A320047 -id:A320047 - cp's OEIS frontend

A028896 6 times triangular numbers: a(n) = 3n(n+1).

0, 6, 18, 36, 60, 90, 126, 168, 216, 270, 330, 396, 468, 546, 630, 720, 816, 918, 1026, 1140, 1260, 1386, 1518, 1656, 1800, 1950, 2106, 2268, 2436, 2610, 2790, 2976, 3168, 3366, 3570, 3780, 3996, 4218, 4446, 4680, 4920, 5166, 5418, 5676

Offset: 0

Joe Keane (jgk(AT)jgk.org), Dec 11 1999

From Floor van Lamoen, Jul 21 2001: (Start)
Write 1,2,3,4,... in a hexagonal spiral around 0; then a(n) is the sequence found by reading the line from 0 in the direction 0, 6, ...
The spiral begins:
                 85--84--83--82--81--80
                 /                     \
               86  56--55--54--53--52  79
               /   /                 \   \
             87  57  33--32--31--30  51  78
             /   /   /             \   \   \
           88  58  34  16--15--14  29  50  77
           /   /   /   /         \   \   \   \
         89  59  35  17   5---4  13  28  49  76
         /   /   /   /   /     \   \   \   \   \
    <==90==60==36==18===6===0   3  12  27  48  75
           /   /   /   /   /   /   /   /   /   /
         61  37  19   7   1---2  11  26  47  74
           \   \   \   \         /   /   /   /
           62  38  20   8---9--10  25  46  73
             \   \   \             /   /   /
             63  39  21--22--23--24  45  72
               \   \                 /   /
               64  40--41--42--43--44  71
                 \                     /
                 65--66--67--68--69--70
(End)
If Y is a 4-subset of an n-set X then, for n >= 5, a(n-5) is the number of (n-4)-subsets of X having exactly two elements in common with Y. - Milan Janjic, Dec 28 2007
a(n) is the maximal number of points of intersection of n+1 distinct triangles drawn in the plane. For example, two triangles can intersect in at most a(1) = 6 points (as illustrated in the Star of David configuration). - Terry Stickels (Terrystickels(AT)aol.com), Jul 12 2008
Also sequence found by reading the line from 0, in the direction 0, 6, ... and the same line from 0, in the direction 0, 18, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Axis perpendicular to A195143 in the same spiral. - Omar E. Pol, Sep 18 2011
Partial sums of A008588. - R. J. Mathar, Aug 28 2014
Also the number of 5-cycles in the (n+5)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jul 27 2017
a(n-4) is the maximum irregularity over all maximal 3-degenerate graphs with n vertices.  The extremal graphs are 3-stars (K_3 joined to n-3 independent vertices).  (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, May 29 2023

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Allan Bickle and Zhongyuan Che, Irregularities of Maximal k-degenerate Graphs, Discrete Applied Math. 331 (2023) 70-87.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Enrique Navarrete and Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Leo Tavares, Illustration: Centroid Hexagons.
Eric Weisstein's World of Mathematics, Graph Cycle.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000567, A003215, A008588, A024966, A028895, A033996, A046092, A049598, A084939, A084940, A084941, A084942, A084943, A084944, A124080.
Cf. A002378 (3-cycles in triangular honeycomb acute knight graph), A045943 (4-cycles), A152773 (6-cycles).
Cf. A007531.
The partial sums give A007531. - Leo Tavares, Jan 22 2022
Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).

