A002412 -id:A002412 - cp's OEIS frontend

A002492 Sum of the first n even squares: 2n(n+1)(2n+1)/3.

0, 4, 20, 56, 120, 220, 364, 560, 816, 1140, 1540, 2024, 2600, 3276, 4060, 4960, 5984, 7140, 8436, 9880, 11480, 13244, 15180, 17296, 19600, 22100, 24804, 27720, 30856, 34220, 37820, 41664, 45760, 50116, 54740, 59640, 64824, 70300, 76076, 82160

Offset: 0

N. J. A. Sloane

Total number of possible bishop moves on an n+1 X n+1 chessboard, if the bishop is placed anywhere. E.g., on a 3 X 3-Board: bishop has 8 X 2 moves and 1 X 4 moves, so a(2)=20. - Ulrich Schimke (ulrschimke(AT)aol.com)
Let M_n denote the n X n matrix M_n(i,j)=(i+j)^2; then the characteristic polynomial of M_n is x^n - a(n)x^(n-1) - .... - Michael Somos, Nov 14 2002
Partial sums of A016742. - Lekraj Beedassy, Jun 19 2004
0,4,20,56,120 gives the number of electrons in closed shells in the double shell periodic system of elements. This is a new interpretation of the periodic system of the elements. The factor 4 in the formula 4*n(n+1)(2n+1)/6 plays a significant role, since it designates the degeneracy of electronic states in this system. Closed shells with more than 120 electrons are not expected to exist. - Karl-Dietrich Neubert (kdn(AT)neubert.net)
Inverse binomial transform of A240434. - Wesley Ivan Hurt, Apr 13 2014
Atomic number of alkaline-earth metals of period 2n. - Natan Arie Consigli, Jul 03 2016
a(n) are the negative cubic coefficients in the expansion of sin(kx) into powers of sin(x) for the odd k: sin(kx) = k sin(x) - c(k) sin^3(x) + O(sin^5(x)); a(n) = c(2n+1) = A000292(2n). - Mathias Zechmeister, Jul 24 2022
Also the number of distinct series-parallel networks under series-parallel reduction on three unlabeled edges of n element kinds. - Michael R. Hayashi, Aug 02 2023

A. O. Barut, Group Structure of the Periodic System, in Wybourne, Ed., The Structure of Matter, University of Canterbury Press, Christchurch, 1972, p. 126.
Edward G. Mazur, Graphic Representation of the Periodic System during One Hundred Years, University of Alabama Press, Alabama, 1974.
W. Permans and J. Kemperman, "Nummeringspribleem van S. Dockx, Mathematisch Centrum. Amsterdam," Rapport ZW; 1949-005, 4 leaves, 19.8 X 34 cm.
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).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jose Manuel Garcia Calcines, Luis Javier Hernandez Paricio, and Maria Teresa Rivas Rodriguez, Semi-simplicial combinatorics of cyclinders and subdivisions, arXiv:2307.13749 [math.CO], 2023. See p. 32.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article #14.3.5.
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
D. Neubert, Double Shell Structure of the Periodic System of the Elements, Z. Naturforschung, 25A (1970), p. 210.
Karl-Dietrich Neubert, Double-Shell PSE: Metals - Nonmetals.
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.
D. Suprijanto and Rusliansyah, Observation on Sums of Powers of Integers Divisible by Four, Applied Mathematical Sciences, Vol. 8, No. 45 (2014), pp. 2219-2226.
Index entries for sequences related to Chebyshev polynomials.
Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A000330, A005408, A006331, A053120, A081277.
Cf. A033586 (King), A035005 (Queen), A035006 (Rook), A035008 (Knight) and A049450 (Pawn).
Cf. A002412, A016061.

[2*n*(n+1)*(2*n+1)/3: n in [0..40]]; // Vincenzo Librandi, Jun 16 2011

A002492:=n->2*n*(n+1)*(2*n+1)/3; seq(A002492(n), n=0..50); # Wesley Ivan Hurt, Apr 04 2014

Table[2n(n+1)(2n+1)/3, {n,0,40}] (* or *) Binomial[2*Range[0,40]+2,3] (* or *) LinearRecurrence[{4,-6,4,-1}, {0,4,20,56},40] (* Harvey P. Dale, Aug 15 2012 *)
Accumulate[(2*Range[0,40])^2] (* Harvey P. Dale, Jun 04 2019 *)

