A016755 -id:A016755 - cp's OEIS frontend

A188050 a(n) = A016755(n) - A001845(n).

0, 20, 100, 280, 600, 1100, 1820, 2800, 4080, 5700, 7700, 10120, 13000, 16380, 20300, 24800, 29920, 35700, 42180, 49400, 57400, 66220, 75900, 86480, 98000, 110500, 124020, 138600, 154280, 171100, 189100, 208320, 228800, 250580, 273700

Offset: 0

Damien Pras, Mar 19 2011

A016755 are odd cubes and A001845 are centered octahedral numbers, so the sequence might be regarded as odd cubes without their octahedral content.

A000330 are square pyramidal numbers.

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. A016755, A001845, A000330.

A016755:=func< n | (2*n+1)^3 >; A001845:=func< n | (2*n+1)*(2*n^2+2*n+3)/3 >; [ A016755(n)-A001845(n): n in [0..40] ]; // Klaus Brockhaus, Mar 20 2011

Maple
```
(10/3)*n*(n+1)*(2*n+1)
```

10n(n+1)(2n+1)/3
LinearRecurrence[{4,-6,4,-1},{0,20,100,280},40] (* Harvey P. Dale, Jul 18 2016 *)

a(n) = (10/3)*n*(n + 1)*(2*n + 1).
a(n) = 20 * A000330(n).
G.f.: 20*x*(1+x)/(1-x)^4. - Klaus Brockhaus, Mar 20 2011

A006331 a(n) = n(n+1)(2*n+1)/3.

Original entry on oeis.org

0, 2, 10, 28, 60, 110, 182, 280, 408, 570, 770, 1012, 1300, 1638, 2030, 2480, 2992, 3570, 4218, 4940, 5740, 6622, 7590, 8648, 9800, 11050, 12402, 13860, 15428, 17110, 18910, 20832, 22880, 25058, 27370, 29820, 32412, 35150, 38038, 41080, 44280

Offset: 0

N. J. A. Sloane

Triangles in rhombic matchstick arrangement of side n.
Maximum accumulated number of electrons at energy level n. - Scott A. Brown, Feb 28 2000
Let M_n denote the n X n matrix M_n(i,j)=i^2+j^2; then the characteristic polynomial of M_n is x^n - a(n)x^(n-1) - .... - Michael Somos, Nov 14 2002
Convolution of odds (A005408) and evens (A005843). - Graeme McRae, Jun 06 2006
a(n) is the number of non-monotonic functions with domain {0,1,2} and codomain {0,1,...,n}. - Dennis P. Walsh, Apr 25 2011
For any odd number 2n+1, find Sum_{aJ. M. Bergot, Jul 16 2011
a(n) gives the number of (n+1) X (n+1) symmetric (0,1)-matrices containing three ones (see [Cameron]). - L. Edson Jeffery, Feb 18 2012
a(n) is the number of 4-tuples (w,x,y,z) with all terms in {0,...,n} and |w - x| < y. - Clark Kimberling, Jun 02 2012
Partial sums of A001105. - Omar E. Pol, Jan 12 2013
Total number of square diagonals (of any size) in an n X n square grid. - Wesley Ivan Hurt, Mar 24 2015
Number of diagonal attacks of two queens on (n+1) X (n+1) chessboard. - Antal Pinter, Sep 20 2015
a(n) is the minimum value obtainable by partitioning either the set {x in the natural numbers | 1 <= x <= 2n} or the set {x in the natural numbers | 0 <= x <= 2n+1} into pairs, taking the product of all such pairs, and taking the sum of all such products. - Thomas Anton, Oct 21 2020
a(n) is the irregularity of the n-th power of a path of length at least 3*n.  (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, Jun 16 2023
a(n) is the maximum possible total number of inversions in all rows and all columns of a Latin square of order n+1. - Ivaylo Kortezov, Jun 28 2025

