A000537 -id:A000537 - cp's OEIS frontend

A334997 Array T read by ascending antidiagonals: T(n, k) = Sum_{d divides n} T(d, k-1) with T(n, 0) = 1.

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 3, 4, 1, 1, 2, 6, 4, 5, 1, 1, 4, 3, 10, 5, 6, 1, 1, 2, 9, 4, 15, 6, 7, 1, 1, 4, 3, 16, 5, 21, 7, 8, 1, 1, 3, 10, 4, 25, 6, 28, 8, 9, 1, 1, 4, 6, 20, 5, 36, 7, 36, 9, 10, 1, 1, 2, 9, 10, 35, 6, 49, 8, 45, 10, 11, 1, 1, 6, 3, 16, 15, 56, 7, 64, 9, 55, 11, 12, 1

Offset: 1

Stefano Spezia, May 19 2020

T(n, k) is called the generalized divisor function (see Beekman).

As an array with offset n=1, k=0, T(n,k) is the number of length-k chains of divisors of n. For example, the T(4,3) = 10 chains are: 111, 211, 221, 222, 411, 421, 422, 441, 442, 444. - Gus Wiseman, Aug 04 2022

			From _Gus Wiseman_, Aug 04 2022: (Start)
Array begins:
       k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
  n=1:  1   1   1   1   1   1   1   1   1
  n=2:  1   2   3   4   5   6   7   8   9
  n=3:  1   2   3   4   5   6   7   8   9
  n=4:  1   3   6  10  15  21  28  36  45
  n=5:  1   2   3   4   5   6   7   8   9
  n=6:  1   4   9  16  25  36  49  64  81
  n=7:  1   2   3   4   5   6   7   8   9
  n=8:  1   4  10  20  35  56  84 120 165
The T(4,5) = 21 chains:
  (1,1,1,1,1)  (4,2,1,1,1)  (4,4,2,2,2)
  (2,1,1,1,1)  (4,2,2,1,1)  (4,4,4,1,1)
  (2,2,1,1,1)  (4,2,2,2,1)  (4,4,4,2,1)
  (2,2,2,1,1)  (4,2,2,2,2)  (4,4,4,2,2)
  (2,2,2,2,1)  (4,4,1,1,1)  (4,4,4,4,1)
  (2,2,2,2,2)  (4,4,2,1,1)  (4,4,4,4,2)
  (4,1,1,1,1)  (4,4,2,2,1)  (4,4,4,4,4)
The T(6,3) = 16 chains:
  (1,1,1)  (3,1,1)  (6,2,1)  (6,6,1)
  (2,1,1)  (3,3,1)  (6,2,2)  (6,6,2)
  (2,2,1)  (3,3,3)  (6,3,1)  (6,6,3)
  (2,2,2)  (6,1,1)  (6,3,3)  (6,6,6)
The triangular form T(n-k,k) gives the number of length k chains of divisors of n - k. It begins:
  1
  1  1
  1  2  1
  1  2  3  1
  1  3  3  4  1
  1  2  6  4  5  1
  1  4  3 10  5  6  1
  1  2  9  4 15  6  7  1
  1  4  3 16  5 21  7  8  1
  1  3 10  4 25  6 28  8  9  1
  1  4  6 20  5 36  7 36  9 10  1
  1  2  9 10 35  6 49  8 45 10 11  1
(End)

Richard Beekman, An Introduction to Number-Theoretic Combinatorics, Lulu Press 2017.

Stefano Spezia, First 150 antidiagonals of the array, flattened
Richard Beekman, A General Solution of the Ferris Wheel Problem.

Cf. A000217 (4th row), A000290 (6th row), A000292 (8th row), A000332 (16th row), A000389 (32nd row), A000537 (36th row), A000578 (30th row), A002411 (12th row), A002417 (24th row), A007318, A027800 (48th row), A335078, A335079.
Column k = 2 of the array is A007425.
Column k = 3 of the array is A007426.
Column k = 4 of the array is A061200.
The transpose of the array is A077592.
The subdiagonal n = k + 1 of the array is A163767.
The version counting all multisets of divisors (not just chains) is A343658.
The strict case is A343662 (row sums: A337256).
Diagonal n = k of the array is A343939.
Antidiagonal sums of the array (or row sums of the triangle) are A343940.
A067824(n) counts strict chains of divisors starting with n.
A074206(n) counts strict chains of divisors from n to 1.
A146291 counts divisors by Omega.
A251683(n,k) counts strict length k + 1 chains of divisors from n to 1.
A253249(n) counts nonempty chains of divisors of n.
A334996(n,k) counts strict length k chains of divisors from n to 1.
A337255(n,k) counts strict length k chains of divisors starting with n.
Cf. A000005, A018892, A051026, A062319, A066959, A176029, A327527, A343652, A343656.

