A211795 -id:A211795 - cp's OEIS frontend

A212959 Number of (w,x,y) such that w,x,y are all in {0,...,n} and |w-x| = |x-y|.

1, 4, 11, 20, 33, 48, 67, 88, 113, 140, 171, 204, 241, 280, 323, 368, 417, 468, 523, 580, 641, 704, 771, 840, 913, 988, 1067, 1148, 1233, 1320, 1411, 1504, 1601, 1700, 1803, 1908, 2017, 2128, 2243, 2360, 2481, 2604, 2731, 2860, 2993, 3128, 3267

Offset: 0

Clark Kimberling, Jun 01 2012

In the following guide to related sequences: M=max(x,y,z), m=min(x,y,z), and R=range=M-m. In some cases, it is an offset of the listed sequence which fits the conditions shown for w,x,y. Each sequence satisfies a linear recurrence relation, some of which are identified in the list by the following code (signature):
  A: 2,  0, -2,  1, i.e., a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4);
  B: 3, -2, -2,  3, -1;
  C: 4, -6,  4, -1;
  D: 1,  2, -2, -1,  1;
  E: 2,  1, -4,  1,  2, -1;
  F: 2, -1,  1, -2,  1;
  G: 2, -1,  0,  1, -2,  1;
  H: 2, -1,  2, -4,  2, -1,  2, -1;
  I: 3, -3,  2, -3,  3, -1;
  J: 4, -7,  8, -7,  4, -1.
  ...
A212959 ... |w-x|=|x-y| ...... recurrence type A
A212960 ... |w-x| != |x-y| ................... B
A212683 ... |w-x| < |x-y| .................... B
A212684 ... |w-x| >= |x-y| ................... B
A212963 ... see entry for definition ......... B
A212964 ... |w-x| < |x-y| < |y-w| ............ B
A006331 ... |w-x| < y ........................ C
A005900 ... |w-x| <= y ....................... C
A212965 ... w = R ............................ D
A212966 ... 2*w = R
A212967 ... w < R ............................ E
A212968 ... w >= R ........................... E
A077043 ... w = x > R ........................ A
A212969 ... w != x and x > R ................. E
A212970 ... w != x and x < R ................. E
A055998 ... w = x + y - 1
A011934 ... w < floor((x+y)/2) ............... B
A182260 ... w > floor((x+y)/2) ............... B
A055232 ... w <= floor((x+y)/2) .............. B
A011934 ... w >= floor((x+y)/2) .............. B
A212971 ... w < floor((x+y)/3) ............... B
A212972 ... w >= floor((x+y)/3) .............. B
A212973 ... w <= floor((x+y)/3) .............. B
A212974 ... w > floor((x+y)/3) ............... B
A212975 ... R is even ........................ E
A212976 ... R is odd ......................... E
A212978 ... R = 2*n - w - x
A212979 ... R = average{w,x,y}
A212980 ... w < x + y and x < y .............. B
A212981 ... w <= x+y and x < y ............... B
A212982 ... w < x + y and x <= y ............. B
A212983 ... w <= x + y and x <= y ............ B
A002623 ... w >= x + y and x <= y ............ B
A087811 ... w = 2*x + y ...................... A
A008805 ... w = 2*x + 2*y .................... D
A000982 ... 2*w = x + y ...................... F
A001318 ... 2*w = 2*x + y .................... F
A001840 ... w = 3*x + y
A212984 ... 3*w = x + y
A212985 ... 3*w = 3*x + y
A001399 ... w = 2*x + 3*y
A212986 ... 2*w = 3*x + y
A008810 ... 3*x = 2*x + y .................... F
A212987 ... 3*w = 2*x + 2*y
A001972 ... w = 4*x + y ...................... G