PARI
```
a(n)=2*n*(n+1)*(2*n+1)/3
```

G.f.: 4*x*(1+x)/(1-x)^4. - Simon Plouffe in his 1992 dissertation
a(-1-n) = -a(n).
a(n) = 4*A000330(n) = 2*A006331(n) = A000292(2*n).
a(n) = (-1)^(n+1)*A053120(2*n+1,3) (fourth unsigned column of Chebyshev T-triangle, zeros omitted).
a(n) = binomial(2*n+2, 3). - Lekraj Beedassy, Jun 19 2004
A035005(n+1) = a(n) + A035006(n+1) since Queen = Bishop + Rook. - Johannes W. Meijer, Feb 04 2010
a(n) - a(n-1) = 4*n^2. - Joerg Arndt, Jun 16 2011
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3. - Harvey P. Dale, Aug 15 2012
a(n) = Sum_{k=0..3} C(n-2+k,n-2)*C(n+3-k,n), for n>2. - J. M. Bergot, Jun 14 2014
a(n) = 2*A006331(n). - R. J. Mathar, May 28 2016
From Natan Arie Consigli Jul 03 2016: (Start)
a(n) = A166464(n) - 1.
a(n) = A168380(2*n). (End)
a(n) = Sum_{i=0..n} A005408(i)*A005408(i-1)+1 with A005408(-1):=-1. - Bruno Berselli, Jan 09 2017
a(n) = A002412(n) + A016061(n). - Bruce J. Nicholson, Nov 12 2017
From Amiram Eldar, Jan 04 2022: (Start)
Sum_{n>=1} 1/a(n) = 9/2 - 6*log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*Pi/2 - 9/2. (End)
a(n) = A081277(3, n-1) = (1+2*n)*binomial(n+1, n-2)*2^2/(n-1) for n > 0. - Mathias Zechmeister, Jul 26 2022
E.g.f.: 2*exp(x)*x*(6 + 9*x + 2*x^2)/3. - Stefano Spezia, Jul 31 2022

Minor errors corrected and edited by Johannes W. Meijer, Feb 04 2010

Title modified by Charles R Greathouse IV at the suggestion of J. M. Bergot, Apr 05 2014

A015084 Carlitz-Riordan q-Catalan numbers for q=3.

Original entry on oeis.org

1, 1, 4, 43, 1252, 104098, 25511272, 18649337311, 40823535032644, 267924955577741566, 5274102955963545775864, 311441054994969341088610030, 55171471477692117486494217498280

Offset: 0

Olivier Gérard

nonn

Limit_{n->inf} a(n)/3^((n-1)(n-2)/2) = Product{k>=1} 1/(1-1/3^k) = 1.785312341998534190367486296013703535718796... - Paul D. Hanna, Jan 24 2005
It appears that the Hankel transform is 3^A002412(n). - Paul Barry, Aug 01 2008
Hankel transform of the aerated sequence is 3^C(n+1,3). - Paul Barry, Oct 31 2008

			G.f. = 1 + x + 4*x^2 + 43*x^3 + 1252*x^4 + 104098*x^5 + 25511272*x^6 + ...
From _Seiichi Manyama_, Dec 05 2016: (Start)
a(1) = 1,
a(2) = 3^1 + 1 = 4,
a(3) = 3^3 + 3^2 + 2*3^1 + 1 = 43,
a(4) = 3^6 + 3^5 + 2*3^4 + 3*3^3 + 3*3^2 + 3*3^1 + 1 = 1252. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..65
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Thesis, University of Vienna, 2013.

Cf. A227543.
Cf. A015108 (q=-11), A015107 (q=-10), A015106 (q=-9), A015105 (q=-8), A015103 (q=-7), A015102 (q=-6), A015100 (q=-5), A015099 (q=-4), A015098 (q=-3), A015097 (q=-2), A090192 (q=-1), A000108 (q=1), A015083 (q=2), this sequence (q=3), A015085 (q=4), A015086 (q=5), A015089 (q=6), A015091 (q=7), A015092 (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).
Column k=3 of A090182, A290759.

A015084 := proc(n)
    option remember;
    if n = 1 then
        1;
    else
    add(3^(i-1)*procname(i)*procname(n-i),i=1..n-1) ;
    end if;
end proc: # R. J. Mathar, Sep 29 2012

a[n_] := a[n] = Sum[3^i*a[i]*a[n -i -1], {i, 0, n -1}]; a[0] = 1; Array[a, 16, 0] (* Robert G. Wilson v, Dec 24 2016 *)
m = 13; ContinuedFractionK[If[i == 1, 1, -3^(i-2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 17 2019 *)

a(n)=if(n==1,1,sum(i=1,n-1,3^(i-1)*a(i)*a(n-i))) \\ Paul D. Hanna

def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
  ary
end
def A015084(n)
  A(3, n)
end # Seiichi Manyama, Dec 24 2016

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=3 and a(0)=1.
G.f. satisfies: A(x) = 1/(1-x*A(3*x)) = 1/(1-x/(1-3*x/(1-3^2*x/(1-3^3*x/(1-...))))) (continued fraction). - Paul D. Hanna, Jan 24 2005
a(n) = the upper left term in M^n, M an infinite production matrix as follows:
  1,  3,  0,  0,  0,  0, ...
  1,  3,  9,  0,  0,  0, ...
  1,  3,  9, 27,  0,  0, ...
  1,  3,  9, 27, 81,  0, ...
  ... - Gary W. Adamson, Jul 14 2011
G.f.: T(0), where T(k) = 1 - x*3^k/(x*3^k - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 18 2013

More terms from Paul D. Hanna, Jan 24 2005

Offset changed to 0 by Seiichi Manyama, Dec 05 2016

A080851 Square array of pyramidal numbers, read by antidiagonals.