T[n_,k_]:=If[n==1,1,Product[Binomial[Extract[Extract[FactorInteger[n],i],2]+k,k],{i,1,Length[FactorInteger[n]]}]]; Table[T[n-k,k],{n,1,13},{k,0,n-1}]//Flatten

T(n, k) = if (k==0, 1, sumdiv(n, d, T(d, k-1)));
matrix(10, 10, n, k, T(n, k-1)) \\ to see the array for n>=1, k >=0; \\ Michel Marcus, May 20 2020

T(n, k) = Sum_{d divides n} T(d, k-1) with T(n, 0) = 1 (see Theorem 3 in Beekman's article).
T(i*j, k) = T(i, k)*T(j, k) if i and j are coprime positive integers (see Lemma 1 in Beekman's article).
T(p^m, k) = binomial(m+k, k) for every prime p (see Lemma 2 in Beekman's article).

Duplicate term removed by Stefano Spezia, Jun 03 2020

A006011 a(n) = n^2*(n^2 - 1)/4.

Original entry on oeis.org

0, 0, 3, 18, 60, 150, 315, 588, 1008, 1620, 2475, 3630, 5148, 7098, 9555, 12600, 16320, 20808, 26163, 32490, 39900, 48510, 58443, 69828, 82800, 97500, 114075, 132678, 153468, 176610, 202275, 230640, 261888, 296208, 333795, 374850, 419580, 468198

Offset: 0

N. J. A. Sloane, Simon Plouffe

Products of two consecutive triangular numbers (A000217).
a(n) is the number of Lyndon words of length 4 on an n-letter alphabet. A Lyndon word is a primitive word that is lexicographically earliest in its cyclic rotation class. For example, a(2)=3 counts 1112, 1122, 1222. - David Callan, Nov 29 2007
For n >= 2 this is the second rightmost column of A163932. - Johannes W. Meijer, Oct 16 2009
Partial sums of A059270. - J. M. Bergot, Jun 27 2013
Using the integers, triangular numbers, and squares plot the points (A001477(n),A001477(n+1)), (A000217(n), A000217(n+1)), and (A000290(n),A000290(n+1)) to create the vertices of a triangle. One-half the area of this triangle = a(n). - J. M. Bergot, Aug 01 2013
a(n) is the Wiener index of the triangular graph T(n+1). - Emeric Deutsch, Aug 26 2013

			From _Bruno Berselli_, Aug 29 2014: (Start)
After the zeros, the sequence is provided by the row sums of the triangle:
   3;
   4, 14;
   5, 16, 39;
   6, 18, 42,  84;
   7, 20, 45,  88, 155;
   8, 22, 48,  92, 160, 258;
   9, 24, 51,  96, 165, 264, 399;
  10, 26, 54, 100, 170, 270, 406, 584;
  11, 28, 57, 104, 175, 276, 413, 592, 819;
  12, 30, 60, 108, 180, 282, 420, 600, 828, 1110; etc.,
where T(r,c) = c*(c^2+r+1), with r = row index, c = column index, r >= c > 0. (End)

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
Miguel Azaola and Francisco Santos, The number of triangulations of the cyclic polytope C(n,n-4), Discrete Comput. Geom., Vol. 27 (2002), pp. 29-48 (see Prop. 4.2(a)).
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber., Vol. 30 (1897), pp. 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber., Vol. 30 (1897), pp. 1917-1926. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, Triangular Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Thrice A002415. Row 4 of A074650.
Cf. A000217, A000290, A000537, A001477, A002415, A008911, A047928, A059270, A083374, A126274, A163932, A228317
A column of A124428.

[n^2*(n^2-1)/4: n in [0..40]]; // Vincenzo Librandi, Sep 14 2011

A006011 := proc(n)
    n^2*(n^2-1)/4 ;
end proc: # R. J. Mathar, Nov 29 2015

Table[n^2 (n^2 - 1)/4, {n, 0, 38}]
Binomial[Range[20]^2, 2]/2 (* Eric W. Weisstein, Sep 08 2017 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 3, 18, 60, 150}, 20] (* Eric W. Weisstein, Sep 08 2017 *)
CoefficientList[Series[-3 x (1 + x)/(-1 + x)^5, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 08 2017 *)
Join[{0},Times@@@Partition[Accumulate[Range[0,40]],2,1]] (* Harvey P. Dale, Aug 08 2025 *)

a(n)=binomial(n^2,2)/2 \\ Charles R Greathouse IV, Jun 27 2013

G.f.: 3*(1 + x) / (1 - x)^5.
a(n) = (n-1)*n/2 * n*(n+1)/2 = A000217(n-1)*A000217(n) = 1/2*(n^2-1)*n^2/2 = 1/2*A000217(n^2-1). - Alexander Adamchuk, Apr 13 2006
a(n) = 3*A002415(n) = A047928(n-1)/4 = A083374(n-1)/2 = A008911(n)*3/2. - Zerinvary Lajos, May 09 2007
a(n) = (A126274(n) - A000537(n+1))/2. - Enrique Pérez Herrero, Mar 11 2013
Ceiling(sqrt(a(n)) + sqrt(a(n-1)))/2 = A000217(n). - Richard R. Forberg, Aug 14 2013
a(n) = Sum_{i=1..n-1} i*(i^2+n) for n > 1 (see Example section). - Bruno Berselli, Aug 29 2014
Sum_{n>=2} 1/a(n) = 7 - 2*Pi^2/3 = 0.42026373260709425411... . - Vaclav Kotesovec, Apr 27 2016
a(n) = A000217(n^2+n) - A000217(n)*A000217(n+1). - Charlie Marion, Feb 15 2020
Sum_{n>=2} (-1)^n/a(n) = Pi^2/3 - 3. - Amiram Eldar, Nov 02 2021
E.g.f.: exp(x)*x^2*(6 + 6*x + x^2)/4. - Stefano Spezia, Mar 12 2024

