Original entry on oeis.org
1, 63, 305, 847, 1809, 3311, 5473, 8415, 12257, 17119, 23121, 30383, 39025, 49167, 60929, 74431, 89793, 107135, 126577, 148239, 172241, 198703, 227745, 259487, 294049, 331551, 372113, 415855, 462897, 513359, 567361, 625023, 686465, 751807, 821169, 894671
Offset: 0
Original entry on oeis.org
0, 17, 154, 531, 1268, 2485, 4302, 6839, 10216, 14553, 19970, 26587, 34524, 43901, 54838, 67455, 81872, 98209, 116586, 137123, 159940, 185157, 212894, 243271, 276408, 312425, 351442, 393579, 438956, 487693, 539910, 595727, 655264, 718641, 785978, 857395
Offset: 0
-
I:=[0,17,154,531]; [n le 4 select I[n] else 4*Self(n-1) - 6*Self(n-2) +4*Self(n-3) -Self(n-4): n in [1..50]]; // G. C. Greubel, May 02 2018
-
Table[LegendreP[3,2n],{n,0,50}]
LinearRecurrence[{4,-6,4,-1},{0,17,154,531},151] (* Harvey P. Dale, May 07 2011 *)
-
a(n)=pollegendre(3,2*n) \\ Charles R Greathouse IV, Mar 19 2017
A000578
The cubes: a(n) = n^3.
Original entry on oeis.org
0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 59319, 64000, 68921, 74088, 79507
Offset: 0
For k=3, b(3) = 2 b(2) - b(1) = 4-1 = 3, so det(S(4,3,(1,1,-1))) = 3*3^2 = 27.
For n=3, a(3) = 3 + (3*0^2 + 3*0 + 3*1^2 + 3*1 + 3*2^2 + 3*2) = 27. - _Patrick J. McNab_, Mar 28 2016
- Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 191.
- John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 43, 64, 81.
- R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 255; 2nd. ed., p. 269. Worpitzky's identity (6.37).
- Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.6 Figurate Numbers, p. 292.
- T. Aaron Gulliver, "Sequences from cubes of integers", International Mathematical Journal, 4 (2003), no. 5, 439 - 445. See http://www.m-hikari.com/z2003.html for information about this journal. [I expanded the reference to make this easier to find. - N. J. A. Sloane, Feb 18 2019]
- J. Propp and A. Propp-Gubin, "Counting Triangles in Triangles", Pi Mu Epsilon Journal (to appear).
- 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).
- James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 6-7.
- D. Wells, You Are A Mathematician, pp. 238-241, Penguin Books 1995.
- N. J. A. Sloane, Table of n, a(n) for n = 0..10000
- H. Bottomley, Illustration of initial terms
- British National Museum, Tablet 92698
- N. Brothers, S. Evans, L. Taalman, L. Van Wyk, D. Witczak, and C. Yarnall, Spiral knots, Missouri J. of Math. Sci., 22 (2010).
- M. DeLong, M. Russell, and J. Schrock, Colorability and determinants of T(m,n,r,s) twisted torus knots for n equiv. +/-1(mod m), Involve, Vol. 8 (2015), No. 3, 361-384.
- Ralph Greenberg, Math For Poets
- R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
- Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets [Cached version at the Wayback Machine]
- Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75. - fixed by _Felix Fröhlich_, Jun 16 2014
- T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (8).
- Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
- Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
- 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
- James Propp and Adam Propp-Gubin, Counting Triangles in Triangles, arXiv:2409.17117 [math.CO], 25 September 2024.
- Kenneth A. Ross, First Digits of Squares and Cubes, Math. Mag. 85 (2012) 36-42. doi:10.4169/math.mag.85.1.36.
- Eric Weisstein's World of Mathematics, Cubic Number, and Hex Pyramidal Number
- Ronald Yannone, Hilbert Matrix Analyses
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
- Index entries for "core" sequences
- Index entries for sequences related to Benford's law
(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives
A004006,
A006527,
A006003,
A005900,
A004068,
A000578,
A004126,
A000447,
A004188,
A004466,
A004467,
A007588,
A062025,
A063521,
A063522,
A063523.
