A059270 -id:A059270 - cp's OEIS frontend

A252117 Irregular triangle read by row: T(n,k), n>=1, k>=1, in which column k lists the numbers of A000716 multiplied by A000330(k), and the first element of column k is in row A000217(k).

1, 3, 9, 5, 22, 15, 51, 45, 108, 110, 14, 221, 255, 42, 429, 540, 126, 810, 1105, 308, 1479, 2145, 714, 30, 2640, 4050, 1512, 90, 4599, 7395, 3094, 270, 7868, 13200, 6006, 660, 13209, 22995, 11340, 1530, 21843, 39340, 20706, 3240, 55, 35581, 66045, 36960, 6630, 165, 57222, 109215, 64386, 12870, 495

Offset: 1

Omar E. Pol, Dec 14 2014

Gives an identity for sigma(n). Alternating sum of row n equals A000203(n), the sum of the divisors of n.
Row n has length A003056(n) hence column k starts in row A000217(k).
Column 1 is A000716, but here the offset is 1 not 0.
The 1st element of column k is A000330(k).
The 2nd element of column k is A059270(k).
The 3rd element of column k is A220443(k).
The partial sums of column k give the k-th column of A249120.
This triangle has been constructed after Mircea Merca's formula for A000203.
From Omar E. Pol, May 05 2022: (Start)
In the Honda-Yoda paper see "3. String theory and Riemann hypothesis". The coefficients that are mentioned in 3.11 are the first 16 terms of A000716, the coefficients that are mentioned in 3.12 are the first 5 terms of A000330, and the coefficients that are mentioned in 3.13 are the first 16 terms of A000203.
Another triangle with the same row lengths and whose alternating row sums give A000203 is A196020. (End)

			Triangle begins:
       1;
       3;
       9,      5;
      22,     15;
      51,     45;
     108,    110,     14;
     221,    255,     42;
     429,    540,    126;
     810,   1105,    308;
    1479,   2145,    714,     30;
    2640,   4050,   1512,     90;
    4599,   7395,   3094,    270;
    7868,  13200,   6006,    660;
   13209,  22995,  11340,   1530;
   21843,  39340,  20706,   3240,    55;
   35581,  66045,  36960,   6630,   165;
   57222, 109215,  64386,  12870,   495;
   90882, 177905, 110152,  24300,  1210;
  142769, 286110, 184926,  44370,  2805;
  221910, 454410, 305802,  79200,  5940;
  341649, 713845, 498134, 137970, 12155, 91;
...
For n = 6 the divisors of 6 are 1, 2, 3, 6, so the sum of the divisors of 6 is 1 + 2 + 3 + 6 = 12. On the other hand, the 6th row of the triangle is 108, 110, 14, so the alternating row sum is 108 - 110 + 14 = 12, equaling the sum of the divisors of 6.
For n = 15 the divisors of 15 are 1, 3, 5, 15, so the sum of the divisors of 15 is 1 + 3 + 5 + 15 = 24. On the other hand, the 15th row of the triangle is 21843, 39340, 20706, 3240, 55, so the alternating row sum is 21843 - 39340 + 20706 - 3240 + 55 = 24, equaling the sum of the divisors of 15.

Masazumi Honda and Takuya Yoda, String theory, N = 4 SYM and Riemann hypothesis, arXiv:2203.17091 [hep-th], (2022), pp. 5-6.
Index entries for sequences related to sigma(n)

Cf. A000203, A000217, A000330, A000716, A003056, A059270, A196020, A220443, A238442, A249120.

A003056(n) = (sqrtint(8*n+1)-1)\2;
A000330(n) = n*(n+1)*(2*n+1)/6;
A000217(n) = n*(n+1)/2;
A000716(n) = polcoef(1/eta('x+O('x^(n+1)))^3, n, x);
T(n, k) = A000330(k)*A000716(n-A000217(k));
row(n) = vector(A003056(n), k, T(n,k)); \\ Michel Marcus, Oct 29 2022

A000203(n) = Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k).

A106846 a(n) = Sum_{k + lm <= n} (k + lm), with 0 <= k,l,m <= n.