A006322 4-dimensional analog of centered polygonal numbers.

Original entry on oeis.org

1, 8, 31, 85, 190, 371, 658, 1086, 1695, 2530, 3641, 5083, 6916, 9205, 12020, 15436, 19533, 24396, 30115, 36785, 44506, 53383, 63526, 75050, 88075, 102726, 119133, 137431, 157760, 180265, 205096, 232408, 262361, 295120, 330855, 369741, 411958, 457691, 507130

Offset: 1

Albert Rich (Albert_Rich(AT)msn.com)

Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005
Partial sums give A006414. - L. Edson Jeffery, Dec 13 2011
Also the number of (w,x,y,z) with all terms in {1,...,n} and w<=x>=y<=z, see A211795. - Clark Kimberling, May 19 2012

			An illustration for a(5)=190: 5*(1+2+3+4+5)+4*(2+3+4+5)+3*(3+4+5)+2*(4+5)+1*(5) gives 75+56+36+18+5=190. - _J. M. Bergot_, Feb 13 2018

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 166, Table 10.4/I/4).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Abderrahim Arabi, Hacène Belbachir, and Jean-Philippe Dubernard, Plateau Polycubes and Lateral Area, arXiv:1811.05707 [math.CO], 2018. See Column 2 Table 2 p. 9.
Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]. See p. 31.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000330, A006414, A050446, A050447.

List([1..40], n->5*Binomial(n+2,4) + Binomial(n+1,2)); # Muniru A Asiru, Feb 13 2018

[n*(n+1)*(5*n^2 +5*n +2)/24: n in [1..40]]; // G. C. Greubel, Sep 02 2019

a:=n->5*binomial(n+2,4) + binomial(n+1,2): seq(a(n), n=1..40); # Muniru A Asiru, Feb 13 2018

Table[5*Binomial[n+2, 4] + Binomial[n+1, 2], {n, 40}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
CoefficientList[Series[(1+3x+x^2)/(1-x)^5, {x,0,40}], x] (* Vincenzo Librandi, Jun 09 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{1,8,31,85,190},40] (* Harvey P. Dale, Sep 27 2016 *)

a(n)=n*(5*n^3+10*n^2+7*n+2)/24 \\ Charles R Greathouse IV, Dec 13 2011, corrected by Altug Alkan, Aug 15 2017

[n*(n+1)*(5*n^2 +5*n +2)/24 for n in (1..40)] # G. C. Greubel, Sep 02 2019

a(n) = 5*C(n+2,4) + C(n+1,2) = (C(5*n+4,4) - 1)/5^3 = n*(n+1)*(5*n^2 + 5*n + 2)/24.
a(n) = (((n+1)^5-n^5) - ((n+1)^3-n^3))/24. - Xavier Acloque, Jan 14 2003, corrected by Eric Rowland, Aug 15 2017
Partial sums of A004068. - Xavier Acloque, Jan 15 2003
G.f.: x*(1+3*x+x^2)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) = Sum_{i=1..n} Sum_{j=1..n} i * min(i,j). - Enrique Pérez Herrero, Jan 30 2013
a(n) = A000537(n) - A000332(n+2). - J. M. Bergot, Jun 03 2017
Sum_{n>=1} 1/a(n) = 42 - 4*sqrt(15)*Pi*tanh(sqrt(3/5)*Pi/2). - Amiram Eldar, May 28 2022
From Elmo R. Oliveira, Aug 14 2025: (Start)
E.g.f.: exp(x)*x*(2 + x)*(12 + 30*x + 5*x^2)/24.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5. (End)

A035287 Number of ways to place a non-attacking white and black rook on n X n chessboard.