Original entry on oeis.org

1, 1, 3, 1, 4, 6, 1, 5, 10, 10, 1, 6, 14, 20, 15, 1, 7, 18, 30, 35, 21, 1, 8, 22, 40, 55, 56, 28, 1, 9, 26, 50, 75, 91, 84, 36, 1, 10, 30, 60, 95, 126, 140, 120, 45, 1, 11, 34, 70, 115, 161, 196, 204, 165, 55, 1, 12, 38, 80, 135, 196, 252, 288, 285, 220, 66, 1, 13, 42, 90, 155, 231, 308, 372, 405, 385, 286, 78

Offset: 0

Paul Barry, Feb 21 2003

The first row contains the triangular numbers, which are really two-dimensional, but can be regarded as degenerate pyramidal numbers. - N. J. A. Sloane, Aug 28 2015

			Array begins (n>=0, k>=0):
1,  3,  6, 10,  15,  21,  28,  36,  45,   55, ... A000217
1,  4, 10, 20,  35,  56,  84, 120, 165,  220, ... A000292
1,  5, 14, 30,  55,  91, 140, 204, 285,  385, ... A000330
1,  6, 18, 40,  75, 126, 196, 288, 405,  550, ... A002411
1,  7, 22, 50,  95, 161, 252, 372, 525,  715, ... A002412
1,  8, 26, 60, 115, 196, 308, 456, 645,  880, ... A002413
1,  9, 30, 70, 135, 231, 364, 540, 765, 1045, ... A002414
1, 10, 34, 80, 155, 266, 420, 624, 885, 1210, ... A007584

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Numerous sequences in the database are to be found in the array. Rows include the pyramidal numbers A000217, A000292, A000330, A002411, A002412, A002413, A002414, A007584, A007585, A007586.
Columns include or are closely related to A017029, A017113, A017017, A017101, A016777, A017305. Diagonals include A006325, A006484, A002417.
Cf. A057145, A027660 (antidiagonal sums).
See A257199 for another version of this array.

vector(vector(poly_coeff(Taylor((1+kx)/(1-x)^4,x,11),x,n),n,0,11),k,-1,10) VECTOR(VECTOR(comb(k+2,2)+comb(k+2,3)n, k, 0, 11), n, 0, 11)

A080851 := proc(n,k)
    binomial(k+3,3)+(n-1)*binomial(k+2,3) ;
end proc:
seq( seq(A080851(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Oct 01 2021

pyramidalFigurative[ ngon_, rank_] := (3 rank^2 + rank^3 (ngon - 2) - rank (ngon - 5))/6; Table[ pyramidalFigurative[n-k-1, k], {n, 4, 15}, {k, n-3}] // Flatten (* Robert G. Wilson v, Sep 15 2015 *)

T(n, k) = binomial(k+3, 3) + (n-1)*binomial(k+2, 3), corrected Oct 01 2021.
T(n, k) = T(n-1, k) + C(k+2, 3) = T(n-1, k) + k*(k+1)*(k+2)/6.
G.f. for rows: (1 + n*x)/(1-x)^4, n>=-1.
T(n,k) = sum_{j=1..k+1} A057145(n+2,j). - R. J. Mathar, Jul 28 2016

A016061 a(n) = n(n+1)(4*n+5)/6.

Original entry on oeis.org

0, 3, 13, 34, 70, 125, 203, 308, 444, 615, 825, 1078, 1378, 1729, 2135, 2600, 3128, 3723, 4389, 5130, 5950, 6853, 7843, 8924, 10100, 11375, 12753, 14238, 15834, 17545, 19375, 21328, 23408, 25619, 27965, 30450, 33078, 35853, 38779, 41860