			For n=2, a(2)=10 since there are 10 non-monotonic functions f from {0,1,2} to {0,1,2}, namely, functions f = <f(1),f(2),f(3)> given by <0,1,0>, <0,2,0>, <0,2,1>, <1,0,1>, <1,0,2>, <1,2,0>, <1,2,1>, <2,0,1>, <2,0,2>, and <2,1,2>. - _Dennis P. Walsh_, Apr 25 2011
Let n=4, 2*n+1 = 9. Since 9 = 1+8 = 3+6 = 5+4 = 7+2, a(4) = 1*8 + 3*6 + 5*4 + 7*2 = 60. - _Vladimir Shevelev_, May 11 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
J. L. Bailey, Jr., A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., Vol. 2 (1931), pp. 355-359.
J. L. Bailey, A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., Vol. 2 (1931), pp. 355-359. [Annotated scanned copy]
Rowan Beckworth, Basic atomic information.
Allan Bickle and Zhongyuan Che, Irregularities of Maximal k-degenerate Graphs, Discrete Applied Math. 331 (2023) 70-87.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
P. Cameron, T. Prellberg and D. Stark, Asymptotics for incidence matrix classes, Electron. J. Combin. 13 (2006), #R85, p. 11.
Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, Semi-simplicial combinatorics of cyclinders and subdivisions, arXiv:2307.13749 [math.CO], 2023. See p. 25.
N. S. S. Gu, H. Prodinger and S. Wagner, Bijections for a class of labeled plane trees, Eur. J. Combinat., Vol. 31 (2010), pp. 720-732, doi|10.1016/j.ejc.2009.10.007, Theorem 2 at n=3.
JBMO 2025, 29th Junior Balkan Mathematical Olympiad, Problem 4, author: Boris Mihov
Germain Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, Vol. 6 (1965), circa p. 82.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Dennis Walsh, Notes on finite monotonic and non-monotonic functions.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

A row of A132339.
Cf. A002378, A048147, A098077, A163102.
Cf. A005900, A084980, A077415, A112524, A188475, A100178, A035597, A096000.
Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).

a006331 n = sum $ zipWith (*) [2*n-1, 2*n-3 .. 1] [2, 4 ..]
-- Reinhard Zumkeller, Feb 11 2012

[n*(n+1)*(2*n+1)/3: n in [0..40]]; // Vincenzo Librandi, Aug 15 2011

A006331 := proc(n)
    n*(n+1)*(2*n+1)/3 ;
end proc:
seq(A006331(n),n=0..80) ; # R. J. Mathar, Sep 27 2013

Table[n(n+1)(2n+1)/3,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,2,10,28},50] (* Harvey P. Dale, Apr 12 2013 *)

PARI
```
a(n)=if(n<0,0,n*(n+1)*(2*n+1)/3)
```

G.f.: 2*x*(1 + x)/(1 - x)^4. - Simon Plouffe (in his 1992 dissertation)
a(n) = 2*binomial(n+1,3) + 2*binomial(n+2,3).
a(n) = 2*A000330(n) = A002492(n)/2.
a(n) = Sum_{i=0..n} T(i,n-i), array T as in A048147. - N. J. A. Sloane, Dec 11 1999
From the formula for the sum of squares of positive integers 1^2 + 2^2 + 3^2 + ... + n^2 = n*(n+1)(2*n+1)/6, if we multiply both sides by 2 we get Sum_{k=0..n} 2*k^2 = n*(n+1)*(2*n+1)/3, which is an alternative formula for this sequence. - Mike Warburton, Sep 08 2007
10*a(n) = A016755(n) - A001845(n); since A016755 are odd cubes and A001845 centered octahedral numbers, 10*a(n) are the "odd cubes without their octahedral contents." - Damien Pras, Mar 19 2011
a(n) = sum(a*b), where the summing is over all unordered partitions 2*n+1=a+b. - Vladimir Shevelev, May 11 2012
a(n) = binomial(2*n+2, 3)/2. - Ronan Flatley, Dec 13 2012
a(n) = A000292(n) + A002411(n). - Omar E. Pol, Jan 11 2013
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3, with a(0)=0, a(1)=2, a(2)=10, a(3)=28. - Harvey P. Dale, Apr 12 2013
a(n) = A208532(n+1,2). - Philippe Deléham, Dec 05 2013
Sum_{n>0} 1/a(n) = 9 - 12*log(2). - Enrique Pérez Herrero, Dec 03 2014
a(n) = A000292(n-1) + (n+1)*A000217(n). - J. M. Bergot, Sep 02 2015
a(n) = 2*(A000332(n+3) - A000332(n+1)). - Antal Pinter, Sep 20 2015
From Bruno Berselli, May 17 2018: (Start)
a(n) = n*A002378(n) - Sum_{k=0..n-1} A002378(k) for n>0, a(0)=0. Also:
A163102(n) = n*a(n) - Sum_{k=0..n-1} a(k) for n>0, A163102(0)=0. (End)
a(n) = A005900(n) - A000290(n) = A096000(n) - A000578(n+1) = A000578(n+1) - A084980(n+1) = A000578(n+1) - A077415(n)-1 = A112524(n) + 1 = A188475(n) - 1 = A061317(n) - A100178(n) = A035597(n+1) - A006331(n+1). - Bruce J. Nicholson, Jun 24 2018
E.g.f.: (1/3)*exp(x)*x*(6 + 9*x + 2*x^2). - Stefano Spezia, Jan 05 2020
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi - 9. - Amiram Eldar, Jan 04 2022

