A006007 -id:A006007 - cp's OEIS frontend

A351153 Triangle read by rows: T(n, k) = n(k - 1) - k(k - 3)/2 with 0 < k <= n.

1, 1, 3, 1, 4, 6, 1, 5, 8, 10, 1, 6, 10, 13, 15, 1, 7, 12, 16, 19, 21, 1, 8, 14, 19, 23, 26, 28, 1, 9, 16, 22, 27, 31, 34, 36, 1, 10, 18, 25, 31, 36, 40, 43, 45, 1, 11, 20, 28, 35, 41, 46, 50, 53, 55, 1, 12, 22, 31, 39, 46, 52, 57, 61, 64, 66, 1, 13, 24, 34, 43, 51, 58, 64, 69, 73, 76, 78

Offset: 1

Stefano Spezia, Feb 02 2022

Except for the number 2, it contains all the positive integers.

			Triangle begins:
  1;
  1, 3;
  1, 4,  6;
  1, 5,  8, 10;
  1, 6, 10, 13, 15;
  1, 7, 12, 16, 19, 21;
  1, 8, 14, 19, 23, 26, 28;
  ...

Cf. A000012 (1st column), A000217 (leading diagonal), A005843 (3rd column), A006007 (sum of the first n rows), A006527 (row sums).

Cf. A087401, A141418, A351154.

Flatten[Table[n(k-1)-k(k-3)/2,{n,12},{k,n}]]

T(n, k) = 1 + Sum_{i=1..k-1} (n - i + 1).
From R. J. Mathar, Feb 07 2022: (Start)
G.f.: x*y*(1 - x + y*x^2 + y^2*x^3)/((1 - x)^2*(1 - y*x)^3).
T(n, k) = 1 + A141418(n+1, k-1) = 1 + A087401(n+1, k-1). (End)

A132366 Partial sum of centered tetrahedral numbers A005894.

Original entry on oeis.org

1, 6, 21, 56, 125, 246, 441, 736, 1161, 1750, 2541, 3576, 4901, 6566, 8625, 11136, 14161, 17766, 22021, 27000, 32781, 39446, 47081, 55776, 65625, 76726, 89181, 103096, 118581, 135750, 154721, 175616, 198561, 223686, 251125, 281016, 313501, 348726, 386841

Offset: 0

Jonathan Vos Post, Nov 09 2007

From Robert A. Russell, Oct 09 2020: (Start)
a(n-1) is the number of achiral colorings of the 5 tetrahedral facets (or vertices) of a regular 4-dimensional simplex using n or fewer colors. An achiral arrangement is identical to its reflection. The 4-dimensional simplex is also called a 5-cell or pentachoron. Its Schläfli symbol is {3,3,3}.
There are 60 elements in the automorphism group of the 4-dimensional simplex that are not in its rotation group. Each is an odd permutation of the vertices and can be associated with a partition of 5 based on the conjugacy class of the permutation. The first formula for a(n-1) is obtained by averaging their cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Partition  Count  Odd Cycle Indices
  41         30     x_1x_4^1
  32         20     x_2^1x_3^1
  2111       10     x_1^3x_2^1 (End)

Nathaniel Johnston, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000292, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.
Cf. A337895 (oriented), A000389(n+4) (unoriented), A000389 (chiral), A331353 (5-cell edges, faces), A337955 (8-cell vertices, 16-cell facets), A337958 (16-cell vertices, 8-cell facets), A338951 (24-cell), A338967 (120-cell, 600-cell).
a(n-1) = A325001(4,n).

Do[Print[n, " ", (n^4 + 4 n^3 + 11 n^2 + 14 n + 6)/6 ], {n, 0, 10000}]
Accumulate[Table[(2n+1)(n^2+n+3)/3,{n,0,40}]] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{1,6,21,56,125},40] (* Harvey P. Dale, Feb 26 2020 *)