Original entry on oeis.org

0, 4, 36, 144, 400, 900, 1764, 3136, 5184, 8100, 12100, 17424, 24336, 33124, 44100, 57600, 73984, 93636, 116964, 144400, 176400, 213444, 256036, 304704, 360000, 422500, 492804, 571536, 659344, 756900, 864900, 984064, 1115136, 1258884

Offset: 1

Erich Friedman

a(n) is equal to the number of functions f:{1,2,3,4}->{1,2,...,n} such that for fixed different x_1, x_2 in {1,2,3,4} and fixed y_1, y_2 in {1,2,...,n} we have f(x_1) <> y_1 and f(x_2) <> y_2. - Milan Janjic, Apr 17 2007
The third differences of certain values of the hypergeometric function 3F2 lead to this sequence, i.e., 3F2([1,n+1,n+1], [n+2,n+2], z=1) - 3*3F2([1,n+2,n+2], [n+3,n+3], z=1) + 3*3F2([1,n+3,n+3], [n+4,n+4], z=1) - 3F2([1,n+4,n+4], [n+5,n+5], z=1) = (1/((n+2)*(n+3)))^2 with n = -1, 0, 1, 2, ... See also A162990. - Johannes W. Meijer, Jul 21 2009
a(n) is the denominator (m*n)^2 of the term (1/m^2 - 1/n^2) = (2*n-1)/(m*n)^2, n = m+1, m > 0 in the Rydberg formula, while A005408 is the numerator 2n-1. So the quotient A005408/A035287 simulates the hydrogen spectral series of all hydrogen-like elements. - Freimut Marschner, Aug 10 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Leo Tavares, Illustration: Square of squares
Wolfram Research, Hypergeometric Function 3F2, The Wolfram Functions site. [From _Johannes W. Meijer_, Jul 21 2009]
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002378.

Cf. A000290.

[n^2 * (n-1)^2: n in [1..40]]; // Vincenzo Librandi, May 21 2011

Table[(n - 1)^2 n^2, {n, 30}] (* Alonso del Arte, May 20 2011 *)

for n in range(100):
  print(((n+1)*n)**2) # John H. Chakkour, Dec 14 2019

[n^2*(n-1)^2 for n in range(1, 35)] # Zerinvary Lajos, Dec 03 2009

a(n) = n^2 * (n-1)^2.
a(n) = A002378(n-1)^2. - Zerinvary Lajos, Apr 11 2006
From Stephen Crowley, Jul 19 2009: (Start)
a(n) = n!*(2*n+1) / lim_{x->0} (d^n/dx^n) (polylog(2,x)*(1-1/x));
Sum_{n >= 2} 1/a(n) = 2*zeta(2) - 3 = A145426. (End) [Comment from Jianing Song, Dec 31 2022: Note that polylog(2,x)*(1-1/x) = -1 + Sum_{n>=1} ((2*n+1)/(n^2*(n+1)^2))*x^n, so (d^n/dx^n) (polylog(2,x)*(1-1/x)) = n!*(2*n+1)/(n^2*(n+1)^2) for n >= 1. - Jianing Song, Dec 31 2022]
a(n) = 4*A000537(n-1) = 2*A163102(n-1). - Omar E. Pol, Nov 29 2011
G.f.: 4*x^2*(1+4*x+x^2)/(1-x)^5. - Colin Barker, Apr 04 2012
a(n) = 4*A000217(n-1)^2. - J. M. Bergot, Nov 01 2012
E.g.f.: x^2*(2 + 4*x + x^2)*exp(x). - Ilya Gutkovskiy, May 24 2016
Sum_{n>=2} (-1)^n/a(n) = 3 - 4*log(2). - Amiram Eldar, Jul 02 2020
Product_{n>=2} (1 - 1/a(n)) = -cos(sqrt(5)*Pi/2)*cosh(sqrt(3)*Pi/2)/Pi^2. - Amiram Eldar, Jan 29 2021
(n^2)^2 + (n^2+1)^2 + ... + (n^2 + n)^2 + a(n) = (n^2 + n + 1)^2 + ... + (n^2 + 2*n)^2. - Charlie Marion, Jun 18 2022
a(n) = A000290(n-1) * A000290(n). - Leo Tavares, Dec 03 2022
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n >= 6. - Jianing Song, Dec 30 2022

A062392 a(n) = n^4 - (n-1)^4 + (n-2)^4 - ... 0^4.

Original entry on oeis.org

0, 1, 15, 66, 190, 435, 861, 1540, 2556, 4005, 5995, 8646, 12090, 16471, 21945, 28680, 36856, 46665, 58311, 72010, 87990, 106491, 127765, 152076, 179700, 210925, 246051, 285390, 329266, 378015, 431985, 491536, 557040, 628881, 707455, 793170, 886446, 987715

Offset: 0

Henry Bottomley, Jun 21 2001

Number of edges in the join of two complete graphs of order n^2 and n, K_n^2 * K_n. - Roberto E. Martinez II, Jan 07 2002
The general formula for alternating sums of powers is in terms of the Swiss-Knife polynomials P(n,x) A153641 2^(-n-1)(P(n,1)-(-1)^k P(n,2k+1)). Thus a(k) = |2^(-5)(P(4,1)-(-1)^k P(4,2k+1))|. - Peter Luschny, Jul 12 2009
Define an infinite symmetric array by T(n,m) = n*(n-1) + m for 0 <= m <= n and T(n,m) = T(m,n), n >= 0. Then a(n) is the sum of terms in the top left (n+1) X (n+1) subarray: a(n) = Sum_{r=0..n} Sum_{c=0..n} T(r,c). - J. M. Bergot, Jul 05 2013
a(n) is the sum of all positive numbers less than A002378(n). - J. M. Bergot, Aug 30 2013
Except the first term, these are triangular numbers that remain triangular when divided by their index, e.g., 66 divided by 11 gives 6. - Waldemar Puszkarz, Sep 14 2017
a(n) is the semiperimeter of the unique primitive Pythagorean triple such that (a-b+c)/2 = T(n) = A000217(n). Its long leg and hypotenuse are consecutive natural numbers and the triple is (2*T(n) - 1, 2*T(n)*(T(n) - 1), 2*T(n)*(T(n) - 1) + 1). - Miguel-Ángel Pérez García-Ortega, May 27 2025