Original entry on oeis.org

0, 4, 22, 64, 144, 269, 461, 720, 1072, 1522, 2092, 2774, 3626, 4614, 5776, 7126, 8694, 10445, 12461, 14684, 17204, 19997, 23077, 26412, 30156, 34206, 38600, 43352, 48532, 54042, 60072, 66458, 73338, 80664, 88450, 96710, 105638, 114999

Offset: 0

Ralf Stephan, May 06 2005

nonn

R. J. Mathar, Table of n, a(n) for n = 0..1000

Cf. A106633, A106634, A106847.

Cf. A055112, A006218, A143127, A319085, A059270, A143127.

A106846 := proc(n)
    local a,k,l,m ;
    a := 0 ;
    for k from 0 to n do
        for l from 0 to n do
            if l = 0 then
                m := n ;
            else
                m := floor((n-k)/l) ;
            end if;
            if m >=0 then
                m := min(m,n) ;
                a := a+(m+1)*k+l*m*(m+1)/2 ;
            end if;
        end do:
    end do:
    a ;
end proc: # R. J. Mathar, Oct 17 2012

A106846[n_] := Module[{a, k, l, m }, a = 0; For[k = 0, k <= n, k++, For[l = 0, l <= n, l++, If[l == 0, m = n, m = Floor[(n - k)/l]]; If[m >= 0, m = Min[m, n]; a = a + (m + 1)*k + l*m*(m + 1)/2 ]]]; a];
Table[A106846[n], {n, 0, 40}] (* Jean-François Alcover, Apr 04 2024, after R. J. Mathar *)

From Ridouane Oudra, Jun 24 2024: (Start)
a(n) = (1/2) * (n*(n+1)*(2*n+1) + Sum_{k=1..n} (n^2 + n + k - k^2) * tau(k)).
a(n) = (1/2) * (A055112(n) + (n^2 + n) * A006218(n) + A143127(n) - A319085(n)).
a(n) = A059270(n) + A143127(n) + A106847(n). (End)

A360849 Array read by antidiagonals: T(m,n) is the number of (undirected) cycles in the complete bipartite graph K_{m,n}.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 6, 15, 6, 0, 0, 10, 42, 42, 10, 0, 0, 15, 90, 204, 90, 15, 0, 0, 21, 165, 660, 660, 165, 21, 0, 0, 28, 273, 1650, 3940, 1650, 273, 28, 0, 0, 36, 420, 3486, 15390, 15390, 3486, 420, 36, 0, 0, 45, 612, 6552, 45150, 113865, 45150, 6552, 612, 45, 0

Offset: 1

Andrew Howroyd, Feb 23 2023

Also, T(m,n) is the number of chordless cycles of length >= 4 in the m X n rook graph.

			Array begins:
========================================================
m\n| 1  2   3    4      5       6        7         8 ...
---+----------------------------------------------------
1  | 0  0   0    0      0       0        0         0 ...
2  | 0  1   3    6     10      15       21        28 ...
3  | 0  3  15   42     90     165      273       420 ...
4  | 0  6  42  204    660    1650     3486      6552 ...
5  | 0 10  90  660   3940   15390    45150    109480 ...
6  | 0 15 165 1650  15390  113865   526155   1776180 ...
7  | 0 21 273 3486  45150  526155  4662231  24864588 ...
8  | 0 28 420 6552 109480 1776180 24864588 256485040 ...
  ...
Lower half of array as triangle T(n,k) for 1 <= k <= n begins:
  0;
  0,  1;
  0,  3,  15;
  0,  6,  42,  204;
  0, 10,  90,  660,  3940;
  0, 15, 165, 1650, 15390, 113865;
  0, 21, 273, 3486, 45150, 526155, 4662231;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Eric Weisstein's World of Mathematics, Chordless Cycle.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Eric Weisstein's World of Mathematics, Graph Cycle.
Eric Weisstein's World of Mathematics, Rook Graph.

Rows 1..3 are A000004, A000217(n-1), A059270(n-1).
Main diagonal is A070968.
Cf. A269562, A286418, A360850 (paths), A360853.

T(m,n) = sum(j=2, min(m,n), binomial(m,j)*binomial(n,j)*j!*(j-1)!/2)

T(m,n) = Sum_{j=2..min(m,n)} binomial(m,j)*binomial(n,j)*j!*(j-1)!/2.

