[(1 + 11 x + 8 x^2) / (1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

Maxima

makelist((n+1)*(20*n^2+19*n+6)/6, n, 0, 20); /* Martin Ettl, Dec 12 2012 */

PARI

a(n)=(n+1)*(20*n^2+19*n+6)/6 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: (1+11*x+8*x^2)/(1-x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), for n>3, a(0)=1, a(1)=15, a(2)=62, a(3)=162. - Harvey P. Dale, Dec 23 2012

a(n) = (n+1)*A000566(n+1) + Sum_{i=0..n} A000566(i). - Bruno Berselli, Dec 18 2013

E.g.f.: exp(x)*(6 + 84*x + 99*x^2 + 20*x^3)/6. - Elmo R. Oliveira, Aug 06 2025