A233091 Decimal expansion of Sum_{i>=0} 1/(2*i+1)^3.

Original entry on oeis.org

1, 0, 5, 1, 7, 9, 9, 7, 9, 0, 2, 6, 4, 6, 4, 4, 9, 9, 9, 7, 2, 4, 7, 7, 0, 8, 9, 1, 3, 2, 2, 5, 1, 8, 7, 4, 1, 9, 1, 9, 3, 6, 3, 0, 0, 5, 7, 9, 7, 9, 3, 6, 5, 2, 1, 5, 6, 8, 2, 3, 7, 6, 1, 0, 9, 2, 4, 1, 0, 8, 4, 3, 0, 0, 6, 3, 0, 2, 3, 9, 5, 3, 9, 1, 3, 1

Offset: 1

Bruno Berselli, Dec 04 2013

This constant is irrational. - Charles R Greathouse IV, Feb 03 2025

			1.0517997902646449997247708913225187419193630057979365215682376109241...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 42.

J. M. Borwein, I.J. Zucker, and J. Boersma, The evaluation of character Euler double sums, The Ramanujan Journal, April 2008, Volume 15, Issue 3, pp 377-405, see p. 17 c(3).
R. J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, Table 22.

Cf. A002117: zeta(3); A197070: 3*zeta(3)/4; A233090: 5*zeta(3)/8.
Cf. A153071: sum( i >= 0, (-1)^i/(2*i+1)^3 ).
Cf. A251809: sum( i >= 0, (-1)^floor(i/2)/(2*i+1)^3 ).
Cf. A016755.

Mathematica
```
RealDigits[7 Zeta[3]/8, 10, 90][[1]]
```

7*zeta(3)/8 \\ Stefano Spezia, Oct 31 2024

Equals 7*zeta(3)/8.
Also equals -(1/16)*PolyGamma(2, 1/2). - Jean-François Alcover, Dec 18 2013
Equals Integral_{x=0..Pi/2} x * log(tan(x)) dx. - Amiram Eldar, Jun 29 2020
Equals Integral_{x=0..1} arcsin(x)*arccos(x)/x dx. - Amiram Eldar, Aug 03 2020

A002593 a(n) = n^2(2n^2 - 1); also Sum_{k=0..n-1} (2k+1)^3.

Original entry on oeis.org

0, 1, 28, 153, 496, 1225, 2556, 4753, 8128, 13041, 19900, 29161, 41328, 56953, 76636, 101025, 130816, 166753, 209628, 260281, 319600, 388521, 468028, 559153, 662976, 780625, 913276, 1062153, 1228528, 1413721, 1619100, 1846081