			From _Bruno Berselli_, Oct 30 2017: (Start)
After 0:
1   =                 -(1) + (2);
15  =             -(1 + 2) + (3 + 4 + 5 + 2*3);
66  =         -(1 + 2 + 3) + (4 + 5 + 6 + 7 + ... + 11 + 3*4);
190 =     -(1 + 2 + 3 + 4) + (5 + 6 + 7 + 8 + ... + 19 + 4*5);
435 = -(1 + 2 + 3 + 4 + 5) + (6 + 7 + 8 + 9 + ... + 29 + 5*6), etc. (End)

T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000538, A000583. A062393 provides the result for 5th powers, A011934 for cubes, A000217 for squares, A001057 (unsigned) for nonnegative integers, A000035 (offset) for 0th powers.
Cf. A000217, A000384, A007588.
Cf. A236770 (see crossrefs).
Cf. A384329, A384369.

a := n -> (2*n^2+n^3-1)*n/2; # Peter Luschny, Jul 12 2009

Table[n (n + 1) (n^2 + n - 1)/2, {n, 0, 40}] (* Harvey P. Dale, Oct 19 2011 *)

{ a=0; for (n=0, 1000, write("b062392.txt", n, " ", a=n^4 - a) ) } \\ Harry J. Smith, Aug 07 2009

a(n) = n*(n+1)*(n^2 + n - 1)/2 = n^4 - a(n-1) = A000583(n) - a(n-1) = A000217(A028387(n-1)) = A000217(n)*A028387(n-1).
a(n) = Sum_{i=0..n} A007588(i) for n > 0. - Jonathan Vos Post, Mar 15 2006
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4. - Harvey P. Dale, Oct 19 2011
G.f.: x*(x*(x + 10) + 1)/(1 - x)^5. - Harvey P. Dale, Oct 19 2011
a(n) = A000384(A000217(n)). - Bruno Berselli, Jan 31 2014
a(n) = A110450(n) - A002378(n). - Gionata Neri, May 13 2015
Sum_{n>=1} 1/a(n) = tan(sqrt(5)*Pi/2)*2*Pi/sqrt(5). - Amiram Eldar, Jan 22 2024
a(n) = sqrt(144*A288876(n-2) + 72*A006542(n+2) + A000537(n)). - Yasser Arath Chavez Reyes, Jul 22 2024
E.g.f.: exp(x)*x*(2 + 13*x + 8*x^2 + x^3)/2. - Stefano Spezia, Apr 27 2025
a(n) = A000217(n)*(2*A000217(n)-1). - Miguel-Ángel Pérez García-Ortega, May 27 2025

A000541 Sum of 7th powers: 1^7 + 2^7 + ... + n^7.

Original entry on oeis.org

0, 1, 129, 2316, 18700, 96825, 376761, 1200304, 3297456, 8080425, 18080425, 37567596, 73399404, 136147921, 241561425, 412420800, 680856256, 1091194929, 1703414961, 2597286700, 3877286700, 5678375241, 8172733129, 11577558576, 16164030000, 22267545625

Offset: 0

N. J. A. Sloane

a(n) is divisible by A000537(n) if and only n is congruent to 1 mod 3 (see A016777) - Artur Jasinski, Oct 10 2007

This sequence is related to A000540 by a(n) = n*A000540(n) - Sum_{i=0..n-1} A000540(i). - Bruno Berselli, Apr 26 2010

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 815.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
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
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Row 7 of array A103438.

Cf. A000217, A000537, A000539, A069092.

[n^2*(n+1)^2*(3*n^4+6*n^3-n^2-4*n+2)/24: n in [0..30]]; // Vincenzo Librandi, Feb 20 2016

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^7 od: seq(a[n], n=0..25); # Zerinvary Lajos, Feb 22 2008