List([0..44],n->3*n*(n+1)); # Muniru A Asiru, Mar 15 2019

[3*n*(n+1): n in [0..50]]; // Wesley Ivan Hurt, Jun 09 2014

[seq(6*binomial(n,2),n=1..44)]; # Zerinvary Lajos, Nov 24 2006

6 Accumulate[Range[0, 50]] (* Harvey P. Dale, Mar 05 2012 *)
6 PolygonalNumber[Range[0, 20]] (* Eric W. Weisstein, Jul 27 2017 *)
LinearRecurrence[{3, -3, 1}, {0, 6, 18}, 20] (* Eric W. Weisstein, Jul 27 2017 *)

a(n)=3*n*(n+1) \\ Charles R Greathouse IV, Sep 24 2015

first(n) = Vec(6*x/(1 - x)^3 + O(x^n), -n) \\ Iain Fox, Feb 14 2018

def A028896(n): return 3*n*(n+1) # Chai Wah Wu, Aug 07 2025

O.g.f.: 6*x/(1 - x)^3.
E.g.f.: 3*x*(x + 2)*exp(x). - G. C. Greubel, Aug 19 2017
a(n) = 6*A000217(n).
a(n) = polygorial(3, n+1). - Daniel Dockery (peritus(AT)gmail.com), Jun 16 2003
From Zerinvary Lajos, Mar 06 2007: (Start)
a(n) = A049598(n)/2.
a(n) = A124080(n) - A046092(n).
a(n) = A033996(n) - A002378(n). (End)
a(n) = A002378(n)*3 = A045943(n)*2. - Omar E. Pol, Dec 12 2008
a(n) = a(n-1) + 6*n for n>0, a(0)=0. - Vincenzo Librandi, Aug 05 2010
a(n) = A003215(n) - 1. - Omar E. Pol, Oct 03 2011
From Philippe Deléham, Mar 26 2013: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=6, a(2)=18.
a(n) = A174709(6*n + 5). (End)
a(n) = A049450(n) + 4*n. - Lear Young, Apr 24 2014
a(n) = Sum_{i = n..2*n} 2*i. - Bruno Berselli, Feb 14 2018
a(n) = A320047(1, n, 1). - Kolosov Petro, Oct 04 2018
a(n) = T(3*n) - T(2*n-2) + T(n-2), where T(n) = A000217(n). In general, T(k)*T(n) = Sum_{i=0..k-1} (-1)^i*T((k-i)*(n-i)). - Charlie Marion, Dec 04 2020
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 1/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/3 - 1/3. (End)
From Amiram Eldar, Feb 21 2023: (Start)
Product_{n>=1} (1 - 1/a(n)) = -(3/Pi)*cos(sqrt(7/3)*Pi/2).
Product_{n>=1} (1 + 1/a(n)) = (3/Pi)*cosh(Pi/(2*sqrt(3))). (End)

A316349 Consider coefficients U(m,L,k) defined by the identity Sum_{k=1..L} Sum_{j=0..m} A302971(m,j)/A304042(m,j) * k^j * (T-k)^j = Sum_{k=0..m} (-1)^(m-k) * U(m,L,k) * T^k that holds for all positive integers L,m,T. This sequence gives 3-column table read by rows, where the n-th row lists coefficients U(2,n,k) for k = 0, 1, 2; n >= 1.

Original entry on oeis.org

31, 60, 30, 512, 540, 150, 2943, 2160, 420, 10624, 6000, 900, 29375, 13500, 1650, 68256, 26460, 2730, 140287, 47040, 4200, 263168, 77760, 6120, 459999, 121500, 8550, 760000, 181500, 11550, 1199231, 261360, 15180, 1821312, 365040, 19500, 2678143, 496860, 24570, 3830624, 661500, 30450

Offset: 1

Kolosov Petro, Jun 29 2018

For L=T, the identity takes form T^(2m+1) = Sum_{k=0..m} (-1)^(m-k)*U(m,T,k)*T^k, which holds for all positive integers T and m.

			       column   column  column
   L     k=0      k=1     k=2
  --  -------  -------  ------
   1       31       60      30
   2      512      540     150
   3     2943     2160     420
   4    10624     6000     900
   5    29375    13500    1650
   6    68256    26460    2730
   7   140287    47040    4200
   8   263168    77760    6120
   9   459999   121500    8550
  10   760000   181500   11550
  11  1199231   261360   15180
  12  1821312   365040   19500
  ...