Offset: 0

Robert G. Wilson v

Number of ZnS molecules in cluster of n layers in zinc blende crystal.
(Zinc sulfide crystallizes in two different forms: wurtzite and zinc blende, the latter is also spelled zincblende.) - Jonathan Vos Post, Jan 22 2013
The Kn4 triangle sums of the Connell-Pol triangle A159797 lead to the sequence given above. For the definitions of the Kn4 and other triangle sums see A180662. - Johannes W. Meijer, May 20 2011
If one generated primitive Pythagorean triangles (2n+1, 2n+3) the collective sum of their perimeters for each n is four times the numbers listed in this sequence. - J. M. Bergot, Jul 18 2011
a(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and nA000292(n)+A000292(n+1)=n^3. - Clark Kimberling, Jun 04 2012
Degrees of the Hilbert polynomials for B_3 and C_3, per p. 13 of Gashi et al. - Jonathan Vos Post, Dec 14 2013
Number of solutions to a + b = c + d when 0 < a <= k, 0 <= b, c, d <= k, k = 0, 1, 2, 3.... Taken from Step 1 2007 problem #1(i) using 4 digit balanced numbers. - Bobby Milazzo, Mar 09 2013
From J. M. Bergot, Jun 18 2013: (Start)
Consider the lower half, including the main diagonal, of the array in A144216 as a triangle.  The rows begin:
 0;
 1, 2;
 3, 4, 6;
 6, 7, 9, 12, ...
The sum of the terms in row(n) is a(n). (End)
This sequence is related to A008865 by a(n) = n*A008865(n+1) - Sum_{i=1..n} A008865(i) for n>0. - Bruno Berselli, Aug 06 2015

P. Jena and S. N. Behera, Clusters and Nanostructured Materials, Nova Science Publishers, 1996.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
David N. Blauch, Crystal Structure of Zinc Blende.
Qëndrim R. Gashi, Travis Schedler and David E. Speyer, Looping of the numbers game and the alcoved hypercube, arXiv:0909.5324v1 [math.RT], 2009.
Johan Kok, Introduction to total chromatic vertex stress of graphs, Open J. Disc. Appl. Math. (ODAM, 2023) Vol. 6, No. 2, 32-38. See p. 34.
T. P. Martin, Shells of atoms, Phys. Reports, Vol. 273 (1996), pp. 199-241, see p. 233.
Gloria Olive, Problem #504, Factorizations and Sums, Two-Year College Math. Jnl., Vol. 25 (1994), pp. 244-245.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Bisection of A002623.
Cf. A000292, A000330, A002412, A008865, A014105, A144216, A159797, A180662.
Row sums of triangle A120070.

I:=[0,3,13,34]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 25 2013

A016061 := proc(n)
    n*(n+1)*(4*n+5)/6 ;
end proc: # R. J. Mathar, Sep 26 2013

CoefficientList[Series[x (3 + x) / (1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 25 2013 *)
Table[n(n+1)(4*n+5)/6, {n,0,100}] (* Wesley Ivan Hurt, Sep 25 2013 *)

v=vector(40,i,t(i)); s=0; forstep(i=2,40,2,s+=v[i]; print1(s","))

G.f.: x*(3+x)/(1-x)^4. - Paul Barry, Feb 27 2003
Partial sums of A014105. - Jon Perry, Jul 23 2003
a(n) = Sum_{i=0..n-1} 2*i^2 + i. - Jani Nurminen (slinky(AT)iki.fi), May 14 2006
a(n) = 2*n^3/3  +3*n^2/2 + 5*n/6. - Jonathan Vos Post, Dec 14 2013
a(n) = (4*n+5)/(2*n+1)*A000330(n). - Alexander R. Povolotsky, Mar 09 2013
a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - Bobby Milazzo, Mar 10 2013
Sum_{n>=1} 1/a(n) = 12*Pi/5 + 72*log(2)/5 - 426/25. - Amiram Eldar, Jan 04 2022
E.g.f.: exp(x)*x*(18 + 21*x + 4*x^2)/6. - Stefano Spezia, Jul 31 2022

A237616 a(n) = n(n + 1)(5*n - 4)/2.

Original entry on oeis.org

0, 1, 18, 66, 160, 315, 546, 868, 1296, 1845, 2530, 3366, 4368, 5551, 6930, 8520, 10336, 12393, 14706, 17290, 20160, 23331, 26818, 30636, 34800, 39325, 44226, 49518, 55216, 61335, 67890, 74896, 82368, 90321, 98770, 107730, 117216, 127243, 137826, 148980, 160720

Offset: 0

Bruno Berselli, Feb 10 2014

Also 17-gonal (or heptadecagonal) pyramidal numbers.

This sequence is related to A226489 by 2*a(n) = n*A226489(n) - Sum_{i=0..n-1} A226489(i).

			After 0, the sequence is provided by the row sums of the triangle:
   1;
   2,  16;
   3,  32,  31;
   4,  48,  62,  46;
   5,  64,  93,  92,  61;
   6,  80, 124, 138, 122,  76;
   7,  96, 155, 184, 183, 152,  91;
   8, 112, 186, 230, 244, 228, 182, 106;
   9, 128, 217, 276, 305, 304, 273, 212, 121;
  10, 144, 248, 322, 366, 380, 364, 318, 242, 136; etc.,
where (r = row index, c = column index):
T(r,r) = T(c,c) = 15*r-14 and T(r,c) = T(r-1,c)+T(r,r) = (r-c+1)*T(r,r), with r>=c>0.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (fifteenth row of the table).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pyramidal Number.
Index to sequences related to pyramidal numbers.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A051869, A104728.