a(n) = (n^4 + 4*n^3 + 11*n^2 + 14*n + 6)/6 = (n^2+2*n+6)*(n+1)^2/6.
G.f.: -(x+1)*(x^2+1) / (x-1)^5. - Colin Barker, May 04 2013
From Robert A. Russell, Oct 09 2020: (Start)
a(n-1) = n^2 * (5 + n^2) / 6.
a(n-1) = binomial(n+4,5) - binomial(n,5) = A000389(n+4) - A000389(n).
a(n-1) = 1*C(n,1) + 4*C(n,2) + 6*C(n,3) + 4*C(n,4), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n-1) = 2*A000389(n+4) - A337895(n) = A337895(n) - 2*A000389(n) .
G.f. for a(n-1): x * (x+1) * (x^2+1) / (1-x)^5. (End)
From Amiram Eldar, Feb 14 2023: (Start)
Sum_{n>=0} 1/a(n) = Pi^2/5 + 3/25 - 3*Pi*coth(sqrt(5)*Pi)/(5*sqrt(5)).
Sum_{n>=0} (-1)^n/a(n) = Pi^2/10 - 3/25 + 3*Pi*cosech(sqrt(5)*Pi)/(5*sqrt(5)). (End)
a(n) = A006007(n) + A006007(n+1) = A002415(n) + A002415(n+2). - R. J. Mathar, Jun 05 2025

Corrected offset, Mathematica program by Tomas J. Bulka (tbulka(AT)rodincoil.com), Sep 02 2009

A236770 a(n) = n(n + 1)(3n^2 + 3n - 2)/8.

Original entry on oeis.org

0, 1, 12, 51, 145, 330, 651, 1162, 1926, 3015, 4510, 6501, 9087, 12376, 16485, 21540, 27676, 35037, 43776, 54055, 66045, 79926, 95887, 114126, 134850, 158275, 184626, 214137, 247051, 283620, 324105, 368776, 417912, 471801, 530740, 595035, 665001, 740962

Offset: 0

Bruno Berselli, Jan 31 2014

After 0, first trisection of A011779 and right border of A177708.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Partial sums of A004188.
Cf. A000217, A000332, A011779, A177708.
Cf. similar sequences on the polygonal numbers: A002817(n) = A000217(A000217(n)); A000537(n) = A000290(A000217(n)); A037270(n) = A000217(A000290(n)); A062392(n) = A000384(A000217(n)).
Cf. sequences of the form A000217(m)+k*A000332(m+2): A062392 (k=12); A264854 (k=11); A264853 (k=10); this sequence (k=9); A006324 (k=8); A006323 (k=7); A000537 (k=6); A006322 (k=5); A006325 (k=4), A002817 (k=3), A006007 (k=2), A006522 (k=1).
Cf. A232713, A260810.

[n*(n+1)*(3*n^2+3*n-2)/8: n in [0..40]];

Table[n (n + 1) (3 n^2 + 3 n - 2)/8, {n, 0, 40}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,12,51,145},40] (* Harvey P. Dale, Aug 22 2016 *)

for(n=0, 40, print1(n*(n+1)*(3*n^2+3*n-2)/8", "));

G.f.: x*(1 + 7*x + x^2)/(1 - x)^5.
a(n) = a(-n-1) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A000326(A000217(n)).
a(n) = A000217(n) + 9*A000332(n+2).
Sum_{n>=1} 1/a(n) = 2 + 4*sqrt(3/11)*Pi*tan(sqrt(11/3)*Pi/2) = 1.11700627139319... . - Vaclav Kotesovec, Apr 27 2016

A383834 Sum of the legs of the unique primitive Pythagorean triple whose inradius is A000217(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

1, 7, 31, 97, 241, 511, 967, 1681, 2737, 4231, 6271, 8977, 12481, 16927, 22471, 29281, 37537, 47431, 59167, 72961, 89041, 107647, 129031, 153457, 181201, 212551, 247807, 287281, 331297, 380191, 434311, 494017, 559681, 631687, 710431, 796321, 889777, 991231, 1101127, 1219921, 1348081

Offset: 0

Miguel-Ángel Pérez García-Ortega, May 11 2025

			For n=1, the short leg is A002061(1) = 3 and the long leg is A212135(2) = 4 so the sum of the legs is then a(1) = 3 + 4 = 7.