Max Alekseyev, Derivation of the general formula for U(m,n,k), MathOverflow, 2018.
Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
Petro Kolosov, More details on derivation of present sequence.
Petro Kolosov, Mathematica program, verifies the identity T^(2m+1) = Sum_{k=0..m} (-1)^(m-k)*U(m,T,k)*T^k for m=0,1,...,12.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.
Petro Kolosov, An efficient method of spline approximation for power function, arXiv:2503.07618 [math.GM], 2025.

The case m=1 is A320047.
The case m=3 is A316387.
Column k=0 is A316457.
Column k=1 is A316458.
Column k=2 is A316459.
Cf. A302971, A304042, A287326, A300656, A300785.

(* Define the R[n,k] := A302971(m,j)/A304042(m,j) *)
R[n_, k_] := 0
R[n_, k_] := (2 k + 1)*Binomial[2 k, k]*
   Sum[R[n, j]*Binomial[j, 2 k + 1]*(-1)^(j - 1)/(j - k)*
   BernoulliB[2 j - 2 k], {j, 2 k + 1, n}] /; 2 k + 1 <= n
R[n_, k_] := (2 n + 1)*Binomial[2 n, n] /; k == n;
(* Define the U(m,l,t) coefficients *)
U[m_, l_, t_] := (-1)^m Sum[Sum[Binomial[j, t] R[m,j] k^(2 j - t) (-1)^j, {j, t, m}], {k, 1, l}];
(* Define the value of the variable 'm', should be m = 2 for A316349 *)
m = 2;
(* Print first 10 rows of U(m,l,t) coefficients over l: 1 <= l <= 10 *)
Column[Table[U[m, l, t], {l, 1, 10}, {t, 0, m}]]

U(2,n,0) = 6*n^5 + 15*n^4 + 10*n^3; U(2,n,1) = 15*n^4 + 30*n^3 + 15*n^2; U(2,n,2) = 10*n^3 + 15*n^2 + 5*n. - Max Alekseyev, Sep 06 2018
From Colin Barker, Jul 06 2018: (Start)
G.f.: x*(31 + 60*x + 30*x^2 + 326*x^3 + 180*x^4 - 30*x^5 + 336*x^6 - 180*x^7 - 30*x^8 + 26*x^9 - 60*x^10 + 30*x^11 + x^12) / ((1 - x)^6*(1 + x + x^2)^6).
a(n) = 6*a(n-3) - 15*a(n-6) + 20*a(n-9) - 15*a(n-12) + 6*a(n-15) - a(n-18) for n>18. (End)
U(m,L,t) = (-1)^m * Sum_{k=1..L} Sum_{j=t..m} binomial(j,t) * R(m,j) * k^{2j-t} * (-1)^j, where m = 1, L >= 1 and R(m,j) = A302971(m,j)/A304042(m,j); after Max Alekseyev, see links.

Edited by Max Alekseyev, Sep 06 2018

A316387 Consider coefficients U(m,L,k) defined by the identity Sum_{k=1..L} Sum_{j=0..m} A302971(m,j)/A304042(m,j) * k^j * (T-k)^j = Sum_{k=0..m} (-1)^(m-k) * U(m,L,k) * T^k that holds for all positive integers L,m,T. This sequence gives 4-column table read by rows, where the n-th row lists coefficients U(3,n,k) for k = 0, 1, 2, 3; n >= 1.

Original entry on oeis.org

125, 406, 420, 140, 9028, 13818, 7140, 1260, 110961, 115836, 41160, 5040, 684176, 545860, 148680, 14000, 2871325, 1858290, 411180, 31500, 9402660, 5124126, 955500, 61740, 25872833, 12182968, 1963920, 109760, 62572096, 25945416, 3684240, 181440, 136972701, 50745870, 6439860, 283500, 276971300, 92745730, 10639860, 423500

Offset: 1

Kolosov Petro, Jul 01 2018

For T=L, the identity takes form T^(2m+1) = Sum_{k=0..m} (-1)^(m-k)*U(m,T,k)*T^k, which holds for all positive integers T and m.

