A155977 -id:A155977 - cp's OEIS frontend

A034262 a(n) = n^3 + n.

0, 2, 10, 30, 68, 130, 222, 350, 520, 738, 1010, 1342, 1740, 2210, 2758, 3390, 4112, 4930, 5850, 6878, 8020, 9282, 10670, 12190, 13848, 15650, 17602, 19710, 21980, 24418, 27030, 29822, 32800, 35970, 39338, 42910, 46692, 50690, 54910, 59358, 64040, 68962, 74130

Offset: 0

Stuart M. Ellerstein (ellerstein(AT)aol.com), Apr 21 2000

k such that x^3 + x + k factors over the integers. - James R. Buddenhagen, Apr 19 2005
If a(n)=X [A155977], Y=b(n) [A071253], Z=c(n) [A034262], then X^2+Y^2 = n*Z^3; e.g., if n=3, a(3)=270, b(3)=90, c(3)=30, then 270^2+90^2=3*30^3. - Vincenzo Librandi, Nov 24 2010
From Bruno Berselli, Sep 06 2018: (Start)
After 0, sum of next n even numbers:
... 2,                           2
... 4,  6,                      10
... 8, 10, 12,                  30
.. 14, 16, 18, 20,              68
.. 22, 24, 26, 28, 30,         130
.. 32, 34, 36, 38, 40, 42,     222 etc. (End)
Sequence occurs in the binomial identity Sum_{k = 0..n} a(k)* binomial(n,k)/binomial(n+k,k) = n*(n + 1)/2. Cf. A092181 and A155977. - Peter Bala, Feb 12 2019
For n >= 2, a(n) is the sum of the numbers in the 1st and last columns of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 17 2021

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

Cf. A006003.
Cf. A000290, A000578, A001477, A002522.
Cf. A071253, A092181, A155977.
Cf. A001621.
Cf. A001620, A248177.

a034262 n = a000578 n + n  -- Reinhard Zumkeller, Sep 26 2014

Table[n^3 + n, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2010 *)

{a(n) = n^3 + n}; /* Michael Somos, Jul 11 2017 */

[n**3+n for n in (0..38)] # Stefano Spezia, Oct 08 2022

a(n) = 2*A006003(n).
a(n) = A002522(n)*A001477(n). - Zerinvary Lajos, Apr 20 2008
For n>1, a(n) = floor(n^5/(n^2-1)). - Gary Detlefs, Feb 10 2010
Sum_{n>=1} 1/a(n) = 0.6718659855... = gamma + Re psi(1+i) = A001620+A248177. [Borwein et al., J. Math. Anal. Appl. 316 (2006) 328]. - R. J. Mathar, Jul 17 2012
a(n) = -a(-n) for all n in Z. - Michael Somos, Jul 11 2017
G.f.: 2*x*(x^2+x+1)/(x-1)^4. - Alois P. Heinz, Oct 08 2022
E.g.f.: x*(2 + 3*x + x^2)*exp(x). - Stefano Spezia, Jun 20 2024

A071253 a(n) = n^2*(n^2+1).

Original entry on oeis.org

0, 2, 20, 90, 272, 650, 1332, 2450, 4160, 6642, 10100, 14762, 20880, 28730, 38612, 50850, 65792, 83810, 105300, 130682, 160400, 194922, 234740, 280370, 332352, 391250, 457652, 532170, 615440, 708122, 810900, 924482, 1049600, 1187010, 1337492, 1501850, 1680912

Offset: 0

N. J. A. Sloane, Jun 12 2002

The identity (n^5 + n^3)^2 + (n^2*(n^2 + 1))^2 = n*(n^3 + n)^3 can be written as A155977(n)^2 + a(n)^2 = n*A034262(n)^3. - Vincenzo Librandi, Aug 08 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000290, A002522, A002378, A034262, A037270, A069187, A155977.