-
a000578 = (^ 3)
a000578_list = 0 : 1 : 8 : zipWith (+)
(map (+ 6) a000578_list)
(map (* 3) $ tail $ zipWith (-) (tail a000578_list) a000578_list)
-- Reinhard Zumkeller, Sep 05 2015, May 24 2012, Oct 22 2011
-
[ n^3 : n in [0..50] ]; // Wesley Ivan Hurt, Jun 14 2014
-
I:=[0,1,8,27]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi, Jul 05 2014
-
A000578 := n->n^3;
seq(A000578(n), n=0..50);
isA000578 := proc(r)
local p;
if r = 0 or r =1 then
true;
else
for p in ifactors(r)[2] do
if op(2, p) mod 3 <> 0 then
return false;
end if;
end do:
true ;
end if;
end proc: # R. J. Mathar, Oct 08 2013
-
Table[n^3, {n, 0, 30}] (* Stefan Steinerberger, Apr 01 2006 *)
CoefficientList[Series[x (1 + 4 x + x^2)/(1 - x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Jul 05 2014 *)
Accumulate[Table[3n^2+3n+1,{n,0,20}]] (* or *) LinearRecurrence[{4,-6,4,-1},{1,8,27,64},20](* Harvey P. Dale, Aug 18 2018 *)
-
A000578(n):=n^3$
makelist(A000578(n),n,0,30); /* Martin Ettl, Nov 03 2012 */
-
A000578(n)=n^3 \\ M. F. Hasler, Apr 12 2008
-
is(n)=ispower(n,3) \\ Charles R Greathouse IV, Feb 20 2012
-
A000578_list, m = [], [6, -6, 1, 0]
for _ in range(10**2):
A000578_list.append(m[-1])
for i in range(3):
m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015
-
(define (A000578 n) (* n n n)) ;; Antti Karttunen, Oct 06 2017
A006003
a(n) = n*(n^2 + 1)/2.
Original entry on oeis.org
0, 1, 5, 15, 34, 65, 111, 175, 260, 369, 505, 671, 870, 1105, 1379, 1695, 2056, 2465, 2925, 3439, 4010, 4641, 5335, 6095, 6924, 7825, 8801, 9855, 10990, 12209, 13515, 14911, 16400, 17985, 19669, 21455, 23346, 25345, 27455, 29679, 32020, 34481, 37065, 39775
Offset: 0
G.f. = x + 5*x^2 + 15*x^3 + 34*x^4 + 65*x^5 + 111*x^6 + 175*x^7 + 260*x^8 + ...
For a(2)=5, the five tetrahedra have faces AAAA, AAAB, AABB, ABBB, and BBBB with colors A and B. - _Robert A. Russell_, Jan 31 2020
- J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 15, p. 5, Ellipses, Paris 2008.
- F.-J. Papp, Colloquium Talk, Department of Mathematics, University of Michigan-Dearborn, March 6, 2005.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- T. D. Noe, Table of n, a(n) for n = 0..1000
- J. D. Bell, A translation of Leonhard Euler's "De Quadratis Magicis", E795, arXiv:math/0408230 [math.CO], 2004-2005.
- James Grime and Brady Haran, Magic Hexagon, Numberphile video (2014).
- Milan Janjic, Two Enumerative Functions.
- Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - _N. J. A. Sloane_, Feb 13 2013
- Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, Journal of Integer Sequences, 17 (2014), Article 14.3.5. - _Felix Fröhlich_, Oct 11 2016
- S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
- S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
- T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
- Ashish Kumar Pandey and Brajesh Kumar Sharma, A Note On Magic Squares And Magic Constants, Appl. Math. E-Notes (2023) Vol. 23, Art. No. 53, 577-582. See p. 577.
- A. J. Turner and J. F. Miller, Recurrent Cartesian Genetic Programming Applied to Famous Mathematical Sequences, preprint, Proceedings of the Companion Publication of the 2015 Annual Conference on Genetic and Evolutionary Computation.
- Eric Weisstein's World of Mathematics, Magic Constant.