T(m,n) = T(n,m).

A185732 Accumulation array of the polygonal number array (A086270), by antidiagonals.

Original entry on oeis.org

1, 4, 2, 10, 9, 3, 20, 24, 15, 4, 35, 50, 42, 22, 5, 56, 90, 90, 64, 30, 6, 84, 147, 165, 140, 90, 39, 7, 120, 224, 273, 260, 200, 120, 49, 8, 165, 324, 420, 434, 375, 270, 154, 60, 9, 220, 450, 612, 672, 630, 510, 350, 192, 72, 10, 286, 605, 855, 984, 980, 861, 665, 440, 234, 85, 11, 364, 792, 1155, 1380, 1440, 1344, 1127, 840, 540, 280, 99, 12, 455, 1014, 1518, 1870, 2025, 1980, 1764, 1428, 1035, 650, 330, 114, 13, 560, 1274, 1950, 2464, 2750, 2790, 2604, 2240

Offset: 1

Clark Kimberling, Feb 01 2011

This is the (first) accumulation array of A086270; the second is A185733. See A144112 for the definition of accumulation array.

			Northwest corner:
1....4....10...20...35
2....9....24...50...90
3....15...42...90...165
4....22...64...140..260
5....30...90...200..375

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Rows 1 to 5: A000292, A006002, A059270, A177814, 5*A002411.
Columns 1 to 4: A000027, A055999, A067728, 10*A000096.
Cf. A144112, A086270, A185732.

f[n_,k_]:=k+n*k(k-1)/2;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]  (* Array A086270 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten  (* A086270 *)
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* acc. arr. of {f(n,k)} *)
Factor[s[n,k]]  (* formula for A185732 *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]] (* acc. arr. A185732 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten (* A185732 *)

T(n,k) = k*(k+1)*n*(n+1)*(k*n-n+k+5)/12.

A316224 a(n) = n(2n + 1)(4n + 1).

Original entry on oeis.org

0, 15, 90, 273, 612, 1155, 1950, 3045, 4488, 6327, 8610, 11385, 14700, 18603, 23142, 28365, 34320, 41055, 48618, 57057, 66420, 76755, 88110, 100533, 114072, 128775, 144690, 161865, 180348, 200187, 221430, 244125, 268320, 294063, 321402, 350385, 381060, 413475, 447678, 483717

Offset: 0

Bruno Berselli, Jun 27 2018

Sums of the consecutive integers from A000384(n) to A000384(n+1)-1. This is the case s=6 of the formula n*(n*(s-2) + 1)*(n*(s-2) + 2)/2 related to s-gonal numbers.

The inverse binomial transform is 0, 15, 60, 48, 0, ... (0 continued).

			Row sums of the triangle:
|  0 |  ................................................................. 0
|  1 |  2  3  4  5  .................................................... 15
|  6 |  7  8  9 10 11 12 13 14  ........................................ 90
| 15 | 16 17 18 19 20 21 22 23 24 25 26 27  ........................... 273
| 28 | 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44  ............... 612
| 45 | 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65  .. 1155
...
where:
. first column is A000384,
. second column is A130883 (without 1),
. third column is A033816,
. diagonal is A014106,
. 0, 2, 8, 18, 32, 50, ... are in A001105.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

First bisection of A059270 and subsequence of A034828, A047866, A109900, A290168.

Sums of the consecutive integers from P(s,n) to P(s,n+1)-1, where P(s,k) is the k-th s-gonal number: A027480 (s=3), A055112 (s=4), A228888 (s=5).