Offset: 0

N. J. A. Sloane

The m-th term, for m = A065549(n), is perfect (A000396). - Lekraj Beedassy, Jun 04 2002
Partial sums of A016755. - Lekraj Beedassy, Jan 06 2004
Also, the k-th triangular number, where k = 2n^2 - 1 = A056220(n), i.e., a(n) = A000217(A056220(n)). - Lekraj Beedassy, Jun 11 2004
Also, the j-th hexagonal number, where j = n^2 = A000290(n), i.e., a(n) = A000384(A000290(n)) and a(n) = A056220(n) * A000290(n) or j * k. This sequence is a subsequence of the hexagonal number sequence and retains the aspect intrinsic to the hexagonal number sequence that each number in this sequence can be found by multiplying its triangular number by its hexagonal number. - Bruce J. Nicholson, Aug 22 2017
Odd numbers and their squares both having the form 2x-+1, we may write (2r+1)^3 = (2r+1)*(2s-1), where s = centered squares = (r+1)^2 + r^2. Since 2r+1 = (r+1)^2 - r^2, it follows immediately from summing telescopingly over n-1, the product 2*{(r+1)^4 - r^4} - {(r+1)^2 - r^2}, that Sum_{r=0..n-1} (2r+1)^3 = 2*n^4 - n^2 = n^2*(2n^2 - 1). - Lekraj Beedassy, Jun 16 2004
a(n) is also the starting term in the sum of a number M(n) of consecutive cubed integers equaling a squared integer (A253724) for M(n) equal to twice a squared integer (A001105). Numbers a(n) such that a^3 + (a+1)^3 + ... + (a+M-1)^3 = c^2 has nontrivial solutions over the integers for M equal to twice a squared integer (A001105). If M is twice a squared integer, there always exists at least one nontrivial solution for the sum of M consecutive cubed integers starting from a^3 and equaling a squared integer c^2. For n >= 1, M(n) = 2n^2 (A001105), a(n) = M(M-1)/2 = n^2(2n^2 - 1), and c(n) = sqrt(M/2) (M(M^2-1)/2) = n^3(4n^4 - 1). The trivial solutions with M < 1 and a < 2 are not considered. - Vladimir Pletser, Jan 10 2015
Binomial transform of the sequence with offset 1 is (1, 27, 98, 120, 48, 0, 0, 0, ...). - Gary W. Adamson, Jul 23 2015

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 169, #31.
F. E. Croxton and D. J. Cowden, Applied General Statistics. 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1955, p. 742.
L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 7.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 47.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
F. E. Croxton and D. J. Cowden, Applied General Statistics, 2nd Ed., Prentice-Hall, Englewood Cliffs, NJ, 1955. [Annotated scans of just pages 742-743]
Neslihan Kilar, Abdelmejid Bayad, and Yilmaz Simsek, Finite sums involving trigonometric functions and special polynomials: analysis of generating functions and p-adic integrals, Appl. Anal. Disc. Math., hal-04535748, 2024. See p. 22.
Vladimir Pletser, File Triplets (M,a,c) for M=2n^2
Vladimir Pletser, General solutions of sums of consecutive cubed integers equal to squared integers, arXiv:1501.06098 [math.NT], 2015.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
R. J. Stroeker, On the sum of consecutive cubes being a perfect square, Compositio Mathematica, 97 no. 1-2 (1995), pp. 295-307.
G. Xiao, Sigma Server, Operate on "(2*n-1)^3".
M. J. Zerger, Proof without words: The sum of consecutive odd cubes is a triangular number, Math. Mag., 68 (1995), 371.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000290, A000384, A000447, A000583, A002309, A253724, A253725, A260810.

[n^2*(2*n^2 - 1): n in [0..40]]; // Vincenzo Librandi, Sep 07 2011

A002593:=-z*(z+1)*(z**2+22*z+1)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation
a:= n-> n^2*(2*n^2-1): seq(a(n), n=0..50);  # Vladimir Pletser, Jan 10 2015

CoefficientList[Series[(-x^4-23x^3-23x^2-x)/(x-1)^5,{x,0, 80}],x]  (* or *)
Table[ n^2 (2n^2-1),{n,0,80}]  (* Harvey P. Dale, Mar 28 2011 *)
Join[{0},Accumulate[Range[1,91,2]^3]] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,28,153,496},40] (* Harvey P. Dale, Mar 22 2017 *)

a(n) = n^2*(2*n^2 - 1) \\ Charles R Greathouse IV, Feb 07 2017

a(n) = A000217(A056220(n)). - Lekraj Beedassy, Jun 11 2004
G.f.: (-x^4 - 23*x^3 - 23*x^2 - x)/(x - 1)^5. - Harvey P. Dale, Mar 28 2011
a(n) = n^2*(2n^2 - 1). - Vladimir Pletser, Jan 10 2015
E.g.f.: exp(x)*x*(1 + 13*x + 24*x^2/2! + 12*x^3/3!). - Wolfdieter Lang, Mar 11 2017
a(n) = A000384(A000290(n)) = A056220(n) * A000290(n). - Bruce J. Nicholson, Aug 22 2017
From Amiram Eldar, Aug 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 1 - Pi^2/6 - cot(Pi/sqrt(2))*Pi/sqrt(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = cosec(Pi/sqrt(2))*Pi/sqrt(2) - Pi^2/12 - 1. (End)

A016947 a(n) = (6*n + 3)^3.