- Wikipedia, Floyd's triangle. - _Paul Muljadi_, Jan 25 2010
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
- Index entries for sequences related to magic squares.
- Index to sequences related to polygonal numbers.
(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives
A004006,
A006527, this sequence,
A005900,
A004068,
A000578,
A004126,
A000447,
A004188,
A004466,
A004467,
A007588,
A062025,
A063521,
A063522,
A063523.
Antidiagonal sums of array in
A000027. Row sums of the triangular view of
A000027.
Other polyhedron colorings:
A337898 (cube faces, octahedron vertices),
A337897 (octahedron faces, cube vertices),
A337962 (dodecahedron faces, icosahedron vertices),
A337960 (icosahedron faces, dodecahedron vertices).
Row 3 of
A325001 (simplex vertices and facets) and
A337886 (simplex faces and peaks).
-
a_n:=List([0..nmax], n->n*(n^2 + 1)/2); # Stefano Spezia, Aug 12 2018
-
a006003 n = n * (n ^ 2 + 1) `div` 2
a006003_list = scanl (+) 0 a005448_list
-- Reinhard Zumkeller, Jun 20 2013
-
% Also works with FreeMat.
for(n=0:nmax); tm=n*(n^2 + 1)/2; fprintf('%d\t%0.f\n', n, tm); end
% Stefano Spezia, Aug 12 2018
-
[n*(n^2 + 1)/2 : n in [0..50]]; // Wesley Ivan Hurt, Sep 11 2015
-
[Binomial(n,3)+Binomial(n-1,3)+Binomial(n-2,3): n in [2..60]]; // Vincenzo Librandi, Sep 12 2015
-
Table[ n(n^2 + 1)/2, {n, 0, 45}]
LinearRecurrence[{4,-6,4,-1}, {0,1,5,15},50] (* Harvey P. Dale, May 16 2012 *)
CoefficientList[Series[x (1 + x + x^2)/(x - 1)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Sep 12 2015 *)
With[{n=50},Total/@TakeList[Range[(n(n^2+1))/2],Range[0,n]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Nov 28 2017 *)
-
a(n):=n*(n^2 + 1)/2$ makelist(a(n), n, 0, nmax); /* Stefano Spezia, Aug 12 2018 */
-
{a(n) = n * (n^2 + 1) / 2}; /* Michael Somos, Dec 24 2011 */
-
concat(0, Vec(x*(1+x+x^2)/(x-1)^4 + O(x^20))) \\ Felix Fröhlich, Oct 11 2016
-
def A006003(n): return n*(n**2+1)>>1 # Chai Wah Wu, Mar 25 2024
Better description from Albert Rich (Albert_Rich(AT)msn.com), Mar 1997
A005900
Octahedral numbers: a(n) = n*(2*n^2 + 1)/3.
Original entry on oeis.org
0, 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894, 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726, 13131, 14644, 16269, 18010, 19871, 21856, 23969, 26214, 28595, 31116, 33781, 36594, 39559, 42680
Offset: 0
G.f. = x + 6*x^2 + 19*x^3 + 44*x^4 + 85*x^5 + 146*x^6 + 231*x^7 + ...
- John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 50.
- H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- T. D. Noe, Table of n, a(n) for n = 0..1000
- X. Acloque, Polynexus Numbers and other mathematical wonders [broken link]
- Karoly Bezdek, Contact numbers for congruent sphere packings, arXiv:1102.1198 [math.MG], 2011.
- Matteo Cavaleri and Alfredo Donno, Some degree and distance-based invariants of wreath products of graphs, arXiv:1805.08989 [math.CO], 2018.
- Y-h. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq (5), m=2.
- Milan Janjic, Two Enumerative Functions
- Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
- Muhammad Fatih Killik and Bünyamin Şahi̇n, Further Results on Level Matrix, Int'l J. Math. Combin. (2024) Vol 4, 68-74. See p. 71.
- Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
- Hankyung Ko, Volodymyr Mazorchuk and Rafael Mrđen, Join operation for the Bruhat order and Verma modules, arXiv:2109.01067 [math.RT], 2021. See Remark 5.10 p. 19.