GAP
```
List([0..40], n -> n*(2*n+1)*(4*n+1));
```

[n*(2*n+1)*(4*n+1) for n in 0:40] |> println

Magma
```
[n*(2*n+1)*(4*n+1): n in [0..40]];
```

seq(n*(2*n+1)*(4*n+1),n=0..40); # Muniru A Asiru, Jun 27 2018

Table[n (2 n + 1) (4 n + 1), {n, 0, 40}]

Maxima
```
makelist(n*(2*n+1)*(4*n+1), n, 0, 40);
```
PARI
```
vector(40, n, n--; n*(2*n+1)*(4*n+1))
```
Python
```
[n*(2*n+1)*(4*n+1) for n in range(40)]
```
Sage
```
[n*(2*n+1)*(4*n+1) for n in (0..40)]
```

O.g.f.: 3*x*(5 + 10*x + x^2)/(1 - x)^4.
E.g.f.: x*(15 + 30*x + 8*x^2)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) =  3*A258582(n).
a(n) = -3*A100157(-n).
Sum_{n>0} 1/a(n) = 2*(3 - log(4)) - Pi.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) + 2*sqrt(2)*log(1+sqrt(2)) + (sqrt(2)-1/2)*Pi - 6. - Amiram Eldar, Sep 17 2022

A234319 Smallest sum of n-th powers of k+1 consecutive positive integers that equals the sum of n-th powers of the next k consecutive integers, or -n if none.

Original entry on oeis.org

0, 3, 25, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, -13, -14, -15, -16, -17, -18, -19, -20, -21, -22, -23, -24, -25, -26, -27, -28, -29, -30, -31, -32, -33, -34, -35, -36, -37, -38, -39, -40, -41, -42, -43, -44, -45, -46, -47, -48, -49, -50, -51, -52, -53, -54

Offset: 0

Jonathan Sondow, Dec 23 2013

a(n) is the smallest solution to m^n + (m+1)^n + ... + (m+k)^n = (m+k+1)^n + (m+k+2)^n + ... + (m+2*k)^n, or -n if no solution.
In 1879 Dostor gave all solutions for n = 2. In particular, a(2) = 25.
In 1906 Collignon proved that no solution exists for n = 3 and 4, so a(3) = -3 and a(4) = -4.
In 2013 Felten and Müller-Stach claimed to prove that no solution exists when n > 2, so if their proof is correct, a(n) = -n for n >= 3.

			m^0 + (m+1)^0 + ... + (m+k)^0 = k+1 > k = (m+k+1)^0 + (m+k+2)^0 + ... + (m+2*k)^0 for m > 0, so a(0) = -0 = 0.
1^1 + 2^1 = 3 = 3^1 is minimal for n = 1, so a(1) = 3.
3^2 + 4^2 = 25 = 5^2 is minimal for n = 2, so a(2) = 25.

Edouard Collignon, Note sur la résolution en entiers de m^2 + (m-r)^2 + ... + (m-kr)^2 = (m+r)^2 + ... + (m+kr)^2, Sphinx-Oedipe, 1 (1906-1907), 129-133.

L. E. Dickson, History of the Theory of Numbers, II, p. 564.
Georges Dostor, Question sur les nombres, Archiv der Mathematik und Physik, 64 (1879), 350-352.
Simon Felten and Stefan Müller-Stach, A diophantine equation for sums of consecutive like powers, arXiv:1312.5943 [math.NT], 2013-2015; Elem. Math., 70 (2015), 117-124. doi: 10.4171/EM/284
Greg Frederickson, Casting Light on Cube Dissections, Math. Mag., 82 (2009), 323-331.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A059255, A059270, A222716, A230718.

CoefficientList[Series[x*(27*x^3 - 50*x^2 + 19*x + 3)/(x - 1)^2, {x, 0, 50}], x] (* Wesley Ivan Hurt, Jun 21 2014 *)

Vec(x*(27*x^3-50*x^2+19*x+3)/(x-1)^2 + O(x^100)) \\ Colin Barker, Apr 23 2014

a(0) = A059270(0) = A059255(0).
a(1) = A059270(1) = A230718(1).
a(2) = A059255(2) = A230718(2).
a(n) = -n for n > 2.
G.f.: x*(27*x^3-50*x^2+19*x+3) / (x-1)^2. - Colin Barker, Apr 23 2014

A300758 a(n) = 2n(n+1)(2n+1).