Table[Sum[k^7, {k, 1, n}], {n, 0, 100}] (* Artur Jasinski, Oct 10 2007 *)
s = 0; lst = {s}; Do[s += n^7; AppendTo[lst, s], {n, 1, 30, 1}]; lst (* Zerinvary Lajos, Jul 12 2009 *)
LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {0, 1, 129, 2316, 18700, 96825, 376761, 1200304, 3297456}, 35] (* Vincenzo Librandi, Feb 20 2016 *)

a(n)=n^2*(n+1)^2*(3*n^4+6*n^3-n^2-4*n+2)/24 \\ Edward Jiang, Sep 10 2014

a(n) = sum(i=1, n, i^7); \\ Michel Marcus, Sep 11 2014

A000541_list, m = [0], [5040, -15120, 16800, -8400, 1806, -126, 1, 0, 0]
for _ in range(10**2):
    for i in range(8):
        m[i+1] += m[i]
    A000541_list.append(m[-1]) # Chai Wah Wu, Nov 05 2014

a(n) = n^2*(n+1)^2*(3*n^4 + 6*n^3 - n^2 - 4*n + 2)/24.
a(n) = sqrt(Sum_{j=1..n} Sum_{i=1..n} (i*j)^7). - Alexander Adamchuk, Oct 26 2004
Jacobi formula: a(n) = 2(A000217(n))^4 - A000539(n). - Artur Jasinski, Oct 10 2007
G.f.: x*(1 + 120*x + 1191*x^2 + 2416*x^3 + 1191*x^4 + 120*x^5 + x^6)/(1-x)^9. - Colin Barker, May 25 2012
a(n) = 8*a(n-1) - 28* a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) + 5040. - Ant King, Sep 24 2013
a(n) = -Sum_{j=1..7} j*Stirling1(n+1,n+1-j)*Stirling2(n+7-j,n). - Mircea Merca, Jan 25 2014
a(n) = 2*A000217(n)^4 - (4/3)*A000217(n)^3 + (1/3)*A000217(n)^2. - Michael Raney, Feb 19 2016
a(n) = 72*A288876(n-2) + 48*A006542(n+2) + A000537(n). - Yasser Arath Chavez Reyes, Apr 27 2024
a(n) = Sum_{i=1..n} J_7(i)*floor(n/i), where J_7 is A069092. - Ridouane Oudra, Jul 17 2025

A006007 4-dimensional analog of centered polygonal numbers: a(n) = n(n+1)*(n^2+n+4)/12.

Original entry on oeis.org

0, 1, 5, 16, 40, 85, 161, 280, 456, 705, 1045, 1496, 2080, 2821, 3745, 4880, 6256, 7905, 9861, 12160, 14840, 17941, 21505, 25576, 30200, 35425, 41301, 47880, 55216, 63365, 72385, 82336, 93280, 105281, 118405, 132720, 148296, 165205, 183521

Offset: 0

N. J. A. Sloane, Simon Plouffe

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
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..710
Per Alexandersson, Sam Hopkins, and Gjergji Zaimi, Restricted Birkhoff polytopes and Ehrhart period collapse, arXiv:2206.02276 [math.CO], 2022.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
D.-N. Verma, Towards Classifying Finite Point-Set Configurations, 1997, Unpublished. [Scanned copy of annotated version of preprint given to me by the author in 1997. - _N. J. A. Sloane_, Oct 04 2021]
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A003215, A000537, A000578, A005898, A027602. - Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Cf. A000217, A061315, A061316, A061318.

[n*(n+1)*(n^2+n+4)/12: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

f[n_]:=n^3;lst={};s=0;Do[s+=(f[n]+f[n+1]+f[n+2]);AppendTo[lst,s/9],{n,0,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 03 2009 *)
Table[2Binomial[n+2,4]+Binomial[n+1,2],{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,5,16,40},40] (* Harvey P. Dale, Sep 30 2011 *)

a(n)=n*(n+1)*(n^2+n+4)/12 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: (1+x^2)/(1-x)^5.
a(n) = 2*binomial(n + 2, 4) + binomial(n + 1, 2).
a(n) = A061316(n)/3 = A061315(n, 3) = sqrt(A061318(n)-A061316(n)).
a(0)=0, a(1)=1, a(2)=5, a(3)=16, a(4)=40, a(n) = 5*a(n-1) -  10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Sep 30 2011
For n>0, a(n) = (A000217(n-1)^2 + A000217(n)^2 + A000217(n+1)^2 - 1)/9. - Richard R. Forberg, Dec 25 2013
Sum_{n>=1} 1/a(n) = 15/4 - tanh(sqrt(15)*Pi/2)*Pi*sqrt(3/5). - Amiram Eldar, Aug 23 2022
E.g.f.: exp(x)*(12 + 48*x + 42*x^2 + 12*x^3 + x^4)/12. - Stefano Spezia, Aug 31 2023

More terms from Henry Bottomley, Apr 24 2001

A101094 a(n) = n(n+1)(n+2)(n+3)(1+3*n+n^2)/120.

Original entry on oeis.org

1, 11, 57, 203, 574, 1386, 2982, 5874, 10791, 18733, 31031, 49413, 76076, 113764, 165852, 236436, 330429, 453663, 612997, 816431, 1073226, 1394030, 1791010, 2277990, 2870595, 3586401, 4445091, 5468617, 6681368, 8110344, 9785336

Offset: 1

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004

Partial sums of A024166. Third partial sums of cubes (A000578).