Original entry on oeis.org

27, 729, 3375, 9261, 19683, 35937, 59319, 91125, 132651, 185193, 250047, 328509, 421875, 531441, 658503, 804357, 970299, 1157625, 1367631, 1601613, 1860867, 2146689, 2460375, 2803221, 3176523, 3581577, 4019679, 4492125, 5000211, 5545233, 6128487, 6751269

Offset: 0

N. J. A. Sloane

			a(0) = (6*0 + 3)^3 = 3^3 = 27.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A002117, A016755, A016911, A016923, A016935, A016945, A016946, A016959, A016971.

[(6*n+3)^3: n in [0..50]]; // Vincenzo Librandi, May 05 2011

Table[(6*n + 3)^3, {n, 0, 25}] (* Amiram Eldar, Oct 02 2020 *)
LinearRecurrence[{4,-6,4,-1},{27,729,3375,9261},40] (* Harvey P. Dale, Jul 02 2025 *)

Sum_{n>=0} 1/a(n) = 7*zeta(3)/216. - Amiram Eldar, Oct 02 2020
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Oct 02 2020
G.f.: 27*(1+x)*(1+22*x+x^2)/(-1+x)^4. - Wesley Ivan Hurt, Oct 02 2020
From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^3.
a(n) = 3^3*A016755(n).
Sum_{n>=0} (-1)^n/a(n) = Pi^3/864. (End)

A016743 Even cubes: a(n) = (2*n)^3.

Original entry on oeis.org

0, 8, 64, 216, 512, 1000, 1728, 2744, 4096, 5832, 8000, 10648, 13824, 17576, 21952, 27000, 32768, 39304, 46656, 54872, 64000, 74088, 85184, 97336, 110592, 125000, 140608, 157464, 175616, 195112, 216000, 238328, 262144, 287496, 314432

Offset: 0

N. J. A. Sloane

a(n) is also the number of non-degenerate triangles that can be drawn with vertices on a cross with n points on each branch. - James P. B. Hall, Nov 22 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Hilko Koning, 216 neodymium magnets for n=3.
Ana Rechtman, Mars 2022, 1er défi, Images des Mathématiques, CNRS, 2022 (in French).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Even bisection of A000578, cf. A016755.

Cf. A016803 (even bisection), A016827 (odd bisection), A033581, A276712.

[(2*n)^3: n in [0..50]]; // Vincenzo Librandi, Sep 05 2011

A016743:=n->(2*n)^3: seq(A016743(n), n=0..50); # Wesley Ivan Hurt, Sep 15 2018

Range[0, 78, 2]^3 (* Alonso del Arte, Apr 06 2013 *)

a(n) = 8*n^3; \\ Joerg Arndt, Apr 07 2013

a(n) = (2*n)^3 = 8*n^3.
G.f.: x*(8+32*x+8*x^2)/(1-4*x+6*x^2-4*x^3+x^4). - Colin Barker, Jan 02 2012
E.g.f.: 8*x*(1 +3*x +x^2)*exp(x). - G. C. Greubel, Sep 15 2018
From Amiram Eldar, Oct 10 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(3)/8 (A276712).
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*zeta(3)/32. (End)

A016756 a(n) = (2*n+1)^4.

Original entry on oeis.org

1, 81, 625, 2401, 6561, 14641, 28561, 50625, 83521, 130321, 194481, 279841, 390625, 531441, 707281, 923521, 1185921, 1500625, 1874161, 2313441, 2825761, 3418801, 4100625, 4879681, 5764801, 6765201, 7890481, 9150625, 10556001, 12117361, 13845841, 15752961, 17850625

Offset: 0

N. J. A. Sloane

a(n) is the number of ordered pairs of lattice points (vectors in R^2 with integer coordinates) that are in or on a square centered at the origin with side length 2*n. - Geoffrey Critzer, Apr 20 2013

			a(1) = 81 because there are 9 lattice points in or on the 2 x 2 square centered at the origin, so there are 9*9 =81 ordered pairs. - _Geoffrey Critzer_, Apr 20 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. A016755, A060187, A154537, A300707.