A212988 ... 4*w = x + y ...................... G
A212989 ... 4*w = 4*x + y
A008812 ... 5*w = 2*x + 3*y
A016061 ... n < w + x + y <= 2*n ............. C
A000292 ... w + x + y <=n .................... C
A000292 ... 2*n < w + x + y <= 3*n ........... C
A212977 ... n/2 < w + x + y <= n
A143785 ... w < R < x ........................ E
A005996 ... w < R <= x ....................... E
A128624 ... w <= R <= x ...................... E
A213041 ... R = 2*|w - x| .................... A
A213045 ... R < 2*|w - x| .................... B
A087035 ... R >= 2*|w - x| ................... B
A213388 ... R <= 2*|w - x| ................... B
A171218 ... M < 2*m .......................... B
A213389 ... R < 2|w - x| ..................... E
A213390 ... M >= 2*m ......................... E
A213391 ... 2*M < 3*m ........................ H
A213392 ... 2*M >= 3*m ....................... H
A213393 ... 2*M > 3*m ........................ H
A213391 ... 2*M <= 3*m ....................... H
A047838 ... w = |x + y - w| .................. A
A213396 ... 2*w < |x + y - w| ................ I
A213397 ... 2*w >= |x + y - w| ............... I
A213400 ... w < R < 2*w
A069894 ... min(|w-x|,|x-y|) = 1
A000384 ... max(|w-x|,|x-y|) = |w-y|
A213395 ... max(|w-x|,|x-y|) = w
A213398 ... min(|w-x|,|x-y|) = x ............. A
A213399 ... max(|w-x|,|x-y|) = x ............. D
A213479 ... max(|w-x|,|x-y|) = w+x+y ......... D
A213480 ... max(|w-x|,|x-y|) != w+x+y ........ E
A006918 ... |w-x| + |x-y| > w+x+y ............ E
A213481 ... |w-x| + |x-y| <= w+x+y ........... E
A213482 ... |w-x| + |x-y| < w+x+y ............ E
A213483 ... |w-x| + |x-y| >= w+x+y ........... E
A213484 ... |w-x|+|x-y|+|y-w| = w+x+y
A213485 ... |w-x|+|x-y|+|y-w| != w+x+y ....... J
A213486 ... |w-x|+|x-y|+|y-w| > w+x+y ........ J
A213487 ... |w-x|+|x-y|+|y-w| >= w+x+y ....... J
A213488 ... |w-x|+|x-y|+|y-w| < w+x+y ........ J
A213489 ... |w-x|+|x-y|+|y-w| <= w+x+y ....... J
A213490 ... w,x,y,|w-x|,|x-y| distinct
A213491 ... w,x,y,|w-x|,|x-y| not distinct
A213493 ... w,x,y,|w-x|,|x-y|,|w-y| distinct
A213495 ... w = min(|w-x|,|x-y|,|w-y|)
A213492 ... w != min(|w-x|,|x-y|,|w-y|)
A213496 ... x != max(|w-x|,|x-y|)
A213498 ... w != max(|w-x|,|x-y|,|w-y|)
A213497 ... w = min(|w-x|,|x-y|)
A213499 ... w != min(|w-x|,|x-y|)
A213501 ... w != max(|w-x|,|x-y|)
A213502 ... x != min(|w-x|,|x-y|)
...
A211795 includes a guide for sequences that count 4-tuples (w,x,y,z) having all terms in {0,...,n} and satisfying selected properties.  Some of the sequences indexed at A211795 satisfy recurrences that are represented in the above list.
Partial sums of the numbers congruent to {1,3} mod 6 (see A047241). - Philippe Deléham, Mar 16 2014

			a(1)=4 counts these (x,y,z): (0,0,0), (1,1,1), (0,1,0), (1,0,1).
Numbers congruent to {1, 3} mod 6: 1, 3, 7, 9, 13, 15, 19, ...
a(0) = 1;
a(1) = 1 + 3 = 4;
a(2) = 1 + 3 + 7 = 11;
a(3) = 1 + 3 + 7 + 9 = 20;
a(4) = 1 + 3 + 7 + 9 + 13 = 33;
a(5) = 1 + 3 + 7 + 9 + 13 + 15 = 48; etc. - _Philippe Deléham_, Mar 16 2014