Cf. sequences with formula n*(n+1)*(k*n-k+3)/6: A000217 (k=0), A000292 (k=1), A000330 (k=2), A002411 (k=3), A002412 (k=4), A002413 (k=5), A002414 (k=6), A007584 (k=7), A007585 (k=8), A007586 (k=9), A007587 (k=10), A050441 (k=11), A172073 (k=12), A177890 (k=13), A172076 (k=14), this sequence (k=15), A172078(k=16), A237617 (k=17), A172082 (k=18), A237618 (k=19), A172117(k=20), A256718 (k=21), A256716 (k=22), A256645 (k=23), A256646(k=24), A256647 (k=25), A256648 (k=26), A256649 (k=27), A256650(k=28).

List([0..40], n-> n*(n+1)*(5*n-4)/2); # G. C. Greubel, Aug 30 2019

Magma
```
[n*(n+1)*(5*n-4)/2: n in [0..40]];
```

I:=[0,1,18,66]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Feb 12 2014

seq(n*(n+1)*(5*n-4)/2, n=0..40); # G. C. Greubel, Aug 30 2019

Table[n(n+1)(5n-4)/2, {n, 0, 40}]
CoefficientList[Series[x (1+14x)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
LinearRecurrence[{4,-6,4,-1},{0,1,18,66},50] (* Harvey P. Dale, Jan 11 2015 *)

a(n)=n*(n+1)*(5*n-4)/2 \\ Charles R Greathouse IV, Sep 24 2015

[n*(n+1)*(5*n-4)/2 for n in (0..40)] # G. C. Greubel, Aug 30 2019

G.f.: x*(1 + 14*x)/(1 - x)^4.
For n>0, a(n) = Sum_{i=0..n-1} (n-i)*(15*i+1). More generally, the sequence with the closed form n*(n+1)*(k*n-k+3)/6 is also given by Sum_{i=0..n-1} (n-i)*(k*i+1) for n>0.
a(n) = A104728(A001844(n-1)) for n>0.
Sum_{n>=1} 1/a(n) = (2*sqrt(5*(5 + 2*sqrt(5)))*Pi + 10*sqrt(5)*arccoth(sqrt(5)) + 25*log(5) - 16)/72 = 1.086617842136293176... . - Vaclav Kotesovec, Dec 07 2016
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. - Wesley Ivan Hurt, Dec 18 2020
E.g.f.: exp(x)*x*(2 + 16*x + 5*x^2)/2. - Elmo R. Oliveira, Aug 04 2025

A004188 a(n) = n(3n^2 - 1)/2.

Original entry on oeis.org

0, 1, 11, 39, 94, 185, 321, 511, 764, 1089, 1495, 1991, 2586, 3289, 4109, 5055, 6136, 7361, 8739, 10279, 11990, 13881, 15961, 18239, 20724, 23425, 26351, 29511, 32914, 36569, 40485, 44671, 49136, 53889, 58939, 64295, 69966, 75961

Offset: 0

Albert D. Rich (Albert_Rich(AT)msn.com)

3-dimensional analog of centered polygonal numbers.

(1), (4+7), (10+13+16), (19+22+25+28), ... - Jon Perry, Sep 10 2004

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
Cf. A002412, A016061, A051682.
Cf. A236770 (partial sums).

[n*(3*n^2-1)/2: n in [0..50]]; //Vincenzo Librandi, May 15 2011

seq(binomial(2*n+1,3)+binomial(n+1,3), n=0..37); # Zerinvary Lajos, Jan 21 2007

Table[n(3n^2-1)/2,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{4,-6,4,-1},{0,1,11,39},40] (* Harvey P. Dale, Jul 19 2019 *)

PARI
```
vector(40, n, n*(3*n^2-1)/2)
```

Partial sums of n-1 3-spaced triangular numbers, e.g., a(4) = t(1) + t(4) + t(7) = 1 + 10 + 28 = 39. - Jon Perry, Jul 23 2003
a(n) = C(2*n+1,3) + C(n+1,3), n >= 0. - Zerinvary Lajos, Jan 21 2007
a(n) = A000447(n) + A000292(n). - Zerinvary Lajos, Jan 21 2007
G.f.: x*(1+7*x+x^2) / (x-1)^4. - R. J. Mathar, Oct 08 2011
From Miquel Cerda, Dec 25 2016: (Start)
a(n) = A000578(n) + A135503(n).
a(n) = A007588(n) - A135503(n). (End)
E.g.f.: (x/2)*(2 + 9*x + 3*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

A002418 4-dimensional figurate numbers: a(n) = (5n-1)binomial(n+2,3)/4.

