A110450 -id:A110450 - cp's OEIS frontend

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

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

A110449 Triangle read by rows: T(n,k) = n((2k+1)*n+1)/2, 0<=k<=n.

Original entry on oeis.org

0, 1, 2, 3, 7, 11, 6, 15, 24, 33, 10, 26, 42, 58, 74, 15, 40, 65, 90, 115, 140, 21, 57, 93, 129, 165, 201, 237, 28, 77, 126, 175, 224, 273, 322, 371, 36, 100, 164, 228, 292, 356, 420, 484, 548, 45, 126, 207, 288, 369, 450, 531, 612, 693, 774, 55, 155, 255, 355, 455, 555, 655, 755, 855, 955, 1055

Offset: 0

Reinhard Zumkeller, Jul 21 2005

Row sums give A110450; central terms give A110451;
T(n,0) = A000217(n);
T(n,1) = A005449(n) for n>0;
T(n,2) = A005475(n) for n>1;
T(n,3) = A022265(n) for n>2;
T(n,4) = A022267(n) for n>3;
T(n,5) = A022269(n) for n>4;
T(n,6) = A022271(n) for n>5;
T(n,7) = A022263(n) for n>6;
T(n+1,n-1) = A059270(n) for n>1;
T(n,n-1) = A081436(n) for n>1;
T(n,n) = A085786(n).

			Triangle starts:
0;
1, 2;
3, 7, 11;
6, 15, 24, 33;
10, 26, 42, 58, 74;
...

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

Cf. A126890.

Table[n*((2*k + 1)*n + 1)/2, {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 23 2017 *)

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(n*((2*k+1)*n+1)/2, ", ");); print(););} \\ Michel Marcus, Jun 22 2015

T(n,k) = n*((2*k + 1)*n + 1)/2, 0 <= k <= n.

A061317 Split positive integers into extending even groups and sum: 1+2, 3+4+5+6, 7+8+9+10+11+12, 13+14+15+16+17+18+19+20, ...

Original entry on oeis.org

0, 3, 18, 57, 132, 255, 438, 693, 1032, 1467, 2010, 2673, 3468, 4407, 5502, 6765, 8208, 9843, 11682, 13737, 16020, 18543, 21318, 24357, 27672, 31275, 35178, 39393, 43932, 48807, 54030, 59613, 65568, 71907, 78642, 85785, 93348, 101343, 109782

Offset: 0

Henry Bottomley, Feb 13 2002

5*a(n+1) is the sum of the products of the 10 distinct combinations of three consecutive numbers starting with n (using 1,2,3 the 10 combinations are 111 112 113 122 123 133 222 223 233 333; 1*1*1 + 1*1*2 + 1*1*3 + 1*2*2 + 1*2*3 + 1*3*3 + 2*2*2 + 2*2*3 + 2*3*3 + 3*3*3 = 90 = 5*a(2)). - J. M. Bergot, Mar 28 2014 [expanded by Jon E. Schoenfield, Feb 22 2015]

			1+2 = 3; 3+4+5+6 = 18; 7+8+9+10+11+12 = 57; 13+14+15+16+17+18+19+20 = 132.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005898, A006003.

Cf. A000217, A002378, A110450.

A061317:=n->2*n^3+n; seq(A061317(n), n=0..100); # Wesley Ivan Hurt, Mar 20 2014

Table[2n^3+n,{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Dec 04 2010 *)
LinearRecurrence[{4,-6,4,-1},{0,3,18,57},40] (* Harvey P. Dale, Aug 23 2015 *)
With[{nn=40},Total/@TakeList[Range[nn+nn^2],2Range[0,nn]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Mar 10 2018 *)

a(n) = { 2*n^3 + n } \\ Harry J. Smith, Jul 21 2009

a(n) = 2*n^3 + n.
a(n) = A000217(A002378(n)) - A000217(A002378(n-1)).
a(n) = 3 * A005900(n).
a(n) = A001477(n) * A058331(n).
a(n) = A000578(n) + A034262(n).
G.f.: 3*x*(1+x)^2/(x-1)^4.
a(n) = A110450(n) - A110450(n-1). - J.S. Seneschal, Jul 01 2025

A085461 Number of 5-tuples (v1,v2,v3,v4,v5) of nonnegative integers less than n such that v1 <= v5, v2 <= v5, v2 <= v4 and v3 <= v4.