A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.
P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A047241, A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] == Abs[x - y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 50]]   (* A212959 *)

a(n)=(6*n^2+8*n+3)\/4 \\ Charles R Greathouse IV, Jul 28 2015

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1+2*x+3*x^2)/((1+x)*(1-x)^3).
a(n) + A212960(n) = (n+1)^3.
a(n) = (6*n^2 + 8*n + 3 + (-1)^n)/4. - Luce ETIENNE, Apr 05 2014
a(n) = 2*A069905(3*(n+1)+2) - 3*(n+1). - Ayoub Saber Rguez, Aug 31 2021

A001296 4-dimensional pyramidal numbers: a(n) = (3n+1)binomial(n+2, 3)/4. Also Stirling2(n+2, n).

Original entry on oeis.org

0, 1, 7, 25, 65, 140, 266, 462, 750, 1155, 1705, 2431, 3367, 4550, 6020, 7820, 9996, 12597, 15675, 19285, 23485, 28336, 33902, 40250, 47450, 55575, 64701, 74907, 86275, 98890, 112840, 128216, 145112, 163625, 183855, 205905, 229881, 255892, 284050, 314470

Offset: 0

N. J. A. Sloane

Permutations avoiding 12-3 that contain the pattern 31-2 exactly once.
Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005
Partial sums of A002411. - Jonathan Vos Post, Mar 16 2006
If Y is a 3-subset of an n-set X then, for n>=6, a(n-5) is the number of 6-subsets of X having at least two elements in common with Y. - Milan Janjic, Nov 23 2007
Starting with 1 = binomial transform of [1, 6, 12, 10, 3, 0, 0, 0, ...]. Equals row sums of triangle A143037. - Gary W. Adamson, Jul 18 2008
Rephrasing the Perry formula of 2003: a(n) is the sum of all products of all two  numbers less than or equal to n, including the squares. Example: for n=3 the sum of these products is 1*1 + 1*2 + 1*3 + 2*2 + 2*3 + 3*3 = 25. - J. M. Bergot, Jul 16 2011
Half of the partial sums of A011379. [Jolley, Summation of Series, Dover (1961), page 12 eq (66).] - R. J. Mathar, Oct 03 2011
Also the number of (w,x,y,z) with all terms in {1,...,n+1} and w < x >= y > z (see A211795). - Clark Kimberling, May 19 2012
Convolution of A000027 with A000326. - Bruno Berselli, Dec 06 2012
This sequence is related to A000292 by a(n) = n*A000292(n) - Sum_{i=0..n-1} A000292(i) for n>0. - Bruno Berselli, Nov 23 2017
a(n-2) is the maximum number of intersections made from the perpendicular bisectors of all pair combinations of n points. - Ian Tam, Dec 22 2020

			G.f. = x + 7*x^2 + 25*x^3 + 65*x^4 + 140*x^5 + 266*x^6 + 462*x^7 + 750*x^8 + 1155*x^9 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16.
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/3).
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
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].
P. Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
S. Butler and P. Karasik, A note on nested sums, J. Int. Seq. 13 (2010), 10.4.4, page 5.
M. Griffiths, Remodified Bessel Functions via Coincidences and Near Coincidences, Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.1.
L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Chanticleer Press, NY, 1950, p. 36.
C. Krishnamachaki, The operator (xD)^n, J. Indian Math. Soc., 15 (1923),3-4. [Annotated scanned copy]
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. 29.
T. Mansour, Restricted permutations by patterns of type 2-1, arXiv:math/0202219 [math.CO], 2002.
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, Stirling numbers of the 2nd kind.
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

a(n)=f(n, 2) where f is given in A034261.
Cf. A000217, A000326, A001297, A001298, A002411, A008277, A008517, A094262.
a(n)= A093560(n+3, 4), (3, 1)-Pascal column.
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.
Cf. similar sequences listed in A241765 and A254142.
Cf. A000914.