Antidiagonal sums of the array A213564. - Clark Kimberling, Jun 18 2012

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Dead link]
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000537, A101097, A101102, A101104.

[n*(n+1)*(n+2)*(n+3)*(1+3*n+n^2)/120 : n in [1..35]]; // Vincenzo Librandi, Apr 23 2015

Table[n*(n + 1)*(n + 2)*(n + 3)*(1 + 3*n + n^2)/120, {n, 31}] (* Michael De Vlieger, Apr 20 2015 *)

[n*(n+1)*(n+2)*(n+3)*(1+3*n+n^2)/120 for n in range(1,32)] # Danny Rorabaugh, Apr 20 2015

This sequence could be obtained from the general formula n*(n+1)*(n+2)*(n+3)*...*(n+k)*(n*(n+k)+(k-1)*k/6)/((k+3)!/6) at k=3. - Alexander R. Povolotsky, May 17 2008
G.f.: -x*(1+4*x+x^2) / (x-1)^7. - R. J. Mathar, Dec 06 2011
Sum_{n>0} 1/a(n) = (8/3)*(25-9*sqrt(5)*Pi*tan(sqrt(5)*Pi/2)). - Enrique Pérez Herrero, Dec 02 2014
a(k) = MagicNKZ(3,k,7) where MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n+1-z,j)*(k-j+1)^n. (Cf. A101104.) - Danny Rorabaugh, Apr 23 2015

Edited by Ralf Stephan, Dec 16 2004

A143127 a(n) = Sum_{k=1..n} k*d(k) where d(k) is the number of divisors of k.

Original entry on oeis.org

1, 5, 11, 23, 33, 57, 71, 103, 130, 170, 192, 264, 290, 346, 406, 486, 520, 628, 666, 786, 870, 958, 1004, 1196, 1271, 1375, 1483, 1651, 1709, 1949, 2011, 2203, 2335, 2471, 2611, 2935, 3009, 3161, 3317, 3637, 3719, 4055, 4141, 4405, 4675, 4859, 4953, 5433

Offset: 1

Gary W. Adamson, Jul 26 2008

a(n) is also the sum of all parts of all partitions of all positive integers <= n into equal parts. - Omar E. Pol, May 29 2017

a(n) is also the sum of the multiples of k, not exceeding n, for k = 1, 2, ..., n. See a formula and an example below. - Wolfdieter Lang, Oct 18 2021

			a(3) = 11 = (1 + 4 + 6), where n*d(n) = (1, 4, 6, 12, 10, 24, ...).
a(4) = 23 = (8 + 7 + 5 + 3), where (8, 7, 5, 3) = row 4 of triangle A110661.
a(4) = 23 is the sum of [1 2 3 4|2 4|3|4] (multiples of k=1..4, not exceeding n). - _Wolfdieter Lang_, Oct 18 2021
a(4) = [1] + [2 + 1 + 1] + [3 + 1 + 1 + 1] + [4 + 2 + 2 + 1 + 1 + 1 + 1] = 23. - _Omar E. Pol_, Oct 18 2021

Enrique Pérez Herrero, Table of n, a(n) for n = 1..1000
Jacob F. F. Bulmer, Javier Martínez-Cifuentes, Bryn A. Bell, and Nicolás Quesada, Simulating lossy and partially distinguishable quantum optical circuits: theory, algorithms and applications to experiment validation and state preparation, arXiv:2412.17742 [quant-ph], 2024. See p. 29.
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)

Partial sums of A038040.
Cf. A000005, A083356.
Row sums of triangle A110661.
Row sums of triangle A143310. - Gary W. Adamson, Aug 06 2008
Cf. A018804.

a143127 n = a143127_list !! (n-1)
a143127_list = scanl1 (+) a038040_list
-- Reinhard Zumkeller, Jan 21 2014

Accumulate[DivisorSigma[0, Range[48]] Range[48]] (* Giovanni Resta, May 29 2018 *)

a(n) = sum(k=1, n, k*numdiv(k)); \\ Michel Marcus, May 29 2018

from math import isqrt
def A143127(n): return -((k:=isqrt(n))*(k+1)>>1)**2+sum(i*(m:=n//i)*(1+m) for i in range(1,k+1)) # Chai Wah Wu, Jul 11 2023