Original entry on oeis.org

0, 1, 9, 35, 95, 210, 406, 714, 1170, 1815, 2695, 3861, 5369, 7280, 9660, 12580, 16116, 20349, 25365, 31255, 38115, 46046, 55154, 65550, 77350, 90675, 105651, 122409, 141085, 161820, 184760, 210056, 237864, 268345, 301665, 337995

Offset: 0

N. J. A. Sloane

Partial sums of A002413.
Principal diagonal of the convolution array A213550, for n>0. - Clark Kimberling, Jun 17 2012
Convolution of A000027 with A000566. - Bruno Berselli, Dec 06 2012
Coefficients in the hypergeometric series identity 1 - 9*(x - 1)/(4*x + 1) + 35*(x - 1)*(x - 2)/((4*x + 1)*(4*x + 2)) - 95*(x - 1)*(x - 2)*(x - 3)/((4*x + 1)*(4*x + 2)*(4*x + 3)) + ... = 0, valid for Re(x) > 1. Cf. A000326 and A002412. Column 4 of A103450. - Peter Bala, Mar 14 2019

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.
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).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5, 1).
Index to sequences related to pyramidal numbers.

Cf. A093562 ((5, 1) Pascal, column m=4).
Cf. A000332, A000566, A001622, A080852, A128064.
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.

List([0..40],n->(5*n-1)*Binomial(n+2,3)/4); # Muniru A Asiru, Mar 18 2019

[(5*n - 1)*Binomial(n + 2, 3)/4: n in [0..40]]; // Vincenzo Librandi, Oct 17 2012

/* A000027 convolved with A000566: */ A000566:=func; [&+[(n-i+1)*A000566(i): i in [0..n]]: n in [0..35]]; // Bruno Berselli, Dec 06 2012

Table[(5n-1) Binomial[n+2,3]/4,{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{0,1,9,35,95},40] (* Harvey P. Dale, Oct 16 2012 *)
CoefficientList[Series[x*(1 + 4*x)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 17 2012 *)

a(n)=(5*n-1)*binomial(n+2,3)/4 \\ Charles R Greathouse IV, Sep 24 2015

[(5*n-1)*binomial(n+2,3)/4 for n in (0..40)] # G. C. Greubel, Jul 03 2019

G.f.: x*(1+4*x)/(1-x)^5. - Simon Plouffe in his 1992 dissertation.
Starting (1, 9, 35, 95, ...), = A128064 * A000332, (A000332 starting 1, 5, 15, 35, 70, ...), such that a(n) = n*C(n+3,4) - (n-1)*C(n+2,4). E.g., a(5) = 210 = 5*C(8,4) - 4*C(7,4) = 5*70 - 4*35. - Gary W. Adamson, Dec 28 2007
Unit digit, A010879(a(n)), is one of {0,1,9,5,6,4} [Eric Desbiaux] because a(n) mod 5 = 0,1,4,0,0, periodic with period 5. [Proof: A002413(n) mod 5 = 1,3,1,0,0 with period 5 and a(n) are the partial sums of A002413.] - R. J. Mathar, Mar 19 2008
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Oct 16 2012
a(n) = A080852(5,n-1). - R. J. Mathar, Jul 28 2016
a(n) = Sum_{i=0..n} (n-i) * Sum_{j=i..n} j. - J. M. Bergot, May 30 2017
E.g.f.: x*(24 + 84*x + 44*x^2 + 5*x^3)*exp(x)/4!. - G. C. Greubel, Jul 03 2019
Sum_{n>=1} 1/a(n) = (50*sqrt(5)*log(phi) + 125*log(5) - 50*sqrt(1+2/sqrt(5))*Pi - 26)/11, where phi is the golden ratio (A001622). - Amiram Eldar, Feb 11 2022

A211790 Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and w^k

Original entry on oeis.org

1, 7, 1, 23, 7, 1, 54, 22, 7, 1, 105, 51, 22, 7, 1, 181, 97, 50, 22, 7, 1, 287, 166, 96, 50, 22, 7, 1, 428, 263, 163, 95, 50, 22, 7, 1, 609, 391, 255, 161, 95, 50, 22, 7, 1, 835, 554, 378, 253, 161, 95, 50, 22, 7, 1, 1111, 756, 534, 374, 252, 161, 95, 50, 22, 7