/* A000027 convolved with A000326: */ A000326:=func; [&+[(n-i+1)*A000326(i): i in [0..n]]: n in [0..40]]; // Bruno Berselli, Dec 06 2012

[(3*n+1)*Binomial(n+2,3)/4: n in [0..40]]; // Vincenzo Librandi, Jul 30 2014

A001296:=-(1+2*z)/(z-1)**5; # Simon Plouffe in his 1992 dissertation for sequence without the leading zero

Table[n*(1+n)*(2+n)*(1+3*n)/24, {n, 0, 100}]
CoefficientList[Series[x (1 + 2 x)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 30 2014 *)
Table[StirlingS2[n+2, n], {n, 0, 40}] (* Jean-François Alcover, Jun 24 2015 *)
Table[ListCorrelate[Accumulate[Range[n]],Range[n]],{n,0,40}]//Flatten (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,7,25,65},40] (* Harvey P. Dale, Aug 14 2017 *)

t(n)=n*(n+1)/2
for(i=1,30,print1(","sum(j=1,i,j*t(j))))

{a(n) = n * (n+1) * (n+2) * (3*n+1) / 24}; /* Michael Somos, Sep 04 2017 */

[stirling_number2(n+2,n) for n in range(0,38)] # Zerinvary Lajos, Mar 14 2009

a(n) = n*(1+n)*(2+n)*(1+3*n)/24. - T. D. Noe, Jan 21 2008
G.f.: x*(1+2*x)/(1-x)^5. - Paul Barry, Jul 23 2003
a(n) = Sum_{j=0..n} j*A000217(j). - Jon Perry, Jul 28 2003
E.g.f. with offset -1: exp(x)*(1*(x^2)/2! + 4*(x^3)/3! + 3*(x^4)/4!). For the coefficients [1, 4, 3] see triangle A112493.
E.g.f. x*exp(x)*(24 + 60*x + 28*x^2 + 3*x^3)/24 (above e.g.f. differentiated).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 3. - Kieren MacMillan, Sep 29 2008
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Jaume Oliver Lafont, Nov 23 2008
O.g.f. is D^2(x/(1-x)) = D^3(x), where D is the operator x/(1-x)*d/dx. - Peter Bala, Jul 02 2012
a(n) = A153978(n)/2. - J. M. Bergot, Aug 09 2013
a(n) = A002817(n) + A000292(n-1). - J. M. Bergot, Aug 29 2013; [corrected by Cyril Damamme, Feb 26 2018]
a(n) = A000914(n+1) - 2 * A000330(n+1). - Antal Pinter, Dec 31 2015
a(n) = A080852(3,n-1). - R. J. Mathar, Jul 28 2016
a(n) = 1*(1+2+...+n) + 2*(2+3+...+n) + ... + n*n. For example, a(6) = 266 = 1(1+2+3+4+5+6) + 2*(2+3+4+5+6) + 3*(3+4+5+6) + 4*(4+5+6) + 5*(5+6) + 6*(6).- J. M. Bergot, Apr 20 2017
a(n) = A000914(-2-n) for all n in Z. - Michael Somos, Sep 04 2017
a(n) = A000292(n) + A050534(n+1). - Cyril Damamme, Feb 26 2018
From Amiram Eldar, Jul 02 2020: (Start)
Sum_{n>=1} 1/a(n) = (6/5) * (47 - 3*sqrt(3)*Pi - 27*log(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = (6/5) * (16*log(2) + 6*sqrt(3)*Pi - 43). (End)

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)

A061201 Partial sums of A007425: (tau<=)_3(n).