[(2*n+1)^4: n in [0..40]]; // Vincenzo Librandi, Sep 07 2011

Table[(2n+1)^4,{n,0,25}]  (* Geoffrey Critzer, Apr 20 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{1,81,625,2401,6561},30] (* Harvey P. Dale, Mar 24 2020 *)

vector(40, n, n--; (2*n+1)^4) \\ G. C. Greubel, Sep 15 2018

From Wolfdieter Lang, Mar 12 2017: (Start)
G.f.: (1+76*x+230*x^2+76*x^3+x^4)/(1-x)^5; see row n=5 of A060187.
E.g.f.: (1 + 80*x + 232*x^2 + 128*x^3 + 16*x^4)*exp(x); see row n=4 of A154537. (End)
Sum_{n>=0} 1/a(n) = Pi^4/96 (A300707). - Amiram Eldar, Oct 10 2020
From Amiram Eldar, Jan 28 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = (cos(Pi/sqrt(2)) + cosh(Pi/sqrt(2)))/2.
Product_{n>=1} (1 - 1/a(n)) = Pi*cosh(Pi/2)/8. (End)

A343640 Coordinate triples (x(n), y(n), z(n); n >= 0) of the 3D square spiral filling space with shells of increasing radius for the sup-norm, in turn filled by squares extending from one pole to the opposite one.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, -1, 1, 1, -1, 0, 1, -1, -1, 1, 0, -1, 1, 1, -1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, -1, 1, 0, -1, 0, 0, -1, -1, 0, 0, -1, 0, 1, -1, 0, 1, 0, -1, 1, 1, -1, 0, 1, -1, -1, 1, -1, -1, 0, -1, -1, -1, -1, 0, -1, -1, 1, -1, -1, 0, 0, -1, 0, 0, -2, 1, 0, -2, 1, 1, -2, 0, 1, -2, -1, 1, -2, -1, 0, -2, -1, -1, -2, 0, -1, -2, 1, -1, -2

Offset: 0

M. F. Hasler, Apr 28 2021

This is a 3D generalization of the 2D square spiral and could be used to produce a 3D variant of Ulam's prime spiral.
See A343630 for an analog using the Euclidean or 2-norm instead of the sup- or oo-norm used here, so points are partitioned in spheres and circles instead of squares and cubes here.
The integer lattice points, Z^3, are listed in order of increasing sup norm R = max(|x|, |y|, |z|). Each "sphere" or shell of given radius R is filled starting at the North or South pole using concentric squares on the top and bottom face and squares of fixed size (2R+1) X (2R+1) at intermediate z-coordinates. Each square (circle for the sup-norm) is filled in the sense of increasing longitude, where the positive x axis corresponds to longitude 0, i.e., the points (r,0,z), (0,r,z), (-r,0,z) and (0,-r,z) are visited in this order. The z-values are alternatively increasing and decreasing (so over a period of two shells they follow the same rectangle-wave shape as the x-values do over the period of each square).
The sequence can be seen as a table with row length of 3, where each row corresponds to the (x,y,z)-coordinates of one point (then the three columns are A343641, A343642 and A343643), or as a table with row lengths 3*A010014, where A010014(r) is the number of points with sup-norm r.
There are (2n+1)^3 integer lattice points with sup norm <= n. Therefore, the point number n (where 0 is the origin) is in the shell r = round(n^(1/3)/2) = floor(...+1/2). Within shell r, which starts with the point number (2r-1)^3 (except for r=0), the first and last (2r+1)^2 points are on square spirals on the top and bottom faces, and the other points are on 2r-1 squares forming "belts" of 8r points each, on the side faces of the cube.