Original entry on oeis.org

1, 13, 70, 246, 671, 1547, 3164, 5916, 10317, 17017, 26818, 40690, 59787, 85463, 119288, 163064, 218841, 288933, 375934, 482734, 612535, 768867, 955604, 1176980, 1437605, 1742481, 2097018, 2507050, 2978851, 3519151, 4135152, 4834544

Offset: 1

Goran Kilibarda, Vladeta Jovovic, Jul 01 2003

Number of monotone n-weightings of a certain connected bipartite digraph. A monotone n-(vertex) weighting of a digraph D=(V,E) is a function w: V -> {0,1,...,n-1} such that w(v1) <= w(v2) for every arc (v1,v2) from E.
Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005
Can be constructed by taking the product of the three members of a Pythagorean triples and dividing by 60. Formula: n*(n^2-1)*(n^2+1)/240 where n runs through the odd numbers >= 3. - Pierre Gayet, Apr 04 2009
Number of composable morphisms in a height-n tower of retractions. A retraction between objects X and Y is a pair of maps s:X->Y and r:Y->X such that r(s(x))=x for all x in X. Given objects X_0,X_1,X_2,...,X_n, we can ask for retractions s_i:X_i->X_{i+1},r_i:X_{i+1}->X_i, for each 0 <= i < n. The total number of morphisms in that category is 0^2 + 1^2 + 2^2 + ... + n^2 (cf. A000330). The total number of composable pairs of morphisms in that category is the sequence given here. - David Spivak, Feb 26 2014

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 168).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Daeseok Lee and H.-K. Ju, An Extension of Hibi's palindromic theorem, arXiv preprint arXiv:1503.05658 [math.CO], 2015.
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 (6,-15,20,-15,6,-1).

Cf. A006322, A006325, A079547, A085462, A085463, A085464, A085465.

Rest[CoefficientList[Series[x*(1 + x)*(1 + 6*x + x^2)/(1 - x)^6, {x, 0, 50}], x]] (* G. C. Greubel, Oct 06 2017 *)

x='x+O('x^50); Vec(x*(1+x)*(1+6*x+x^2)/(1-x)^6) \\ G. C. Greubel, Oct 06 2017

a(n) = n + 11*binomial(n, 2) + 34*binomial(n, 3) + 40*binomial(n, 4) + 16*binomial(n, 5) = 1/30*n*(n+1)*(2*n+1)*(2*n^2 + 2*n + 1).
From Bruno Berselli, Dec 27 2010: (Start)
G.f.: x*(1+x)*(1+6*x+x^2)/(1-x)^6.
a(n) = ( n*A110450(n) - Sum_{i=0..n-1} A110450(i) )/3. (End)

A342719 Array read by ascending antidiagonals: T(k, n) is the sum of the consecutive positive integers from 1 to (n - 1)*k placed along the perimeter of an n-th order perimeter-magic k-gon.

Original entry on oeis.org

21, 36, 45, 55, 78, 78, 78, 120, 136, 120, 105, 171, 210, 210, 171, 136, 231, 300, 325, 300, 231, 171, 300, 406, 465, 465, 406, 300, 210, 378, 528, 630, 666, 630, 528, 378, 253, 465, 666, 820, 903, 903, 820, 666, 465, 300, 561, 820, 1035, 1176, 1225, 1176, 1035, 820, 561

Offset: 3

Stefano Spezia, Mar 19 2021

			The array begins:
k\n|   3    4    5    6    7 ...
---+------------------------
3  |  21   45   78  120  171 ...
4  |  36   78  136  210  300 ...
5  |  55  120  210  325  465 ...
6  |  78  171  300  465  666 ...
7  | 105  231  406  630  903 ...
...

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equation 3).