Original entry on oeis.org

1, 4, 7, 13, 16, 25, 28, 38, 44, 53, 56, 74, 77, 86, 95, 110, 113, 131, 134, 152, 161, 170, 173, 203, 209, 218, 228, 246, 249, 276, 279, 300, 309, 318, 327, 363, 366, 375, 384, 414, 417, 444, 447, 465, 483, 492, 495, 540, 546, 564, 573, 591, 594, 624, 633, 663

Offset: 1

Vladeta Jovovic, Apr 21 2001

(tau<=)_k(n) = |{(x_1,x_2,...,x_k): x_1*x_2*...*x_k<=n}|, i.e., tau<=_k(n) is number of solutions to x_1*x_2*...*x_k<=n, x_i > 0.
A061201(n) is the number of 4-tuples (w,x,y,z) having all terms in {1,...,n} and w=x*y*z; see A211795 for a list of related counting sequences. - Clark Kimberling, Apr 28 2012
The formula for Sum_{k=1..n} d3(k) in the Benoit Cloitre article on page 15 is incorrect. For correct asymptotic formula see below or generate it in the Mathematica: Residue[Zeta[s]^3 * n^s/s, {s, 1}] // Expand. - Vaclav Kotesovec, Aug 19 2021

M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 239.

Vaclav Kotesovec, Table of n, a(n) for n = 1..100000 (terms 1..1000 from Harry J. Smith)
Benoit Cloitre, On the divisor and circle problems
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)

Cf. tau_2(n): A000005, tau_3(n): A007425, tau_4(n): A007426, tau_5(n): A061200, tau_6(n): A034695, (unordered) 2-factorizations of n: A038548, (unordered) 3-factorizations of n: A034836, A001055, (tau<=)_2(n): A006218, (tau<=)_4(n): A061202, (tau<=)_5(n): A061203, (tau<=)_6(n): A061204.

[&+[NumberOfDivisors(k)*Floor(n/k): k in [1..n]]: n in [1..56]];  // Bruno Berselli, Apr 13 2011

b:= proc(k, n) option remember; uses numtheory;
     `if`(k=1, 1, add(b(k-1, d), d=divisors(n)))
    end:
a:= proc(n) option remember; `if`(n=0, 0, b(3, n)+a(n-1)) end:
seq(a(n), n=1..76);  # Alois P. Heinz, Oct 23 2023

a[n_] := Sum[ DivisorSigma[0, k]*Floor[n/k], {k, 1, n}]; Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Sep 20 2011, after Benoit Cloitre *)
(* Asymptotics: *) n*(Log[n]^2/2 + (3*EulerGamma - 1)*Log[n] + 3*EulerGamma^2 - 3*EulerGamma - 3*StieltjesGamma[1] + 1) (* Vaclav Kotesovec, Sep 09 2018 *)
Accumulate[a[n_]:=DivisorSum[n, DivisorSigma[0, #]&]; Array[a, 60]] (* Vincenzo Librandi, Jan 12 2020 *)

a(n)=sum(k=1,n,numdiv(k)*floor(n/k)) \\ Benoit Cloitre, Apr 19 2007

{ for (n=1, 1000, write("b061201.txt", n, " ", sum(k=1, n, numdiv(k)*(n\k))) ) } \\ Harry J. Smith, Jul 18 2009

my(N=60, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k)*x^k/(1-x^k))/(1-x)) \\ Seiichi Manyama, Jul 24 2022

from math import isqrt
from sympy import integer_nthroot
def A061201(n): return (m:=integer_nthroot(n,3)[0])**3+3*sum(-(s:=isqrt(r:=n//i))**2+(sum(r//k for k in range(1,s+1))<<1)-sum(n//(i*j) for j in range(1,m+1)) for i in range(1,m+1)) # Chai Wah Wu, Oct 23 2023