			Shell r = 0 is the origin, {(0,0,0)}.
Shell r = 1 contains the 3*3 + 4*2 + 3*3 = 26 points with oo-norm 1, i.e., all points with coordinates within {-1, 0, 1} except for the origin. They are listed in a square spiral starting at the North Pole: (0,0,1), (1,0,1), (1,1,1), (0,1,1), (-1,1,1), (-1,0,1), (-1,-1,1), (0,-1,1), (1,-1,1); then on the equator:  (1,0,0), (1,1,0), (0,1,0), (-1,1,0), (-1,0,0), (-1,-1,0), (0,-1,0), (1,-1,0), and then on the South face using an inward spiral: (1,0,-1), (1,1,-1), (0,1,-1), (-1,1,-1), (-1,0,-1), (-1,-1,-1), (0,-1,-1), (1,-1,-1), (0,0,-1).
Since there are no empty shells, the z-coordinate is always increasing for even r and decreasing for odd r.

Hugo Pfoertner, Visualization of shells 0, 1, 2.
Hugo Pfoertner, List of terms up to r = 10 (n = 9260), [n, x, y, z].

Cf. A343641, A343642, A343643 (list of x, y resp. z-coordinates only).
Cf. A343631, A343632, A343633 (variant using the Euclidean norm => circle shaped spirals), A342561, A343632, A342563 (another variant).
Cf. A010014 (number of points on a shell with given radius), A016755.
Cf. A174344, A268038, A274923 (2-dimensional square spiral).

A343640_row(n)={local(L=List(), a(r, z, d=I)= if(r, for(i=1,8*r, listput(L,[real(r),imag(r),z]); r+=d; abs(real(r))==abs(imag(r)) && d*=I), listput(L,[0,0,z])), s=(-1)^n /* flip South <-> North for odd n */); /* main prog: (1) square spiral on South face from center to board */ for(d=!n,n, a(d,-s*n)); /* (2) "equatorial(?) bands" from South to North */ for(z=1-n,n-1, a(n,s*z)); /* (3) square spiral on North face ending in pole */ for(d=0,n, a(n-d,s*n)); Vec(L)} \\ row n of the table = list of points (x,y,z) in the shell n, i.e., with sup norm n. [Missing "s*" in a(n,s*z) added on May 27 2021]
A343640_vec=concat([A343640_row(r) | r<-[0..2]]) \\ From r=0 up to n there are (2n+1)^3 points with 3 coordinates each!

A046142 Haüy rhombic dodecahedral numbers.

Original entry on oeis.org

1, 33, 185, 553, 1233, 2321, 3913, 6105, 8993, 12673, 17241, 22793, 29425, 37233, 46313, 56761, 68673, 82145, 97273, 114153, 132881, 153553, 176265, 201113, 228193, 257601, 289433, 323785, 360753, 400433, 442921, 488313, 536705, 588193, 642873, 700841

Offset: 1

Eric W. Weisstein

The Haüy rhombic dodecahedral formula is remarkably similar to that of A254473, the 24-hedral numbers: a(n) = (2*n+1)*(8*n^2+14*n+7). Compare with (2*n-1)*(8*n^2-14*n+7); the differences are simple: (1) the first factor of the dodecahedral formula has "+1" and the 24-hedral formula has "-1"; (2) the second factor of the former has "-14n" and the latter has "+14n". Note that the rhombic dodecahedron has 24 edges. The difference between these sequences is diff(n) = 72*n^2 + 14. - Peter M. Chema, Jan 09 2016

Named after the French priest and mineralogist René Just Haüy (1743-1822). - Amiram Eldar, Jun 22 2021

H. Steinhaus, Mathematical Snapshots, 3rd ed. New York: Dover, pp. 185-186, 1999.

G. C. Greubel, Table of n, a(n) for n = 1..5000
R.-J. Haüy, Essai d'une théorie sur la structure des crystaux appliquée à plusieurs genres de substances crystallisées, 1784.
Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000 which lists Haüy rhombic dodecahedral numbers as "RhoDod(n)."
Eric Weisstein's World of Mathematics, Haüy Construction.
Eric Weisstein's World of Mathematics, Rhombic Dodecahedral Number.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000447, A001845, A016755, A069072, A254473.