- A. Lascoux and M.-P. Schützenberger, Treillis et bases des groupes de Coxeter, Electron. J. Combin. 3 (1996), #R27.
- T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
- J. K. Merikoski, R. Kumar and R. A. Rajput, Upper bounds for the largest eigenvalue of a bipartite graph, Electronic Journal of Linear Algebra ISSN 1081-3810, A publication of the International Linear Algebra Society, Volume 26, pp. 168-176, April 2013.
- 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
- B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
- Eric Weisstein's World of Mathematics, Octahedral Number.
- Index entries for two-way infinite sequences
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives
A004006,
A006527,
A006003,
A005900,
A004068,
A000578,
A004126,
A000447,
A004188,
A004466,
A004467,
A007588,
A062025,
A063521,
A063522,
A063523.
-
a005900 n = sum $ zipWith (*) odds $ reverse odds
where odds = take n a005408_list
a005900_list = scanl (+) 0 a001844_list
-- Reinhard Zumkeller, Jun 16 2013, Apr 04 2012
-
[n*(2*n^2+1)/3: n in [0..50]]; // Wesley Ivan Hurt, Sep 11 2015
-
I:=[0,1,6,19]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Sep 12 2015
-
al:=proc(s,n) binomial(n+s-1,s); end; be:=proc(d,n) local r; add( (-1)^r*binomial(d-1,r)*2^(d-1-r)*al(d-r,n), r=0..d-1); end; [seq(be(3,n), n=0..100)];
A005900:=(z+1)**2/(z-1)**4; # Simon Plouffe in his 1992 dissertation
with(combinat): seq(fibonacci(4,2*n)/12, n=0..40); # Zerinvary Lajos, Apr 21 2008
-
Table[(2n^3+n)/3, {n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1}, {0,1,6,19},50] (* Harvey P. Dale, Oct 10 2013 *)
CoefficientList[Series[x (1 + x)^2/(1 - x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Sep 12 2015 *)
-
makelist(n*(2*n^2+1)/3, n, 0, 20); /* Martin Ettl, Jan 07 2013 */
-
{a(n) = n*(2*n^2+1)/3};
-
concat([0],Vec(x*(1 + x)^2/(1 - x)^4 + O(x^50))) \\ Indranil Ghosh, Mar 16 2017
-
def a(n): return n*(2*n*n + 1)//3
print([a(n) for n in range(41)]) # Michael S. Branicky, Sep 03 2021
A000447
a(n) = 1^2 + 3^2 + 5^2 + 7^2 + ... + (2*n-1)^2 = n*(4*n^2 - 1)/3.
Original entry on oeis.org
0, 1, 10, 35, 84, 165, 286, 455, 680, 969, 1330, 1771, 2300, 2925, 3654, 4495, 5456, 6545, 7770, 9139, 10660, 12341, 14190, 16215, 18424, 20825, 23426, 26235, 29260, 32509, 35990, 39711, 43680, 47905, 52394, 57155, 62196, 67525, 73150, 79079, 85320, 91881, 98770, 105995, 113564, 121485
Offset: 0
G.f. = x + 10*x^2 + 35*x^3 + 84*x^4 + 165*x^5 + 286*x^6 + 455*x^7 + 680*x^8 + ...
a(2) = 10 since (-1, -1, -1), (-1, -1, 0), (-1, -1, 1), (-1, 0, 0), (-1, 0, 1), (-1, 1, 1), (0, 0, 0), (0, 0, 1), (0, 1, 1), (1, 1, 1) are the 10 solutions (x, y, z) of -1 <= x <= y <= z <= 1.
a(0) = 0, which corresponds to the empty sum.
- G. Chrystal, Textbook of Algebra, Vol. 1, A. & C. Black, 1886, Chap. XX, Sect. 10, Example 2.
- F. E. Croxton and D. J. Cowden, Applied General Statistics. 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1955, p. 742.
- E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.
- C. V. Durell, Advanced Algebra, Volume 1, G. Bell & Son, 1932, Exercise IIIe, No. 4.