(tau<=)k(n) = Sum{i=1..n} tau_k(i).
a(n) = n * ( log(n)^2/2 + (3*g-1)*log(n) + 3*g^2-3*g-3*g1+1 ) + O(sqrt(n)), where g is the Euler-Mascheroni number ~ 0.57721... (see A001620), and g1 is the first Stieltjes constant ~ -0.072816 (see A082633). The determination of the precise size of the error term is an unsolved problem - see references. - Andrew Lelechenko, Apr 15 2011 [corrected by Vaclav Kotesovec, Sep 09 2018]
a(n) = Sum_{k=1..n} A000005(k)*floor(n/k). - Benoit Cloitre, Apr 19 2007
To compute a(n) for huge n (see A180365) in sublinear use a(n) = 3*Sum_{i=1..n3} A006218(n/i) - Sum_{j=1..n3} floor(n/(i*j)) + n3^3, where n3 = floor(n^(1/3)). - Andrew Lelechenko, Apr 15 2011
a(n) = Sum_{k=1..n} Sum_{i=1..n} floor(n/(i*k)). - Wesley Ivan Hurt, Sep 14 2017
G.f.: (1/(1-x)) * Sum_{k>=1} A000005(k) * x^k/(1 - x^k). - Seiichi Manyama, Jul 24 2022

A212508 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y<3z.

Original entry on oeis.org

0, 1, 12, 56, 168, 418, 837, 1554, 2640, 4209, 6375, 9373, 13176, 18161, 24402, 32110, 41472, 52948, 66339, 82384, 101100, 122801, 147741, 176665, 209088, 246225, 287976, 334764, 386904, 445486, 509625, 581126, 659712, 745921

Offset: 0

Clark Kimberling, May 19 2012

For a guide to related sequences, see A211795.

Index entries for linear recurrences with constant coefficients, signature (0, 2, 2, -1, -4, 0, 2, 0, -2, 0, 4, 1, -2, -2, 0, 1).

Cf. A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[w < 2 x && y < 3 z, s = s + 1], {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]]; Map[t[#] &, Range[0, 50]]   (* A212508 *)
Table[n^2/24 + n^3/3 + 5*n^4/8 - 1/12*Floor[n/6] - 1/4*n^2*Floor[n/3] - (n/12 + 5*n^2/12) * Floor[n/2] + 1/12*Floor[(1 + n)/6] + 1/4*n^2*Floor[(1 + n)/3], {n, 0, 50}] (* Vaclav Kotesovec, Dec 11 2015 *)

a(n)=2a(n-2)+2a(n-3)-a(n-4)-4a(n-5)+2a(n-7)-2a(n-9)+4a(n-11)+a(n-12)-2a(n-13)-2a(n-14)+a(n-16).

G.f.: -x*((1 + 12*x + 54*x^2 + 142*x^3 + 283*x^4 + 405*x^5 + 486*x^6 + 520*x^7 + 493*x^8 + 386*x^9 + 265*x^10 + 136*x^11 + 47*x^12 + 9*x^13 + x^14)/((-1 + x)^5*(1 + x)^3 * (1 - x + x^2)*(1 + x + x^2)^3)). - Vaclav Kotesovec, Dec 11 2015

A212133 Number of (w,x,y,z) with all terms in {1,...,n} and median=mean.

Original entry on oeis.org

0, 1, 8, 33, 88, 185, 336, 553, 848, 1233, 1720, 2321, 3048, 3913, 4928, 6105, 7456, 8993, 10728, 12673, 14840, 17241, 19888, 22793, 25968, 29425, 33176, 37233, 41608, 46313, 51360, 56761, 62528, 68673, 75208, 82145, 89496, 97273, 105488, 114153, 123280

Offset: 0

Clark Kimberling, May 04 2012

For a guide to related sequences, see A211795.