A071253:= func< n | 2*Binomial(n^2+1,2) >;
[A071253(n): n in [0..40]]; // G. C. Greubel, Sep 12 2024

with(combinat):seq(lcm(fibonacci(3,n),n^2),n=0..35); # Zerinvary Lajos, Apr 20 2008
a:=n->add(n+add(n+add(n, j=1..n-1),j=1..n),j=1..n):seq(a(n), n=0..41); # Zerinvary Lajos, Aug 27 2008

Table[(1/4) Round[N[Sinh[2 ArcSinh[n]]^2, 100]], {n,0,40}] (* Artur Jasinski, Feb 10 2010 *)
Table[n^2*(n^2+1),{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
CoefficientList[Series[2 x (1+x) (1+4 x+x^2)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 29 2014 *)

a(n)=n^2*(n^2+1) \\ Charles R Greathouse IV, Sep 24 2015

def A071253(n): return 2*binomial(n^2+1,2)
[A071253(n) for n in range(41)] # G. C. Greubel, Sep 12 2024

a(n) = A002522(n)*A000290(n). - Zerinvary Lajos, Apr 20 2008
a(n) = (1/4)*sinh(2*arcsinh(n))^2. - Artur Jasinski, Feb 10 2010
G.f.: 2*x*(1+x)*(1+4*x+x^2)/(1-x)^5. - Colin Barker, Jan 08 2012
a(n) = A002378(A000290(n)). - Rick L. Shepherd, Sep 22 2014
Sum_{n>=1} 1/a(n) = 0.5682... = Pi^2/6- (Pi*coth Pi-1)/2 = A013661 - A259171 [J. Math. Anal. Appl. 316 (2006) 328]. - R. J. Mathar, Oct 18 2019
a(n) = 2*A037270(n). - R. J. Mathar, Oct 18 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - 1/2 + Pi*cosech(Pi)/2. - Amiram Eldar, Nov 05 2020
E.g.f.: exp(x)*x*(2 + 8*x + 6*x^2 + x^3). - Stefano Spezia, Oct 08 2022
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Apr 16 2023

A092181 Figurate numbers based on the 24-cell (4-D polytope with Schlaefli symbol {3,4,3}).

Original entry on oeis.org

1, 24, 153, 544, 1425, 3096, 5929, 10368, 16929, 26200, 38841, 55584, 77233, 104664, 138825, 180736, 231489, 292248, 364249, 448800, 547281, 661144, 791913, 941184, 1110625, 1301976, 1517049, 1757728, 2025969, 2323800, 2653321, 3016704

Offset: 1

Michael J. Welch (mjw1(AT)ntlworld.com), Mar 31 2004

This is the 4-dimensional regular convex polytope called the 24-cell, hyperdiamond or icositetrachoron.

			a(3)= 3^2*((3*3^2)-(4*3)+2) = 9*(27-12+2) = 9*17 = 153

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
Eric Weisstein's World of Mathematics, 24-Cell
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1). [_R. J. Mathar_, Jun 21 2010]

Cf. A000332, A000583, A014820, A092182, A092183.

[n^2*((3*n^2)-(4*n)+2): n in [1..40]]; // Vincenzo Librandi, May 22 2011

Table[SeriesCoefficient[x (1 + 19 x + 43 x^2 + 9 x^3)/(1 - x)^5, {x, 0, n}], {n, 32}] (* Michael De Vlieger, Dec 14 2015 *)
LinearRecurrence[{5,-10,10,-5,1},{1,24,153,544,1425},40] (* Harvey P. Dale, May 25 2022 *)

a(n) = n^2*(3*n^2-4*n+2); \\ Michel Marcus, Dec 14 2015

a(n) = n^2*((3*n^2)-(4*n)+2).
a(n) = C(n+3,4) + 19 C(n+2,4) + 43 C(n+1,4) + 9 C(n,4).
a(n) = +5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5). G.f.: x*(1+19*x+43*x^2+9*x^3)/(1-x)^5. [R. J. Mathar, Jun 21 2010]
a(n) = Sum_{k = 1..n} (k^3 + k^7)* binomial(n,k)/binomial(n+k,k). Cf. A034262 and A155977. - Peter Bala, Feb 12 2019

A168178 a(n) = n^3*(n^2 + 1)/2.

Original entry on oeis.org

0, 1, 20, 135, 544, 1625, 3996, 8575, 16640, 29889, 50500, 81191, 125280, 186745, 270284, 381375, 526336, 712385, 947700, 1241479, 1604000, 2046681, 2582140, 3224255, 3988224, 4890625, 5949476, 7184295, 8616160, 10267769, 12163500

Offset: 0

N. J. A. Sloane, Dec 11 2009

Number of unoriented rows of length 5 using up to n colors. For a(0)=0, there are no rows using no colors. For a(1)=1, there is one row using that one color for all positions. For a(2)=20, there are 8 achiral (AAAAA, AABAA, ABABA, ABBBA, BAAAB, BABAB, BBABB, BBBBB) and 12 chiral pairs (AAAAB-BAAAA, AAABA-ABAAA, AAABB-BBAAA, AABAB-BABAA, AABBA-ABBAA, AABBB-BBBAA, ABAAB-BAABA, ABABB-BBABA, ABBAB-BABBA, ABBBB-BBBBA, BAABB-BBAAB, BABBB-BBBAB). - Robert A. Russell, Nov 14 2018