Original entry on oeis.org

0, 12, 60, 168, 360, 660, 1092, 1680, 2448, 3420, 4620, 6072, 7800, 9828, 12180, 14880, 17952, 21420, 25308, 29640, 34440, 39732, 45540, 51888, 58800, 66300, 74412, 83160, 92568, 102660, 113460, 124992, 137280, 150348, 164220, 178920, 194472, 210900, 228228

Offset: 0

Christopher Purcell, Mar 12 2018

The altitude h(n) = a(n)/A001844(n) of the (A005408(n), A046092(n) and A001844(n)) rectangular triangle is an irreducible fraction. - Ralf Steiner, Feb 25 2020

In this case, area A = a(n)/2 = A055112(n). - Bernard Schott, Feb 27 2020

S. P. Borgatti and M. G. Everett, Notions of Position in Social Network Analysis, Sociological Methodology, 22 (1992), 1-35.
C. Purcell and P. Rombach, Role colouring graphs in hereditary classes, arXiv:1802.10180 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A002492, A006331, A055112, A059270, A005408, A046092, A001844, A000384, A016754, A060300.

Cf. A027480.

a(n) = 12*A000330(n).
G.f.: 12*x*(1+x)/(1-x)^4. - Colin Barker, Mar 12 2018
a(n) = 6*A006331(n) = 4*A059270(n) = 3*A002492(n) = 2*A055112(n). - Omar E. Pol, Apr 04 2018
From Ralf Steiner, Feb 27 2020: (Start)
a(n) = 2*n*A000384(n+1).
a(n) = sqrt(A016754(n)*A060300(n)).
(End)
a(n) = A005408(n) * A046092(n). - Bruce J. Nicholson, Apr 24 2020

Edited by N. J. A. Sloane, Aug 01 2019

A343053 Table read by ascending antidiagonals: T(k, n) is the maximum vertex sum in a perimeter-magic k-gon of order n.

Original entry on oeis.org

15, 24, 24, 40, 42, 33, 54, 65, 56, 42, 77, 93, 90, 74, 51, 96, 126, 126, 115, 88, 60, 126, 164, 175, 165, 140, 106, 69, 150, 207, 224, 224, 198, 165, 120, 78, 187, 255, 288, 292, 273, 237, 190, 138, 87, 216, 308, 350, 369, 352, 322, 270, 215, 152, 96, 260, 366, 429, 455, 450, 420, 371, 309, 240, 170, 105

Offset: 3

Stefano Spezia, Apr 03 2021

			The table begins:
k\n|   3    4    5    6    7 ...
---+------------------------
3  |  15   24   33   42   51 ...
4  |  24   42   56   74   88 ...
5  |  40   65   90  115  140 ...
6  |  54   93  126  165  198 ...
7  |  77  126  175  224  273 ...
...

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equations 6 and 8).

Cf. A005475 (n = 4), A022267 (n = 6), A059270, A179805 (k = 3), A343052 (minimum).

T[k_,n_]:=k(1+k(2n-3)-Mod[n,2](1-Mod[k,2]))/2; Table[T[k+3-n,n],{k,3,14},{n,3,k}]//Flatten

T(k, n) = k*(1 + k*(2n - 3) - (n mod 2)*(1 - (k mod 2)))/2.

T(n, n) = A059270(n-1).

A102094 a(n) = (2n-1)(2*n+1)^2.

Original entry on oeis.org

9, 75, 245, 567, 1089, 1859, 2925, 4335, 6137, 8379, 11109, 14375, 18225, 22707, 27869, 33759, 40425, 47915, 56277, 65559, 75809, 87075, 99405, 112847, 127449, 143259, 160325, 178695, 198417, 219539, 242109, 266175, 291785, 318987, 347829, 378359, 410625

Offset: 1

Gerald McGarvey, Feb 13 2005

Numbers which are both the sum of 2n+1 consecutive odd integers and, after skipping one odd integer, the sum of the 2n-1 immediately higher consecutive odd integers. See A082108(n-1) for the smallest of the 2n+1 odd integers, and A054569(n+1) for the skipped number. Odd integer counterpart to A059270. - Charlie Marion, Apr 30 2020

G. Boros and V. Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004, p. 123.
J. Ewell, An Eulerian Method for Representing Pi^2 by Series, The Rocky Mountain Journal of Mathematics 1992 v.22, pp. 165-168.