For n>=1, a(n) is the number of cells in the n-th rhombic-dodecahedral polycube. - George Sicherman, Jan 22 2024

			a(2) counts these 4-tuples:  (1,1,1,1), (1,1,2,2), (1,2,1,2), (2,1,1,2), (1,2,2,1), (2,1,2,1), (2,2,1,1), (2,2,2,2).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Z. Janelidze, F. van Niekerk, and J. Viljoen, What is the maximal connected partial symmetry index of a connected graph of a given size?, arXiv:2502.00704 [math.CO], 2025. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A211795.
Cf. A226449. - Bruno Berselli, Jun 09 2013
Cf. A005917.

a212133 n = if n == 0 then 0 else (a005917 n + 1) `div` 2
-- Reinhard Zumkeller, Nov 13 2014

t = Compile[{{n, _Integer}},
Module[{s = 0}, (Do[If[Apply[Plus, Rest[Most[Sort[{w, x, y, z}]]]]/2 == (w + x + y + z)/4, s = s + 1], {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Flatten[Map[{t[#]} &, Range[0, 50]]] (* A212133 *)
(* Peter J. C. Moses, May 01 2012 *)

a(n)=2*n^3-3*n^2+2*n; \\ Joerg Arndt, Jun 22 2012

concat(0, Vec(x*(1 + 4*x + 7*x^2) / (1 - x)^4 + O(x^40))) \\ Colin Barker, Dec 02 2017

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = n * (2*n^2 - 3*n + 2). - J. M. Bergot, Jun 22 2012
a(n) = A000384(n) + n*A000384(n-1). - Bruno Berselli, Jun 07 2013
a(n) = (A005917(n) + 1) / 2 for n > 0. - Reinhard Zumkeller, Nov 13 2014
G.f.: x*(1 + 4*x + 7*x^2) / (1 - x)^4. - Colin Barker, Dec 02 2017

Closed form adapted to the offset by Bruno Berselli, Jun 07 2013

A212570 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x|=|x-y|+|y-z|.

Original entry on oeis.org

0, 1, 6, 23, 52, 105, 178, 287, 424, 609, 830, 1111, 1436, 1833, 2282, 2815, 3408, 4097, 4854, 5719, 6660, 7721, 8866, 10143, 11512, 13025, 14638, 16407, 18284, 20329, 22490, 24831, 27296, 29953, 32742, 35735, 38868, 42217, 45714, 49439

Offset: 0

Clark Kimberling, May 22 2012

For a guide to related sequences, see A211795.

Apart from the first term, partial sums of A220082. [Bruno Berselli, Dec 05 2012]

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] == Abs[x - y] + Abs[y - z], s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212570 *)
LinearRecurrence[{2,1,-4,1,2,-1},{0,1,6,23,52,105},40] (* Harvey P. Dale, Oct 02 2021 *)

a(n) = 2a(n-1)+a(n-2)-4a(n-3)+a(n-4)+2a(n-5)-a(n-6).

a(n) = n*(-1-3*(-1)^n+10*n^2)/12. G.f.: x*(x^4+4*x^3+10*x^2+4*x+1)/((x-1)^4*(x+1)^2). [Colin Barker, Oct 04 2012]

A212103 Number of (w,x,y,z) with all terms in {1,...,n} and w = harmonic mean of {x,y,z}.

Original entry on oeis.org

0, 1, 2, 3, 10, 11, 30, 31, 38, 39, 52, 53, 84, 85, 86, 117, 124, 125, 144, 145, 200, 225, 226, 227, 282, 283, 284, 285, 334, 335, 420, 421, 428, 435, 436, 491, 546, 547, 548, 555, 634, 635, 726, 727, 758, 837, 838, 839, 936, 937, 956, 957, 970, 971

Offset: 0

Clark Kimberling, May 03 2012

nonn

Also, the number of (w,x,y,z) with all terms in {1,...,n} and H(w,x,y)=H(w,x,y,z) where H denotes harmonic mean. For a guide to related sequences, see A211795.

			a(4) counts these:  (1,1,1,1), (2,1,4,4), (2,2,2,2), (2,4,1,4), (2,4,4,1), (3,2,4,4), (3,3,3,3), (3,4,2,4), (3,4,4,2), (4,4,4,4); e.g., (3,2,4,4) is included because it satisfies 3/w=1/x+1/y+1/z.

Cf. A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w*(y*z + z*x + x*y) == 3 x*y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 60]] (* A212103 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A212560 Number of (w,x,y,z) with all terms in {1,...,n} and w+x<=y+z.