Offset: 1

Clark Kimberling, Apr 21 2012

...
Let R be the array in A211790 and let R' be the array in A211793.  Then R(k,n) + R'(k,n) = 3^(n-1).  Moreover, (row k of R) =(row k of A211796) for k>2, by Fermat's last theorem; likewise, (row k of R')=(row k of A211799) for k>2.
...
Generalizations:  Suppose that b,c,d are nonzero integers, and let U(k,n) be the number of ordered triples (w,x,y) with all terms in {1,...,n} and b*w*k  c*x^k+d*y^k, where the relation  is one of these: <, >=, <=, >.  What additional assumptions force the limiting row sequence to be essentially one of these: A002412, A000330, A016061, A174723, A051925?
In the following guide to related arrays and sequences, U(k,n) denotes the number of (w,x,y) as described in the preceding paragraph:
                                 first 3 rows          limiting row sequence
A211790:  w^k < x^k+y^k    A004068, A211635, A211650       A002412
A211793:  w^k >= x^k+y^k   A000292, A211636, A211651       A000330
A211796:  w^k <= x^k+y^k   A002413, A211634, A211650       A002412
A211799:  w^k > x^k+y^k    A000292, A211637, A211651       A000330
A211802:  2w^k < x^k+y^k   A182260, A211800, A211801       A016061
A211805:  2w^k >= x^k+y^k  A055232, A211803, A211804       A000330
A211808:  2w^k <= x^k+y^k  A055232, A211806, A211807       A174723
A182259:  2w^k > x^k+y^k   A182260, A211810, A211811       A051925

			Northwest corner:
  1, 7, 23, 54, 105, 181, 287, 428, 609
  1, 7, 22, 51,  97, 166, 263, 391, 554
  1, 7, 22, 50,  96, 163, 255, 378, 534
  1, 7, 22, 50,  95, 161, 253, 374, 528
  1, 7, 22, 50,  95, 161, 252, 373, 527
For n=2 and k>=1, the 7 triples (w,x,y) are (1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,2), (2,2,1), (2,2,2).

Cf. A004068 (row1), A211635 (row2), A211650 (row3), A002412.

Cf. A211793, A211796, A211799, A211802, A211805, A211808, A182259

z = 48;
t[k_, n_] := Module[{s = 0},
   (Do[If[w^k < x^k + y^k, s = s + 1],
       {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)];
Table[t[1, n], {n, 1, z}]  (* A004068 *)
Table[t[2, n], {n, 1, z}]  (* A211635 *)
Table[t[3, n], {n, 1, z}]  (* A211650 *)
TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]] (* A211790 *)
Table[n (n + 1) (4 n - 1)/6,
  {n, 1, z}] (* row-limit sequence, A002412 *)
(* Peter J. C. Moses, Apr 13 2012 *)

R(k,n) = n(n-1)(4n+1)/6 for 1<=k<=n, and

R(k,n) = Sum{Sum{floor[(x^k+y^k)^(1/k)] : 1<=x<=n, 1<=y<=n}} for 1<=k<=n.

A094728 Triangle read by rows: T(n,k) = n^2 - k^2, 0 <= k < n.

Original entry on oeis.org

1, 4, 3, 9, 8, 5, 16, 15, 12, 7, 25, 24, 21, 16, 9, 36, 35, 32, 27, 20, 11, 49, 48, 45, 40, 33, 24, 13, 64, 63, 60, 55, 48, 39, 28, 15, 81, 80, 77, 72, 65, 56, 45, 32, 17, 100, 99, 96, 91, 84, 75, 64, 51, 36, 19, 121, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21

Offset: 1

Reinhard Zumkeller, May 24 2004

(T(n,k) mod 4) <> 2, see A042965, A016825.
All numbers m occur A034178(m) times.
The row polynomials T(n,x) appear in the calculation of the column g.f.s of triangle A120070 (used to find the frequencies of the spectral lines of the hydrogen atom).

			n=3: T(3,x) = 9+8*x+5*x^2.
Triangle begins:
   1;
   4,  3;
   9,  8,  5;
  16, 15, 12,  7;
  25, 24, 21, 16,  9;
  36, 35, 32, 27, 20, 11;
  49, 48, 45, 40, 33, 24, 13;
  64, 63, 60, 55, 48, 39, 28, 15;
  81, 80, 77, 72, 65, 56, 45, 32, 17;
  ... etc. - _Philippe Deléham_, Mar 07 2013

G. C. Greubel, Table of n, a(n) for n = 1..5050

Cf. A000290, A000567, A004736, A005408, A005563, A008586, A008590.
Cf. A008594, A008598, A016825, A016945, A017329, A028347, A028560.
Cf. A028566, A034178, A042965, A094727, A120070, A128624.
Cf. A000384 (alternating row sums), A002412 (row sums).

