A015237 a(n) = (2*n - 1)*n^2.

0, 1, 12, 45, 112, 225, 396, 637, 960, 1377, 1900, 2541, 3312, 4225, 5292, 6525, 7936, 9537, 11340, 13357, 15600, 18081, 20812, 23805, 27072, 30625, 34476, 38637, 43120, 47937, 53100, 58621, 64512

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Structured hexagonal prism numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Number of divisors of 60^(n-1) for n>0. - J. Lowell, Aug 30 2008

The sum of the 2*n+1 numbers between n*(n+1) and (n+1)*(n+2) gives a(n+1). - J. M. Bergot, Apr 17 2013

Partial sums of A080859. - J. M. Bergot, Jul 03 2013

a(n) = number of 2 X 2 matrices having all elements in {0..n} with determinant = permanent. - Indranil Ghosh, Dec 26 2016

Number of additions and multiplications needed to multiply two n X n matrices naively. - Charles R Greathouse IV, Jan 19 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000
• M. Janjic and B. Petkovic, A counting function, arXiv preprint arXiv:1301.4550 [math.CO], 2013.
• M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5
• Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A100177 (structured prisms); A100145 (more on structured numbers).

Cf. A000578, A045991, A000384, A080859 (first diffs), A103220 (partial sums).

Cf. similar sequences, with the formula (k*n-k+2)*n^2/2, listed in A262000.

Cf. A036970, A272378.

Programs

GAP

List([0..40],n->(2*n-1)*n^2); # Muniru A Asiru, Feb 05 2019

Magma

[(2*n-1)*n^2: n in [0..40]]; // Vincenzo Librandi, Jul 19 2011

Maple

[(2*n-1)*n^2$n=0..40]; # Muniru A Asiru, Feb 05 2019

Mathematica

RecurrenceTable[{a[0]==0, a[1]==1, a[2]==12, a[n]== 3*a[n-1] - 3*a[n-2] + a[n-3] + 12}, a, {n, 30}] (* G. C. Greubel, Jul 31 2015 *)
Table[(2 n - 1) n^2, {n, 0, 40}] (* Bruno Berselli, Sep 08 2015 *)

PARI

a(n)=(2*n-1)*n^2 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = A000578(n) + A045991(n). - Zerinvary Lajos, Jun 11 2008

a(n) = A199771(2*n-1) for n > 0. - Reinhard Zumkeller, Nov 23 2011

G.f.: x*(1+8*x+3*x^2)/(1-x)^4. - Colin Barker, Jun 08 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 12, a(0)=1, a(1)=1, a(2)=12. - G. C. Greubel, Jul 31 2015

E.g.f.: x*(2*x^2 + 5*x + 1)*exp(x). - G. C. Greubel, Jul 31 2015

a(n) = Sum_{i=0..n-1} n*(4*i+1) for n>0. - Bruno Berselli, Sep 08 2015

Sum_{n>=1} 1/a(n) = 4*log(2) - Pi^2/6. - Vaclav Kotesovec, Oct 04 2016

a(n) = Sum_{i=n^2-n+1..n^2+n-1} i. - Wesley Ivan Hurt, Dec 27 2016

From Peter Bala, Jan 30 2019: (Start)

Let a(n,x) = Product_{k = 0..n} (x - k)/(x + k). Then for positive integer x we have (2*x - 1)*x^2 = Sum_{n >= 0} ((n+1)^5 + n^5)*a(n,x) and (2*x - 1)*x = Sum_{n >= 0} ((n+1)^4 - n^4)*a(n,x). Both identities are also valid for complex x in the half-plane Re(x) > 2. See the Bala link in A036970. Cf. A272378. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi - Pi^2/12 - 2*log(2). - Amiram Eldar, Jul 12 2020