[(2*n-1)*(8*n^2-14*n+7): n in [1..40]]; // Vincenzo Librandi, Mar 29 2015

A046142:=n->(2*n-1)*(8*n^2-14*n+7): seq(A046142(n), n=1..50); # Wesley Ivan Hurt, Mar 02 2016

Table[(2 n - 1) (8 n^2 - 14 n + 7), {n, 40}] (* Vincenzo Librandi, Mar 29 2015 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 33, 185, 553}, 20] (* Eric W. Weisstein, Sep 27 2017 *)
CoefficientList[Series[(1 + 29 x + 59 x^2 + 7 x^3)/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 27 2017 *)

Vec(x*(7*x^3+59*x^2+29*x+1)/(x-1)^4 + O(x^50)) \\ Michel Marcus, Mar 24 2015

a(n) = (2*n - 1)*(8*n^2 - 14*n + 7).
G.f.: x*(7*x^3 +59*x^2 +29*x +1)/(1-x)^4. - Colin Barker, Oct 26 2012
a(n) = A016755(n) + A069072(n-1). - Luciano Ancora, Mar 23 2015
a(n) = A016755(n) + 6*A000447(n-1). - Luciano Ancora, Mar 23 2015
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4. - Wesley Ivan Hurt, Mar 02 2016
E.g.f.: (-7 +8*x +12*x^2 +16*x^3)*exp(x) + 7. - G. C. Greubel, Nov 04 2017

A309337 a(n) = n^3 if n odd, 3*n^3/4 if n even.

Original entry on oeis.org

0, 1, 6, 27, 48, 125, 162, 343, 384, 729, 750, 1331, 1296, 2197, 2058, 3375, 3072, 4913, 4374, 6859, 6000, 9261, 7986, 12167, 10368, 15625, 13182, 19683, 16464, 24389, 20250, 29791, 24576, 35937, 29478, 42875, 34992, 50653, 41154, 59319, 48000, 68921, 55566, 79507, 63888, 91125

Offset: 0

Ilya Gutkovskiy, Jul 24 2019

Moebius transform of A078307.

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

Cf. A000578, A016755, A059376, A078307, A129194, A193356, A309338.

a[n_] := If[OddQ[n], n^3, 3 n^3/4]; Table[a[n], {n, 0, 45}]
nmax = 45; CoefficientList[Series[x (1 + 6 x + 23 x^2 + 24 x^3 + 23 x^4 + 6 x^5 + x^6)/(1 - x^2)^4, {x, 0, nmax}], x]
LinearRecurrence[{0, 4, 0, -6, 0, 4, 0, -1}, {0, 1, 6, 27, 48, 125, 162, 343}, 46]
Table[n^3 (7 - (-1)^n)/8, {n, 0, 45}]

G.f.: x * (1 + 6*x + 23*x^2 + 24*x^3 + 23*x^4 + 6*x^5 + x^6)/(1 - x^2)^4.
G.f.: Sum_{k>=1} J_3(k) * x^k/(1 + x^k), where J_3() is the Jordan function (A059376).
Dirichlet g.f.: zeta(s-3) * (1 - 2^(1-s)).
a(n) = n^3 * (7 - (-1)^n)/8.
a(n) = Sum_{d|n} (-1)^(n/d + 1) * J_3(d).
Sum_{n>=1} 1/a(n) = 25*zeta(3)/24 = 1.252142607457910713958...
Multiplicative with a(2^e) = 3*2^(3*e-2), and a(p^e) = p^(3*e) for odd primes p. - Amiram Eldar, Oct 26 2020
a(n) = Sum_{1 <= i, j, k <= n} (-1)^(1 + gcd(i,j,k,n)) = Sum_{d | n} (-1)^(d+1) * J_3(n/d). Cf. A129194. - Peter Bala, Jan 16 2024

cp's OEIS Frontend

A188050 a(n) = A016755(n) - A001845(n).

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A006331 a(n) = n*(n+1)*(2*n+1)/3.

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A233091 Decimal expansion of Sum_{i>=0} 1/(2*i+1)^3.

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A002593 a(n) = n^2*(2*n^2 - 1); also Sum_{k=0..n-1} (2k+1)^3.

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A016947 a(n) = (6*n + 3)^3.

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A016743 Even cubes: a(n) = (2*n)^3.

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A006331 a(n) = n(n+1)(2*n+1)/3.

A002593 a(n) = n^2(2n^2 - 1); also Sum_{k=0..n-1} (2k+1)^3.