- L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 7.
- J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
- 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).
- T. D. Noe, Table of n, a(n) for n = 0..1000
- 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]
- Valentin Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, Vol. 275 (2004), pp. 17-41. - _Valentin Bakoev_, Mar 03 2009
- 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]
- Milan Janjic, Two Enumerative Functions.
- T. P. Martin, Shells of atoms, Phys. Reports, Vol. 273 (1996), pp. 199-241, eq. (11).
- 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.
- Eric Weisstein's World of Mathematics, Haüy Construction.
- Eric Weisstein's World of Mathematics, Square Pyramid.
- Index entries for two-way infinite sequences.
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives
A004006,
A006527,
A006003,
A005900,
A004068,
A000578,
A004126,
A000447,
A004188,
A004466,
A004467,
A007588,
A062025,
A063521,
A063522,
A063523.
A000447 is related to partitions of 2^n into powers of 2, as it is shown in the formula, example and cross-references of
A002577. -
Valentin Bakoev, Mar 03 2009
-
a000447 n = a000447_list !! n
a000447_list = scanl1 (+) a016754_list
-- Reinhard Zumkeller, Apr 02 2012
-
[n*(4*n^2-1)/3: n in [0..50]]; // Vincenzo Librandi, Jan 12 2016
-
A000447:=z*(1+6*z+z**2)/(z-1)**4; # Simon Plouffe, 1992 dissertation.
A000447:=n->n*(4*n^2 - 1)/3; seq(A000447(n), n=0..50); # Wesley Ivan Hurt, Mar 30 2014
-
Table[n (4 n^2 - 1)/3, {n, 0, 80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 10, 35}, 80] (* Harvey P. Dale, May 25 2012 *)
Join[{0}, Accumulate[Range[1, 81, 2]^2]] (* Harvey P. Dale, Jul 18 2013 *)
CoefficientList[Series[x (1 + 6 x + x^2)/(-1 + x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 27 2017 *)
-
A000447(n):=n*(4*n^2 - 1)/3$ makelist(A000447(n),n,0,20); /* Martin Ettl, Jan 07 2013 */
-
{a(n) = n * (4*n^2 - 1) / 3};
-
concat(0, Vec(x*(1+6*x+x^2)/(1-x)^4 + O(x^100))) \\ Altug Alkan, Jan 11 2016
-
def A000447(n): return n*((n**2<<2)-1)//3 # Chai Wah Wu, Feb 12 2023
Chrystal and Durell references from
R. K. Guy, Apr 02 2004
A004006
a(n) = C(n,1) + C(n,2) + C(n,3), or n*(n^2 + 5)/6.
Original entry on oeis.org
0, 1, 3, 7, 14, 25, 41, 63, 92, 129, 175, 231, 298, 377, 469, 575, 696, 833, 987, 1159, 1350, 1561, 1793, 2047, 2324, 2625, 2951, 3303, 3682, 4089, 4525, 4991, 5488, 6017, 6579, 7175, 7806, 8473, 9177, 9919, 10700, 11521, 12383, 13287, 14234, 15225
Offset: 0
Albert D. Rich (Albert_Rich(AT)msn.com)
G.f. = x + 3*x^2 + 7*x^3 + 14*x^4 + 25*x^5 + 41*x^6 + 63*x^7 + 92*x^8 + ... - _Michael Somos_, Dec 29 2019
- W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Wiley, 1966, see p. 380.
- Vincenzo Librandi, Table of n, a(n) for n = 0..5000
- Michael Boardman, The Egg-Drop Numbers, Mathematics Magazine, 77 (2004), 368-372. [From Parthasarathy Nambi, Sep 30 2009]
- F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
- Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics, 2017, Vol. 41 Issue 6, pp. 849-853.
- N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
- N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
- Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
- T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
- Laurent Saloff-Coste, Random walks on finite groups, in Probability on discrete structures, 263-346, Encyclopaedia Math. Sci., 110, Springer, 2004.
- Bridget Eileen Tenner, Reduced word manipulation: patterns and enumeration, J. Algebr. Comb. 46, No. 1, 189-217 (2017), w in S_n(231) l(w)=3.