For n > 0, a(2n+1) is the number of non-isomorphic kC_m-snakes, where m = 2n+1 or m = 2n (for n>=2). A kC_n-snake is a connected graph in which the k>=2 blocks are isomorphic to the cycle C_n and the block-cutpoint graph is a path. - Christian Barrientos, May 16 2019

C. Barrientos, Graceful labelings of cyclic snakes, Ars Combin., 60(2001), 85-96.

Vincenzo Librandi, Table of n, a(n) for n = 0..595
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A155977.
Row 5 of A277504.
Cf. A000584 (oriented), A000578 (achiral).

List([0..40], n -> n^3*(1 +n^2)/2); # G. C. Greubel, Nov 15 2018

[n^3*(n^2+1)/2: n in [0..50]]; // Vincenzo Librandi, Apr 25 2011

Table[(n^5+n^3)/2,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1},{0, 1, 20, 135, 544, 1625},40] (* Robert A. Russell, Nov 14 2018 *)

vector(40, n, n--; n^3*(1+n^2)/2) \\ G. C. Greubel, Nov 15 2018

[n^3*(1 + n^2)/2 for n in range(40)] # G. C. Greubel, Nov 15 2018

From Robert A. Russell, Nov 14 2018: (Start)
a(n) = (A000584(n) + A000578(n)) / 2 = (n^5 + n^3) / 2.
G.f.: (Sum_{j=1..5} S2(5,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=1..3} S2(3,j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f.: x*Sum_{k=0..4} A145882(5,k) * x^k / (1-x)^6.
E.g.f.: (Sum_{k=1..5} S2(5,k)*x^k + Sum_{k=1..3} S2(3,k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
For n>5, a(n) = Sum_{j=1..6} -binomial(j-7,j) * a(n-j). (End)
From G. C. Greubel, Nov 15 2018: (Start)
G.f.: x*(1 + 14*x + 30*x^2 + 14*x^3 + x^4)/(1-x)^6.
E.g.f.: x*(2 + 18*x + 26*x^2 + 10*x^3 + x^4)*exp(x)/2. (End)

A133754 a(n) = n^5 - n^3.

Original entry on oeis.org

0, 0, 24, 216, 960, 3000, 7560, 16464, 32256, 58320, 99000, 159720, 247104, 369096, 535080, 756000, 1044480, 1414944, 1883736, 2469240, 3192000, 4074840, 5142984, 6424176, 7948800, 9750000, 11863800, 14329224, 17188416, 20486760, 24273000, 28599360, 33521664

Offset: 0

Rolf Pleisch, Mar 16 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000578, A000584, A155977.

List([0..50], n-> n^5 - n^3); # G. C. Greubel, Sep 02 2019

[n^5-n^3: n in [0..50]]; // Vincenzo Librandi, Feb 20 2012

seq(n^5 - n^3, n=0..50); # G. C. Greubel, Sep 02 2019

Table[n^5-n^3, {n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)

a(n)=n^5-n^3 \\ Charles R Greathouse IV, Feb 20 2012

[n^5 - n^3 for n in (0..50)] # G. C. Greubel, Sep 02 2019

a(n) = 12*n*(2*binomial(n+2,4)- binomial(n+1,3)). - Gary Detlefs, Mar 25 2012
Sum_{n>=2} 1/a(n) = 5/4 - zeta(3). - Daniel Suteu, Feb 06 2017
From G. C. Greubel, Sep 02 2019: (Start)
G.f.: 24*x^2*(1 + 3*x + x^2)/(1-x)^6.
E.g.f.: x^2*(12 + 24*x + 10*x^2 + x^3)*exp(x). (End)
Sum_{n>=2} (-1)^n/a(n) = 3*zeta(3)/4 + 2*log(2) - 9/4. - Amiram Eldar, Jan 09 2021

cp's OEIS Frontend

A034262 a(n) = n^3 + n.

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A071253 a(n) = n^2*(n^2+1).

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A092181 Figurate numbers based on the 24-cell (4-D polytope with Schlaefli symbol {3,4,3}).

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A168178 a(n) = n^3*(n^2 + 1)/2.

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A133754 a(n) = n^5 - n^3.

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