a(n) = Sum_{k=1..n} A038040(k).
a(n) = Sum_{m=1..floor(sqrt(n))} m*(m+floor(n/m))*(floor(n/m)+1-m) - A000330(floor(sqrt(n))) = 2*A083356(n) - A000330(floor(sqrt(n))). - Max Alekseyev, Jan 31 2012
G.f.: x*f'(x)/(1 - x), where f(x) = Sum_{k>=1} x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 13 2017 [Sum_{k>=1} k*x^k/((1-x)*(1-x^k)^2), see A038040. - Wolfdieter Lang, Oct 18 2021]
a(n) = Sum_{k=1..n} k/2 * floor(n/k) * floor(1 + n/k). - Daniel Suteu, May 28 2018
a(n) ~ log(n) * n^2 / 2 + (gamma - 1/4)*n^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 08 2018
From Daniel Hoying, May 21 2020: (Start)
a(n) = (Sum_{i=1..floor(sqrt(n))} i*floor(n/i)*(1+floor(n/i))) - (floor(sqrt(n))*(1+floor(sqrt(n)))/2)^2;
     = (Sum_{i=1..floor(sqrt(n))} i*floor(n/i)*(1+floor(n/i))) - A000537(floor(sqrt(n))).
a(n) = A000537(floor(sqrt(n))) ; n=1;
     = A000537(floor(sqrt(n))) + n*(n+1) - floor(n/2)*(floor(n/2)+1) ; 1     = A000537(floor(sqrt(n))) + n*(n+1) - floor(n/2)*(floor(n/2)+1) + Sum_{i=floor(sqrt(n))+1..floor(n/2)} i*floor(n/i)*(1+floor(n/i)) ; n>=6. (End)
a(n) = Sum_{i=1..n} A018804(i)*floor(n/i). - Ridouane Oudra, Mar 15 2021
a(n) = Sum_{k=1..n} b(n,k), with b(n, k) = Sum_{j=1..floor(n/k)} j*k = k * floor(n/k) * (floor(n/k) + 1)/2. See the formula by Daniel Suteu above. - Wolfdieter Lang, Oct 18 2021

More terms from Carl Najafi, Dec 24 2011

Edited by Max Alekseyev, Jan 31 2012

A085780 Numbers that are a product of 2 triangular numbers.

Original entry on oeis.org

0, 1, 3, 6, 9, 10, 15, 18, 21, 28, 30, 36, 45, 55, 60, 63, 66, 78, 84, 90, 91, 100, 105, 108, 120, 126, 135, 136, 150, 153, 165, 168, 171, 190, 198, 210, 216, 225, 231, 234, 253, 270, 273, 276, 280, 300, 315, 325, 330, 351, 360, 378, 396, 406, 408, 420, 435, 441

Offset: 1

Jon Perry, Jul 23 2003

nonn

Is there a fast algorithm for detecting these numbers? - Charles R Greathouse IV, Jan 26 2013

The number of rectangles with positive width 1<=w<=i and positive height 1<=h<=j contained in an i*j rectangle is t(i)*t(j), where t(k)=A000217(k), see A096948. - Dimitri Boscainos, Aug 27 2015

			18 = 3*6 = t(2)*t(3) is a product of two triangular numbers and therefore in the sequence.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000217, A085782, A068143, A000537 (subsequence), A006011 (subsequence), A033487 (subsequence), A188630 (subsequence).

Cf. A072389 (this times 4).

isA085780 := proc(n)
     local d;
     for d in numtheory[divisors](n) do
        if d^2 > n then
            return false;
        end if;
        if isA000217(d) then
            if isA000217(n/d) then
                return true;
            end if;
        end if;
    end do:
    return false;
end proc:
for n from 1 to 1000 do
    if isA085780(n) then
        printf("%d,",n) ;
    end if ;
end do: # R. J. Mathar, Nov 29 2015

t1 = Table[n (n+1)/2, {n, 0, 100}];Select[Union[Flatten[Outer[Times, t1, t1]]], # <= t1[[-1]] &] (* T. D. Noe, Jun 04 2012 *)

A003056(n)=(sqrtint(8*n+1)-1)\2
list(lim)=my(v=List([0]),t); for(a=1, A003056(lim\1), t=a*(a+1)/2; for(b=a, A003056(lim\t), listput(v,t*b*(b+1)/2))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Jan 26 2013

from itertools import count, islice
from sympy import divisors, integer_nthroot
def A085780_gen(startvalue=0): # generator of terms
    if startvalue <= 0:
        yield 0
    for n in count(max(startvalue,1)):
        for d in divisors(m:=n<<2):
            if d**2 > m:
                break
            if integer_nthroot((d<<2)+1,2)[1] and integer_nthroot((m//d<<2)+1,2)[1]:
                yield n
                break
A085780_list = list(islice(A085780_gen(),10)) # Chai Wah Wu, Aug 28 2022

Conjecture: There are about sqrt(x)*log(x) terms up to x. - Charles R Greathouse IV, Jul 11 2024

More terms from Max Alekseyev and Jon E. Schoenfield, Sep 04 2009

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A334997 Array T read by ascending antidiagonals: T(n, k) = Sum_{d divides n} T(d, k-1) with T(n, 0) = 1.

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A006011 a(n) = n^2*(n^2 - 1)/4.

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A006322 4-dimensional analog of centered polygonal numbers.

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A035287 Number of ways to place a non-attacking white and black rook on n X n chessboard.

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A062392 a(n) = n^4 - (n-1)^4 + (n-2)^4 - ... 0^4.

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A000541 Sum of 7th powers: 1^7 + 2^7 + ... + n^7.

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A101094 a(n) = n(n+1)(n+2)(n+3)(1+3*n+n^2)/120.