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives
A004006,
A006527,
A006003,
A005900,
A004068,
A000578,
A004126,
A000447,
A004188,
A004466,
A004467,
A007588,
A062025,
A063521,
A063522,
A063523.
-
List([0..50], n-> n*(n^2+5)/6); # G. C. Greubel, Aug 27 2019
-
a004006 n = a000292 n + n + 1 -- Reinhard Zumkeller, Mar 31 2012
-
[n*(n^2+5)/6: n in [0..50]]; // Vincenzo Librandi, May 15 2011
-
A004006 := proc(n) n*(n^2+5)/6 ;end proc:
seq(A004006(n),n=0..10) ; # R. J. Mathar, Jun 05 2011
-
Table[Total[Table[Binomial[n,i], {i,3}]], {n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1}, {0,1,3,7}, 50] (* Harvey P. Dale, Aug 21 2011 *)
-
A004006(n):=n*(n^2+5)/6$ makelist(A004006(n),n,0,30); /* Martin Ettl, Jan 08 2013 */
-
{a(n) = n*(n^2 + 5)/6}; /* Michael Somos, May 04 2007 */
-
[n*(n^2+5)/6 for n in (0..50)] # G. C. Greubel, Aug 27 2019
A006527
a(n) = (n^3 + 2*n)/3.
Original entry on oeis.org
0, 1, 4, 11, 24, 45, 76, 119, 176, 249, 340, 451, 584, 741, 924, 1135, 1376, 1649, 1956, 2299, 2680, 3101, 3564, 4071, 4624, 5225, 5876, 6579, 7336, 8149, 9020, 9951, 10944, 12001, 13124, 14315, 15576, 16909, 18316, 19799, 21360, 23001, 24724, 26531, 28424, 30405
Offset: 0
- M. Gardner, New Mathematical Diversions from Scientific American. Simon and Schuster, NY, 1966, p. 246.
- S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.
- 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..5000
- B. Babcock and A. van Tuyl, Revisiting the spreading and covering numbers, arXiv preprint arXiv:1109.5847 [math.AC], 2011.
- Richard A. Brualdi and Geir Dahl, Alternating Sign Matrices and Hypermatrices, and a Generalization of Latin Square, arXiv:1704.07752 [math.CO], 2017. See p. 8.
- Peter Esser?, Guenter Stertenbrink, Triangles with Mac Mahon's pieces, digest of 14 messages in polyforms Yahoo group, Apr 14 - May 2, 2002.
- Th. Grüner, A. Kerber, R. Laue and M. Meringer, Mathematics for Combinatorial Chemistry
- T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
- 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
- polyforms list, Triangles with MacMahon's pieces.
- Taskcentre, McMahon's Triangles 2
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives
A004006,
A006527,
A006003,
A005900,
A004068,
A000578,
A004126,
A000447,
A004188,
A004466,
A004467,
A007588,
A062025,
A063521,
A063522,
A063523.
Row 2 of
A324999 (simplex vertices and facets) and
A327083 (simplex edges and ridges).
-
a006527 n = n * (n ^ 2 + 2) `div` 3 -- Reinhard Zumkeller, Jan 06 2014
-
[(n^3 + 2*n)/3: n in [0..50]]; // Vincenzo Librandi, May 15 2011
-
A006527:=z*(1+z**2)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation
with(combinat):seq(lcm(fibonacci(4,n),fibonacci(2,n))/3,n=0..42); # Zerinvary Lajos, Apr 20 2008
-
Table[ (n^3 + 2*n)/3, {n, 0, 45} ]
LinearRecurrence[{4,-6,4,-1},{0,1,4,11},46] (* or *) CoefficientList[ Series[(x+x^3)/(x-1)^4,{x,0,49}],x] (* Harvey P. Dale, Jun 13 2011 *)
-
a(n)=n*(n^2+2)/3 \\ Charles R Greathouse IV, Jul 25 2011
A007588
Stella octangula numbers: a(n) = n*(2*n^2 - 1).
