A262000 -id:A262000 - cp's OEIS frontend

A015237 a(n) = (2n - 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

N. J. A. Sloane

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

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).

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.

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

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

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

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 *)

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

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

A103532 Number of divisors of 240^n.

Original entry on oeis.org

1, 20, 81, 208, 425, 756, 1225, 1856, 2673, 3700, 4961, 6480, 8281, 10388, 12825, 15616, 18785, 22356, 26353, 30800, 35721, 41140, 47081, 53568, 60625, 68276, 76545, 85456, 95033, 105300, 116281, 128000, 140481, 153748, 167825, 182736

Offset: 0

J. Lowell, Aug 30 2008

Geometric interpretation: Take a simple cubical grid of size (2n+1). Number the coordinates along each axis from 1 to (2n+1). Select only the cells that have at least two odd coordinates, and discard the rest. The number of selected cells is a(n). - Arun Giridhar, Mar 27 2015

			a(2) = 81 because 240^2 has 81 divisors.
a(2) = 81 because a 5 X 5 X 5 grid has 81 cells with at least two odd coordinates each, coordinate numbering starting at 1.

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

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

[(4*n+1)*(n+1)^2: n in [0..45]]; // Vincenzo Librandi, Feb 10 2016

A103532 := proc(n) (4*n+1)*(n+1)^2 ; end proc: # R. J. Mathar, Aug 31 2008

Table[(4 n + 1) (n + 1)^2, {n, 0, 40}] (* Stefan Steinerberger, Aug 31 2008 *)
DivisorSigma[0,240^Range[0,40]] (* or *) LinearRecurrence[{4,-6,4,-1},{1,20,81,208},40] (* Harvey P. Dale, Jan 21 2013 *)

From R. J. Mathar and Stefan Steinerberger, Aug 31 2008: (Start)
a(n) = (4*n+1)*(n+1)^2.
G.f.: (1+16x+7x^2)/(1-x)^4.
Inverse binomial transform: 1, 19, 42, 24, 0 (0 continued). (End)
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3. - Harvey P. Dale, Jan 21 2013
a(n) = (n+1)*A001107(n+1), where A001107 are the partial sums of A017007. - J. M. Bergot, Jul 08 2013
a(n) = Sum_{i=0..n} (n+1)*(8*i+1). [Bruno Berselli, Sep 08 2015]
Sum_{n>=0} 1/a(n) = 2*Pi/9 - Pi^2/18 + 4*log(2)/3 = 1.07401658592825... . - Vaclav Kotesovec, Oct 04 2016
E.g.f.: exp(x)*(1 + 19*x + 21*x^2 + 4*x^3). - Stefano Spezia, Jan 31 2025
Sum_{n>=0} (-1)^n/a(n) = 2*sqrt(2)*Pi/9 - Pi^2/36 - (4/9)*(log(2) + sqrt(2)*log(sqrt(2)-1)). - Amiram Eldar, Aug 15 2025

More terms from Stefan Steinerberger and R. J. Mathar, Aug 31 2008

Example corrected by Harvey P. Dale, Jan 21 2013

A006597 a(n) = n^2(5n-3)/2.

Original entry on oeis.org

0, 1, 14, 54, 136, 275, 486, 784, 1184, 1701, 2350, 3146, 4104, 5239, 6566, 8100, 9856, 11849, 14094, 16606, 19400, 22491, 25894, 29624, 33696, 38125, 42926, 48114, 53704, 59711, 66150, 73036, 80384, 88209, 96526, 105350, 114696, 124579, 135014, 146016, 157600

Offset: 0

N. J. A. Sloane

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

Apart from 0, partial sums of A220083. - Bruno Berselli, Dec 11 2012

W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 29.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A100177 - structured prisms; A100145 for more on structured numbers.
Cf. similar sequences, with the formula (k*n - k + 2)*n^2/2, listed in A262000.
Cf. A006592, A220083.

[n^2*(5*n-3)/2: n in [0..40]]; // Vincenzo Librandi, Jul 20 2011

A006597:=n->n^2*(5*n-3)/2; seq(A006597(n), n=0..40); # Wesley Ivan Hurt, Mar 11 2014

Table[n^2*(5*n-3)/2, {n, 0, 40}] (* Wesley Ivan Hurt, Mar 11 2014 *)

a(n)=n^2*(5*n-3)/2; \\ Joerg Arndt, Jul 20 2011

a(n) = (1/6)*(15*n^3 - 9*n^2). - James A. Record (james.record(AT)gmail.com), Nov 07 2004
G.f.: x*(1+10*x+4*x^2)/(1-x)^4. - Colin Barker, Jun 08 2012
a(n) = Sum_{i=0..n-1} n*(5*i+1) for n>0. - Bruno Berselli, Sep 08 2015
Sum_{n>=1} 1/a(n) = 1.1080093773051638036... = (sqrt(5*(5 - 2*sqrt(5)))*Pi - Pi^2 - 5*sqrt(5)*arccoth(sqrt(5)) + (25*log(5))/2)/9. - Vaclav Kotesovec, Oct 04 2016
From Elmo R. Oliveira, Aug 06 2025: (Start)
E.g.f.: exp(x)*x*(2 + 12*x + 5*x^2)/2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = A006592(n)/4. (End)

Name corrected by Arkadiusz Wesolowski, Jul 20 2011

A050509 House numbers (version 2): a(n) = (n+1)^3 + (n+1)*Sum_{i=0..n} i.