			            column      column      column   column
   L          k=0         k=1         k=2      k=3
  --  ------------  ----------  ----------  -------
   1           125         406         420      140
   2          9028       13818        7140     1260
   3        110961      115836       41160     5040
   4        684176      545860      148680    14000
   5       2871325     1858290      411180    31500
   6       9402660     5124126      955500    61740
   7      25872833    12182968     1963920   109760
   8      62572096    25945416     3684240   181440
   9     136972701    50745870     6439860   283500
  10     276971300    92745730    10639860   423500
  11     524988145   160386996    16789080   609840
  12     943023888   264896268    25498200   851760
  13    1618774781   420839146    37493820  1159340
  14    2672907076   646725030    53628540  1543500
  15    4267591425   965662320    74891040  2016000
  16    6616398080  1406064016   102416160  2589440
  17    9995653693  2002403718   137494980  3277260
  18   14757360516  2796022026   181584900  4093740
  19   21343778801  3835983340   236319720  5054000
  20   30303773200  5179983060   303519720  6174000
  ...

Max Alekseyev, Derivation of the general formula for U(m,n,k), MathOverflow, 2018.
Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
Petro Kolosov, More details on derivation of present sequence.
Petro Kolosov, Mathematica program, verifies the identity T^(2m+1) = Sum_{k=0..m} (-1)^(m-k)*U(m,T,k)*T^k for m=0,1,...,12.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.
Petro Kolosov, An efficient method of spline approximation for power function, arXiv:2503.07618 [math.GM], 2025.

The case m=1 is A320047.
The case m=2 is A316349.
Column k=0 is A317981.
Column k=1 is A317982.
Column k=2 is A317983.
Column k=3 is A317984.
Cf. A287326, A300656, A300785, A302971, A304042.

(* Define the R[n,k] := A302971(m,j)/A304042(m,j) *)
R[n_, k_] := 0
R[n_, k_] := (2 k + 1)*Binomial[2 k, k]*
   Sum[R[n, j]*Binomial[j, 2 k + 1]*(-1)^(j - 1)/(j - k)*
   BernoulliB[2 j - 2 k], {j, 2 k + 1, n}] /; 2 k + 1 <= n
R[n_, k_] := (2 n + 1)*Binomial[2 n, n] /; k == n;
(* Define the U(m,l,t) coefficients *)
U[m_, l_, t_] := (-1)^m Sum[Sum[Binomial[j, t] R[m,j] k^(2 j - t) (-1)^j, {j, t, m}], {k, 1, l}];
(* Define the value of the variable 'm' *)
m = 3;
(* Print first 10 rows of U(m,l,t) coefficients for 'm' defined above *)
Column[Table[U[m, l, t], {l, 1, 10}, {t, 0, m}]]

U(3,n,0) = 20*n^7 + 70*n^6 + 70*n^5 - 28*n^3 - 7*n^2; U(3,n,1) = 70*n^6 + 210*n^5 + 175*n^4 - 42*n^2 - 7*n; U(3,n,2) = 84*n^5 + 210*n^4 + 140*n^3 - 14*n; U(3,n,3) = 35*n^4 + 70*n^3 + 35*n^2. - Max Alekseyev, Sep 06 2018
From Colin Barker, Jul 09 2018; corrected by Max Alekseyev, Sep 06 2018: (Start)
G.f.: x*(125 + 406*x + 420*x^2 + 140*x^3 + 8028*x^4 + 10570*x^5 + 3780*x^6 + 140*x^7 + 42237*x^8 + 16660*x^9 - 4200*x^10 - 1120*x^11 + 42272*x^12 - 16660*x^13 - 4200*x^14 + 1120*x^15 + 8007*x^16 - 10570*x^17 + 3780*x^18 - 140*x^19 + 132*x^20 - 406*x^21 + 420*x^22 - 140*x^23 - x^24) / ((1 - x)^8*(1 + x)^8*(1 + x^2)^8).
a(n) = 8*a(n-4) - 28*a(n-8) + 56*a(n-12) - 70*a(n-16) + 56*a(n-20) - 28*a(n-24) + 8*a(n-28) - a(n-32) for n>32. (End)
U(m,L,t) = (-1)^m * Sum_{k=1..L} Sum_{j=t..m} binomial(j,t) * R(m,j) * k^{2j-t} * (-1)^j, where m = 1, L >= 1 and R(m,j) = A302971(m,j)/A304042(m,j); after Max Alekseyev, see links. - Kolosov Petro, Oct 04 2018