Cf. A014105 (n = 3), A033585 (n = 5), A037270 (1st superdiagonal), A081266 (n = 4), A083374 (1st subdiagonal), A110450 (diagonal), A144312 (n = 6), A144314 (n = 7), A342757, A342758.

T[k_,n_]:=(n-1)k((n-1)k+1)/2; Table[T[k+3-n,n],{k,3,12},{n,3,k}]//Flatten

O.g.f.: (x^2 - 3*x^2*y + x*y^2 + 3*x^2*y^2)/((1 - x)^3*(1 - y)^3).
E.g.f.: exp(x+y)*x*(x - x*y + y^2 + x*y^2)/2.
T(k, n) = (n - 1)*k*((n - 1)*k + 1)/2.

A216154 Triangle read by rows, T(n,k) n>=0, k>=0, generalization of A000255.

Original entry on oeis.org

1, 1, 1, 3, 4, 1, 11, 21, 9, 1, 53, 128, 78, 16, 1, 309, 905, 710, 210, 25, 1, 2119, 7284, 6975, 2680, 465, 36, 1, 16687, 65821, 74319, 35035, 7945, 903, 49, 1, 148329, 660064, 857836, 478464, 133630, 19936, 1596, 64, 1, 1468457, 7275537, 10690812, 6879684, 2279214, 419958, 44268, 2628, 81, 1

Offset: 0

Peter Luschny, Sep 19 2012

			     1,
     1,      1,
     3,      4,      1,
    11,     21,      9,      1,
    53,    128,     78,     16,      1,
   309,    905,    710,    210,     25,      1,
  2119,   7284,   6975,   2680,    465,     36,      1,
16687,  65821,  74319,  35035,   7945,    903,     49,      1,
148329, 660064, 857836, 478464, 133630,  19936,   1596,     64,      1,

A000255 (col. 0), A110450 (diag. n,n-2).

Cf. A111596, A271703.

A216154 := proc(n,k) local L, Z;
L := (n,k) -> `if`(k<0 or k>n,0,(n-k)!*C(n,n-k)*C(n-1,n-k)):
Z := (n,k) -> add(C(-j,-n)*L(j,k), j=0..n);
Z(n+1, k+1) end:
seq(seq(A216154(n,k), k=0..n), n=0..9); # Peter Luschny, Apr 13 2016

T[0, 0] = 1; T[0, ] = 0; T[n, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k-1] + (2k+1) T[n-1, k] + (k+1) (k+2) T[n-1, k+1]; T[, ] = 0;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 02 2019 *)

def A216154_triangle(dim):
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+(1+2*k)*M[n-1,k]+(k+1)*(k+2)*M[n-1,k+1]
    return M
A216154_triangle(9)

Recurrence: T(0,0)=1, T(0,k)=0 for k>0 and for n>=1 T(n,k) = T(n-1,k-1)+(1+2*k)*T(n-1,k)+(k+1)*(k+2)*T(n-1,k+1).

Let Z(n, k) = Sum_{j=0..n} C(-j, -n)*L(j, k) where L denotes the unsigned Lah numbers A271703. Then T(n, k) = Z(n+1, k+1). - Peter Luschny, Apr 13 2016

cp's OEIS Frontend

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

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A110449 Triangle read by rows: T(n,k) = n*((2*k+1)*n+1)/2, 0<=k<=n.

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A061317 Split positive integers into extending even groups and sum: 1+2, 3+4+5+6, 7+8+9+10+11+12, 13+14+15+16+17+18+19+20, ...

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A085461 Number of 5-tuples (v1,v2,v3,v4,v5) of nonnegative integers less than n such that v1 <= v5, v2 <= v5, v2 <= v4 and v3 <= v4.

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A342719 Array read by ascending antidiagonals: T(k, n) is the sum of the consecutive positive integers from 1 to (n - 1)*k placed along the perimeter of an n-th order perimeter-magic k-gon.

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A216154 Triangle read by rows, T(n,k) n>=0, k>=0, generalization of A000255.

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A110449 Triangle read by rows: T(n,k) = n((2k+1)*n+1)/2, 0<=k<=n.