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%I #99 Sep 08 2022 08:44:39
%S 0,1,12,45,112,225,396,637,960,1377,1900,2541,3312,4225,5292,6525,
%T 7936,9537,11340,13357,15600,18081,20812,23805,27072,30625,34476,
%U 38637,43120,47937,53100,58621,64512
%N a(n) = (2*n - 1)*n^2.
%C Structured hexagonal prism numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
%C Number of divisors of 60^(n-1) for n>0. - _J. Lowell_, Aug 30 2008
%C 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
%C Partial sums of A080859. - _J. M. Bergot_, Jul 03 2013
%C a(n) = number of 2 X 2 matrices having all elements in {0..n} with determinant = permanent. - _Indranil Ghosh_, Dec 26 2016
%C Number of additions and multiplications needed to multiply two n X n matrices naively. - _Charles R Greathouse IV_, Jan 19 2018
%H Vincenzo Librandi, <a href="/A015237/b015237.txt">Table of n, a(n) for n = 0..5000</a>
%H M. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A counting function</a>, arXiv preprint arXiv:1301.4550 [math.CO], 2013.
%H M. Janjic, B. Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5
%H Milan Janjić, <a href="https://www.emis.de/journals/JIS/VOL21/Janjic2/janjic103.html">Pascal Matrices and Restricted Words</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = A000578(n) + A045991(n). - _Zerinvary Lajos_, Jun 11 2008
%F a(n) = A199771(2*n-1) for n > 0. - _Reinhard Zumkeller_, Nov 23 2011
%F G.f.: x*(1+8*x+3*x^2)/(1-x)^4. - _Colin Barker_, Jun 08 2012
%F 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
%F E.g.f.: x*(2*x^2 + 5*x + 1)*exp(x). - _G. C. Greubel_, Jul 31 2015
%F a(n) = Sum_{i=0..n-1} n*(4*i+1) for n>0. - _Bruno Berselli_, Sep 08 2015
%F Sum_{n>=1} 1/a(n) = 4*log(2) - Pi^2/6. - _Vaclav Kotesovec_, Oct 04 2016
%F a(n) = Sum_{i=n^2-n+1..n^2+n-1} i. - _Wesley Ivan Hurt_, Dec 27 2016
%F From _Peter Bala_, Jan 30 2019: (Start)
%F 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)
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi - Pi^2/12 - 2*log(2). - _Amiram Eldar_, Jul 12 2020
%p [(2*n-1)*n^2$n=0..40]; # _Muniru A Asiru_, Feb 05 2019
%t 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 *)
%t Table[(2 n - 1) n^2, {n, 0, 40}] (* _Bruno Berselli_, Sep 08 2015 *)
%o (Magma) [(2*n-1)*n^2: n in [0..40]]; // _Vincenzo Librandi_, Jul 19 2011
%o (PARI) a(n)=(2*n-1)*n^2 \\ _Charles R Greathouse IV_, Oct 07 2015
%o (GAP) List([0..40],n->(2*n-1)*n^2); # _Muniru A Asiru_, Feb 05 2019
%Y Cf. A100177 (structured prisms); A100145 (more on structured numbers).
%Y Cf. A000578, A045991, A000384, A080859 (first diffs), A103220 (partial sums).
%Y Cf. similar sequences, with the formula (k*n-k+2)*n^2/2, listed in A262000.
%Y Cf. A036970, A272378.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
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