Original entry on oeis.org

1, 10, 36, 88, 175, 306, 490, 736, 1053, 1450, 1936, 2520, 3211, 4018, 4950, 6016, 7225, 8586, 10108, 11800, 13671, 15730, 17986, 20448, 23125, 26026, 29160, 32536, 36163, 40050, 44206, 48640, 53361, 58378, 63700, 69336, 75295, 81586, 88218, 95200, 102541

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 28 1999

Also as a(n) = (1/6)*(9*n^3 - 3*n^2), n>0: structured pentagonal prism numbers (Cf. A100177 - structured prisms; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Number of inequivalent tetrahedral edge colorings using at most n+1 colors so that no color appears only once. - David Nacin, Feb 22 2017

			        *     *
a(2) = * * + * * = 10.
       * *   * *

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000217, A000578, A051662, A100145, A100177.

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

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

Table[((1+n)^2*(2+3n))/2,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,10,36,88},40] (* Harvey P. Dale, Jun 26 2011 *)

a(n)=(1/2)*(3*n+2)*(n+1)^2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A000578(n+1) + (n+1)*A000217(n).
a(n) = (1/2)*(3*n+2)*(n+1)^2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=10, a(2)=36, a(3)=88. - Harvey P. Dale, Jun 26 2011
G.f.: (1+6*x+2*x^2)/(1-x)^4. - Colin Barker, Jun 08 2012
a(n) = Sum_{i=0..n} (n+1)*(3*i+1). - Bruno Berselli, Sep 08 2015
Sum_{n>=0} 1/a(n) = 9*log(3) - sqrt(3)*Pi - Pi^2/3 = 1.15624437161388... . - Vaclav Kotesovec, Oct 04 2016
E.g.f.: exp(x)*(2 + 18*x + 17*x^2 + 3*x^3)/2. - Elmo R. Oliveira, Aug 06 2025

A100176 Structured octagonal prism numbers.

Original entry on oeis.org

1, 16, 63, 160, 325, 576, 931, 1408, 2025, 2800, 3751, 4896, 6253, 7840, 9675, 11776, 14161, 16848, 19855, 23200, 26901, 30976, 35443, 40320, 45625, 51376, 57591, 64288, 71485, 79200, 87451, 96256, 105633, 115600, 126175, 137376, 149221, 161728, 174915, 188800

Offset: 1

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Number of divisors of 120^(n-1). - J. Lowell, Aug 30 2008

Partial sums of A214675. - J. M. Bergot, Jul 08 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A100177 (structured prisms), A100145 (for more on structured numbers).
Cf. similar sequences, with the formula (k*n - k + 2)*n^2/2, listed in A262000.
Cf. A000567, A016777, A214675.

[3*n^3-2*n^2: n in [1..50] ]; // Vincenzo Librandi, Aug 02 2011

[seq(3*n^3-2*n^2,n=1..47)]; # Zerinvary Lajos, Jun 29 2006

f[n_] := 3 n^3 - 2 n^2; Table[f[n], {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Apr 27 2010 *)

a(n)=3*n^3-2*n^2 \\ Charles R Greathouse IV, Aug 10 2017

a(n) = 3*n^3 - 2*n^2.
G.f.: x*(1+12*x+5*x^2)/(1-x)^4. - Colin Barker, Jun 08 2012
a(n) = Sum_{i=0..n-1} n*(6*i+1). - Bruno Berselli, Sep 08 2015
Sum_{n>=1} 1/a(n) = sqrt(3)*Pi/8 - Pi^2/12 + 9*log(3)/8 = 1.0936465529153418... . - Vaclav Kotesovec, Oct 04 2016
a(n) = n*A000567(n) = n^2 * A016777(n-1). - Bruce J. Nicholson, Aug 10 2017
From Elmo R. Oliveira, Aug 06 2025: (Start)
E.g.f.: exp(x)*x*(1 + 7*x + 3*x^2).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

More terms from Zerinvary Lajos, Jun 29 2006

cp's OEIS Frontend

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

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A103532 Number of divisors of 240^n.

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A006597 a(n) = n^2*(5*n-3)/2.

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A050509 House numbers (version 2): a(n) = (n+1)^3 + (n+1)*Sum_{i=0..n} i.

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A100176 Structured octagonal prism numbers.

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

A006597 a(n) = n^2(5n-3)/2.