G. C. Greubel, Table of n, a(n) for n = 1..1000
J. Ewell, An Eulerian Method for Representing Pi^2 by Series, The Rocky Mountain Journal of Mathematics 1992 v.22, pp. 165-168.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002388.

List([1..40], n-> (2*n-1)*(2*n+1)^2); # G. C. Greubel, Oct 27 2019

[(2*n-1)*(2*n+1)^2: n in [1..40]]; // G. C. Greubel, Oct 27 2019

seq((2*n-1)*(2*n+1)^2, n=1..40); # G. C. Greubel, Oct 27 2019

Table[(2n-1)(2n+1)^2,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{9,75,245,567},40] (* Harvey P. Dale, Jul 24 2012 *)

vector(40, n, (2*n-1)*(2*n+1)^2) \\ G. C. Greubel, Oct 27 2019

[(2*n-1)*(2*n+1)^2 for n in (1..40)] # G. C. Greubel, Oct 27 2019

Sum_{n>=1} 1/a(n) = (12 - Pi^2)/16.
Sum_{n>=1} n/a(n) = (Pi^2 - 4)/32. - Sign flipped by Bernard Schott, May 06 2020
From Harvey P. Dale, Jul 24 2012: (Start)
a(1)=9, a(2)=75, a(3)=245, a(4)=567, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: (9 + 39*x - x^2 + x^3)/(1-x)^4. (End)
E.g.f.: 1 + (-1 + 10*x + 28*x^2 + 8*x^3)*exp(x). - G. C. Greubel, Oct 27 2019

More terms from Harvey P. Dale, Jul 24 2012

A181773 Molecular topological indices of the cocktail party graphs.

Original entry on oeis.org

0, 48, 240, 672, 1440, 2640, 4368, 6720, 9792, 13680, 18480, 24288, 31200, 39312, 48720, 59520, 71808, 85680, 101232, 118560, 137760, 158928, 182160, 207552, 235200, 265200, 297648, 332640, 370272, 410640

Offset: 1

Eric W. Weisstein, Jul 10 2011

a(n) is the number of 2 X 2 matrices (all four elements distinct) having entries in {-n,...,0,...,n} with determinant equal to the permanent. - Indranil Ghosh, Dec 25 2016

Eric Weisstein's World of Mathematics, Molecular Topological Index
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A280059 (2 X 2 matrices, elements can be repeated).

A181773:=n->8*(n-1)*n*(2*n-1): seq(A181773(n), n=1..50); # Wesley Ivan Hurt, Apr 11 2017

a(n) = 8*(n-1)*n*(2n-1).
a(n) = 16*A059270(n-1).
G.f.: 48*x^2*(x+1)/(x-1)^4. - Colin Barker, Oct 17 2012
a(n) = 48*A000330(n-1). - R. J. Mathar, Jan 04 2017
From Omar E. Pol, Jan 05 2017: (Start)
a(n) = 24*A006331(n-1) = 12*A002492(n-1) = 8*A055112(n-1).
a(n) = 2*A069074(n-2), n >= 2. (End)

cp's OEIS Frontend

A252117 Irregular triangle read by row: T(n,k), n>=1, k>=1, in which column k lists the numbers of A000716 multiplied by A000330(k), and the first element of column k is in row A000217(k).

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PARI

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A106846 a(n) = Sum_{k + l*m <= n} (k + l*m), with 0 <= k,l,m <= n.

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Maple

Mathematica

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A360849 Array read by antidiagonals: T(m,n) is the number of (undirected) cycles in the complete bipartite graph K_{m,n}.

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A185732 Accumulation array of the polygonal number array (A086270), by antidiagonals.

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Mathematica

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A316224 a(n) = n*(2*n + 1)*(4*n + 1).

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Julia

Magma

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Maxima

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A234319 Smallest sum of n-th powers of k+1 consecutive positive integers that equals the sum of n-th powers of the next k consecutive integers, or -n if none.

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A300758 a(n) = 2n*(n+1)*(2n+1).

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A106846 a(n) = Sum_{k + lm <= n} (k + lm), with 0 <= k,l,m <= n.

A316224 a(n) = n(2n + 1)(4n + 1).

A300758 a(n) = 2n(n+1)(2n+1).

A102094 a(n) = (2n-1)(2*n+1)^2.