Original entry on oeis.org

0, 1, 11, 50, 150, 355, 721, 1316, 2220, 3525, 5335, 7766, 10946, 15015, 20125, 26440, 34136, 43401, 54435, 67450, 82670, 100331, 120681, 143980, 170500, 200525, 234351, 272286, 314650, 361775, 414005, 471696, 535216, 604945, 681275

Offset: 0

Clark Kimberling, May 21 2012

For a guide to related sequences, see A211795.

For n>=1, a(n) is the n-th antidiagonal sums of the convolution array A213831. - Clark Kimberling, Jul 04 2012

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

Cf. A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w + x <= y + z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212560 *)

a(n)=(n+2*n^3+3*n^4)/6 \\ Charles R Greathouse IV, Oct 21 2022

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5).
a(n) = (n + 2*n^3 + 3*n^4)/6. - Clark Kimberling, Jul 10 2012
G.f.: x*(1 + x)*(1 + 5*x)/(1 - x)^5. - Clark Kimberling, Jul 10 2012
a(n) = Sum_{k=0..n} A059722(k). - J. M. Bergot, Nov 02 2012

A212714 Number of (w,x,y,z) with all terms in {1,...,n} and |w-x| >= w + |y-z|.

Original entry on oeis.org

0, 0, 2, 10, 32, 78, 162, 300, 512, 820, 1250, 1830, 2592, 3570, 4802, 6328, 8192, 10440, 13122, 16290, 20000, 24310, 29282, 34980, 41472, 48828, 57122, 66430, 76832, 88410, 101250, 115440, 131072, 148240, 167042, 187578, 209952

Offset: 0

Clark Kimberling, May 24 2012

For a guide to related sequences, see A211795.

a(n) is also the number of inequivalent (modulo C_4 rotations) square n X n grids with squares coming in two colors and two squares have one of the colors. See the formula from A054772. - Wolfdieter Lang, Oct 03 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Cf. A211795, A054772.

I:=[0,0,2,10,32,78]; [n le 6 select I[n] else 4*Self(n-1)-5*Self(n-2)+5*Self(n-4)-4*Self(n-5)+Self(n-6): n in [1..40]]; // Vincenzo Librandi, Aug 02 2013

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[Abs[w - x] >= w + Abs[y - z], s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212714 *)
%/2  (* A011864 except for offset *)
LinearRecurrence[{4, -5, 0, 5, -4, 1}, {0, 0, 2, 10, 32, 78}, 40]
CoefficientList[Series[(2 x^2 + 2 x^3 + 2 x^4) / (1 - 4 x + 5 x^2 - 5 x^4 + 4 x^5 - x^6), {x, 0, 80}], x] (* Vincenzo Librandi, Aug 02 2013 *)

a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6).
G.f.: (2*x^2 + 2*x^3 + 2*x^4)/(1 - 4*x + 5*x^2 - 5*x^4 + 4*x^5 - x^6).
a(n) = floor(n^4/8). - Wesley Ivan Hurt, Jul 14 2013
a(n) = A054772(n, 2) = A054772(n, n^2-2), n >= 2. - Wolfdieter Lang, Oct 03 2016

cp's OEIS Frontend

A212959 Number of (w,x,y) such that w,x,y are all in {0,...,n} and |w-x| = |x-y|.

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A001296 4-dimensional pyramidal numbers: a(n) = (3*n+1)*binomial(n+2, 3)/4. Also Stirling2(n+2, n).

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A006322 4-dimensional analog of centered polygonal numbers.

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A061201 Partial sums of A007425: (tau<=)_3(n).

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A212508 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y<3z.

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A212133 Number of (w,x,y,z) with all terms in {1,...,n} and median=mean.

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A001296 4-dimensional pyramidal numbers: a(n) = (3n+1)binomial(n+2, 3)/4. Also Stirling2(n+2, n).