Edited by Max Alekseyev, Sep 06 2018

A275709 a(n) = 2n^3 + 3n^2.

Original entry on oeis.org

0, 5, 28, 81, 176, 325, 540, 833, 1216, 1701, 2300, 3025, 3888, 4901, 6076, 7425, 8960, 10693, 12636, 14801, 17200, 19845, 22748, 25921, 29376, 33125, 37180, 41553, 46256, 51301, 56700, 62465, 68608, 75141, 82076, 89425, 97200, 105413, 114076, 123201, 132800, 142885

Offset: 0

Joshua Giambalvo, Aug 06 2016

Apart from the initial zero this sequence gives the 2nd pentagonal number, the 4th hexagonal number, the 6th heptagonal number, the 8th octagonal number, the 10th nonagonal number, etc. as well as the 5th nonnegative number, the 7th triangular number, the 9th square, the 11th pentagonal number, the 13th hexagonal number, etc. This is a reliable pattern that does not seem to appear on any other pairs of polygonal numbers (see link).

a(n) is the maximal determinant of a 3 X 3 matrix with integer elements from {1, ..., n+1}, so (for example) the maximum determinant of a 3 X 3 matrix with integer elements from {1, ..., 5} = det(1, 5, 5; 5, 1, 5; 5, 5, 1) = a(4) = 176. - Matthew Scroggs, Dec 31 2022

Carauleanu Marc and Colin Barker, Table of n, a(n) for n = 0..3030 (first 1000 terms from Colin Barker)
Joshua Giambalvo, Illustration of initial terms in a square array, Imgur, (2016).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000290, A005408, A005563, A033428, A033431, A057145, A139600.

[n^2*(2*n + 3): n in [0..30]]; // G. C. Greubel, Oct 19 2018

seq(2*n^3+3*n^2, n=0..30); # Robert Israel, Aug 09 2016

Table[2 n^3 + 3 n^2, {n, 0, 41}] (* or *)
CoefficientList[Series[x (5 + 8 x - x^2)/(1 - x)^4, {x, 0, 41}], x] (* Michael De Vlieger, Aug 11 2016 *)

concat(0, Vec(x*(5+8*x-x^2)/(1-x)^4 + O(x^50))) \\ Colin Barker, Aug 28 2016

a(n)=n^2*(2*n+3) \\ Charles R Greathouse IV, Aug 28 2016

for n in range(0,50): print(n**2*(2*n+3), end=' ') # Stefano Spezia, Oct 19 2018

From Colin Barker, Aug 06 2016: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.
G.f.: x*(5+8*x-x^2) / (1-x)^4. (End)
a(n) = A033431(n) + A033428(n). - Omar E. Pol, Aug 09 2016
a(n) = A000290(n) * A005408(n+1). - Robert Israel, Aug 09 2016
a(n) = A320047(1, n, 0). - Kolosov Petro, Oct 04 2018
E.g.f.: x*(5 + 9*x + 2*x^2)*exp(x). - G. C. Greubel, Oct 19 2018
From Amiram Eldar, Jan 21 2023: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/18 + 4*log(2)/9 - 16/27.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/36 + Pi/9 -2*log(2)/9 - 8/27. (End)

cp's OEIS Frontend

A028896 6 times triangular numbers: a(n) = 3*n*(n+1).

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A275709 a(n) = 2*n^3 + 3*n^2.

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A028896 6 times triangular numbers: a(n) = 3n(n+1).

A275709 a(n) = 2n^3 + 3n^2.