Original entry on oeis.org
0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, 2651, 3444, 4381, 5474, 6735, 8176, 9809, 11646, 13699, 15980, 18501, 21274, 24311, 27624, 31225, 35126, 39339, 43876, 48749, 53970, 59551, 65504, 71841, 78574, 85715, 93276, 101269, 109706, 118599, 127960
Offset: 0
- J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 51.
- E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.
- W. Ljunggren, Zur Theorie der Gleichung x^2 + 1 = Dy^4, Avh. Norske Vid. Akad. Oslo. I. 1942 (5): 27.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Alexander Adamchuk and Vincenzo Librandi, Table of n, a(n) for n = 0..10000 [Alexander Adamchuk computed terms 0 - 169, Jun 02, 2008; Vincenzo Librandi computed the first 10000 terms, Aug 18 2011]
- A. Bremner, R. Høibakk and D. Lukkassen, Crossed ladders and Euler’s quartic, Annales Mathematicae et Informaticae, 36 (2009) pp. 29-41. See p. 33.
- John Elias, Nesting Cubes of the Surface Points of a Hexagonal Prism
- T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
- Amelia Carolina Sparavigna, Generalized Sum of Stella Octangula Numbers, Politecnico di Torino (Italy, 2021).
- Amelia Carolina Sparavigna, Cardano Formula and Some Figurate Numbers, Politecnico di Torino (Italy, 2021).
- Eric Weisstein's World of Mathematics, Stella Octangula Number
- Wikipedia, Stella octangula number
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Backwards differences give star numbers
A003154:
A003154(n)=a(n)-a(n-1).
1/12*t*(n^3-n)+ n for t = 2, 4, 6, ... gives
A004006,
A006527,
A006003,
A005900,
A004068,
A000578,
A004126,
A000447,
A004188,
A004466,
A004467,
A007588,
A062025,
A063521,
A063522,
A063523.
Cf.
A001653 = Numbers n such that 2*n^2 - 1 is a square.
-
[n*(2*n^2 - 1): n in [0..40]]; // Vincenzo Librandi, Aug 18 2011
-
A007588:=n->n*(2*n^2 - 1); seq(A007588(n), n=0..40); # Wesley Ivan Hurt, Mar 10 2014
-
Table[ n(2n^2-1), {n,0,169} ] (* Alexander Adamchuk, Jun 02 2008 *)
LinearRecurrence[{4,-6,4,-1},{0,1,14,51},50] (* Harvey P. Dale, Sep 16 2011 *)
-
a(n)=n*(2*n^2-1)
-
def A007588(n): return n*(2*n**2-1) # Chai Wah Wu, Feb 18 2022
In the formula given in the 1995 Encyclopedia of Integer Sequences, the second 2 should be an exponent.
A004068
Number of atoms in a decahedron with n shells.
Original entry on oeis.org
0, 1, 7, 23, 54, 105, 181, 287, 428, 609, 835, 1111, 1442, 1833, 2289, 2815, 3416, 4097, 4863, 5719, 6670, 7721, 8877, 10143, 11524, 13025, 14651, 16407, 18298, 20329, 22505, 24831, 27312, 29953, 32759, 35735, 38886, 42217, 45733, 49439
Offset: 0
Albert D. Rich (Albert_Rich(AT)msn.com)
- Vincenzo Librandi, Table of n, a(n) for n = 0..5000
- T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (3).
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
(1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives
A004006,
A006527,
A006003,
A005900,
A004068,
A000578,
A004126,
A000447,
A004188,
A004466,
A004467,
A007588,
A062025,
A063521,
A063522,
A063523.
-
[5*n^3/6+n/6: n in [0..50]]; // Vincenzo Librandi, May 15 2011
-
Table[5*n^3/6+n/6,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
-
A004068(n):=5*n^3/6+n/6$ makelist(A004068(n),n,0,20); /* Martin Ettl, Jan 07 2013 */
-
a(n)=5*n^3/6+n/6 \\ Charles R Greathouse IV, Sep 24 2015
-
def A004068(n): return n*(5*n**2+1)//6 # Chai Wah Wu, Mar 25 2025
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