[n^2-k^2: k in [0..n-1], n in [1..15]]; // G. C. Greubel, Mar 12 2024

Table[n^2 - k^2, {n,12}, {k,0,n-1}]//Flatten (* Michael De Vlieger, Nov 25 2015 *)

flatten([[n^2-k^2 for k in range(n)] for n in range(1,16)]) # G. C. Greubel, Mar 12 2024

Row polynomials: T(n,x) = n^2*Sum_{m=0..n} x^m - Sum_{m=0..n} m^2*x^m = Sum_{k=0..n-1} T(n,k)*x^k, n >= 1.
T(n, k) = A004736(n,k)*A094727(n,k).
T(n, 0) = A000290(n).
T(n, 1) = A005563(n-1) for n>1.
T(n, 2) = A028347(n) for n>2.
T(n, 3) = A028560(n-3) for n>3.
T(n, 4) = A028566(n-4) for n>4.
T(n, n-1) = A005408(n).
T(n, n-2) = A008586(n-1) for n>1.
T(n, n-3) = A016945(n-2) for n>2.
T(n, n-4) = A008590(n-2) for n>3.
T(n, n-5) = A017329(n-3) for n>4.
T(n, n-6) = A008594(n-3) for n>5.
T(n, n-8) = A008598(n-2) for n>7.
T(A005408(k), k) = A000567(k).
Sum_{k=0..n} T(n, k) = A002412(n) (row sums).
From G. C. Greubel, Mar 12 2024: (Start)
Sum_{k=0..n-1} (-1)^k * T(n, k) = A000384(floor((n+1)/2)).
Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A128624(n).
Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = (1/2)*n*(n+1 - (-1)^n*cos(n*Pi/2)). (End)
G.f.: x*(1 - 3*x^2*y + x*(1 + y))/((1 - x)^3*(1 - x*y)^2). - Stefano Spezia, Aug 04 2025

A033994 a(n) = n(n+1)(5*n+1)/6.

Original entry on oeis.org

2, 11, 32, 70, 130, 217, 336, 492, 690, 935, 1232, 1586, 2002, 2485, 3040, 3672, 4386, 5187, 6080, 7070, 8162, 9361, 10672, 12100, 13650, 15327, 17136, 19082, 21170, 23405, 25792, 28336, 31042, 33915, 36960, 40182, 43586, 47177, 50960, 54940

Offset: 1

Barry E. Williams, Dec 16 1999

Partial sums of A005476.

a(n) is the dot product of the vectors of the first n positive integers and the next n integers. - Michel Marcus, Sep 02 2020

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005476, A016873, A000330, A132124, A132112, A050409.

Cf. A000292, A001622, A002411, A002412.

a:=List([1..40],n->n*(n+1)*(5*n+1)/6);; Print(a); # Muniru A Asiru, Jan 01 2019

[n*(n+1)*(5*n+1)/6 : n in [1..40]]; // Vincenzo Librandi, Jan 01 2019

[n*(n+1)*(5*n+1)/6$n=1..40]; # Muniru A Asiru, Jan 01 2019

Table[Range[x].Range[x+1,2x],{x,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{2,11,32,70},40] (* Harvey P. Dale, Jun 01 2018 *)

PARI
```
a(n) = n*(n+1)*(5*n+1)/6;
```

G.f.: x*(2+3*x)/(1-x)^4.
a(n) = A132121(n,1). - Reinhard Zumkeller, Aug 12 2007
a(n) = A000292(n) + A002412(n) = A000330(n) + A002411(n). - Omar E. Pol, Jan 11 2013
a(n) = Sum_{i=1..n} Sum_{j=1..n} i+min(i,j). - Enrique Pérez Herrero, Jan 15 2013
a(n) = Sum_{i=1..n} i*(n+i). - Charlie Marion, Apr 10 2013
Sum_{n>=1} 1/a(n) = 36 - 3*Pi*5^(3/4)*phi^(3/2)/4 - 15*sqrt(5)*log(phi)/4 - 75*log(5)/8 = 0.66131826232008423794478..., where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 01 2018
E.g.f.: exp(x)*x*(12 + 21*x + 5*x^2)/6. - Stefano Spezia, Feb 21 2024

More terms from James Sellers, Jan 19 2000

cp's OEIS Frontend

A002492 Sum of the first n even squares: 2*n*(n+1)*(2*n+1)/3.

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A015084 Carlitz-Riordan q-Catalan numbers for q=3.

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A080851 Square array of pyramidal numbers, read by antidiagonals.

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A016061 a(n) = n*(n+1)*(4*n+5)/6.

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A237616 a(n) = n*(n + 1)*(5*n - 4)/2.

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A004188 a(n) = n*(3*n^2 - 1)/2.

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A002492 Sum of the first n even squares: 2n(n+1)(2n+1)/3.

A016061 a(n) = n(n+1)(4*n+5)/6.

A237616 a(n) = n(n + 1)(5*n - 4)/2.

A004188 a(n) = n(3n^2 - 1)/2.

A002418 4-dimensional figurate numbers: a(n) = (5n-1)binomial(n+2,3)/4.

A033994 a(n) = n(n+1)(5*n+1)/6.