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

Cf. A000217, A383833, A002061, A006007, A058919, A336535.

a=Table[(n(n+1))/2,{n,0,40}];Apply[Join,Map[{2#^2+4#+1}&,a]]

a(n) = 2*(A000217(n))^2 + 4*A000217(n) + 1.

a(n) = 6*A006007(n) + 1

A047662 Square array a(n,k) read by antidiagonals: a(n,1)=n, a(1,k)=k, a(n,k)=a(n-1,k-1)+a(n-1,k)+a(n,k-1)+1.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 12, 12, 4, 5, 20, 31, 20, 5, 6, 30, 64, 64, 30, 6, 7, 42, 115, 160, 115, 42, 7, 8, 56, 188, 340, 340, 188, 56, 8, 9, 72, 287, 644, 841, 644, 287, 72, 9, 10, 90, 416, 1120, 1826, 1826, 1120, 416, 90, 10, 11, 110, 579, 1824, 3591

Offset: 1

Don Knuth, N. J. A. Sloane

			The array begins:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, ...
3, 12, 31, 64, 115, 188, 287, 416, 579, 780, 1023, 1312, ...
4, 20, 64, 160, 340, 644, 1120, 1824, 2820, 4180, 5984, 8320, ...
5, 30, 115, 340, 841, 1826, 3591, 6536, 11181, 18182, 28347, 42652, ...
6, 42, 188, 644, 1826, 4494, 9912, 20040, 37758, 67122, 113652, 184652, ...
7, 56, 287, 1120, 3591, 9912, 24319, 54272, 112071, 216952, 397727, 696032, ...
8, 72, 416, 1824, 6536, 20040, 54272, 132864, 299208, 628232, 1242912, 2336672, ...
...
The first few antidiagonals are:
1,
2, 2,
3, 6, 3,
4, 12, 12, 4,
5, 20, 31, 20, 5,
6, 30, 64, 64, 30, 6,
7, 42, 115, 160, 115, 42, 7,
8, 56, 188, 340, 340, 188, 56, 8,
9, 72, 287, 644, 841, 644, 287, 72, 9,
10, 90, 416, 1120, 1826, 1826, 1120, 416, 90, 10,
...

Vincenzo Librandi, Rows n = 1..100, flattened
M. L. Fredman, The complexity of maintaining an array and its partial sums, J. Assoc. Comp. Machin., 29 (1982), 250-260.
D. E. Knuth and N. J. A. Sloane, Correspondence, December 1999
Matthew Roughan, Surreal Birthdays and Their Arithmetic, arXiv:1810.10373 [math.HO], 2018.

Rows give A037237, 4*A006007, A047661, A047663, A047664, main diagonal is A047665 (see also A001850).

A061316 a(n) = n(n+1)(n^2 + n + 4)/4.

Original entry on oeis.org

0, 3, 15, 48, 120, 255, 483, 840, 1368, 2115, 3135, 4488, 6240, 8463, 11235, 14640, 18768, 23715, 29583, 36480, 44520, 53823, 64515, 76728, 90600, 106275, 123903, 143640, 165648, 190095, 217155, 247008, 279840, 315843, 355215, 398160, 444888

Offset: 0

Henry Bottomley, Apr 24 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A005563, A006007, A061314.

a:=n->sum((n+j^3),j=0..n): seq(a(n),n=0..36); # Zerinvary Lajos, Jul 27 2006
with(combinat):a:=n->sum(fibonacci(4,i), i=0..n): seq(a(n), n=0..36); # Zerinvary Lajos, Mar 20 2008

s=0;lst={};Do[s+=n^3+n*2;AppendTo[lst,s],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 04 2009 *)
Table[n(n+1)(n^2+n+4)/4,{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1}, {0,3,15,48,120},40] (* Harvey P. Dale, May 03 2011 *)

a(n) = n*(n + 1)*(n^2 + n + 4)/4 \\ Harry J. Smith, Jul 21 2009

a(n) = n*(n+1)*(n^2 + n + 4)/4.
a(n) = A005563(A000217(n)) = 3*A006007(n) = A061314(n, 2).
a(0)=0, a(1)=3, a(2)=15, a(3)=48, a(4)=120, a(n) = 5a(n-1) - 10a(n-2) + 10a(n-3) - 5a(n-4) + a(n-5).
G.f.: (-3 (x + x^3))/(-1 + x)^5. - Harvey P. Dale, May 03 2011
Sum_{n>=1} 1/a(n) = 5/4 - tanh(sqrt(15)*Pi/2)*Pi/sqrt(15). - Amiram Eldar, Aug 20 2022
E.g.f.: exp(x)*x*(12 + 18*x + 8*x^2 + x^3)/4. - Stefano Spezia, Aug 31 2023

A153976 a(n) = n^3 + (n+2)^3.

Original entry on oeis.org

8, 28, 72, 152, 280, 468, 728, 1072, 1512, 2060, 2728, 3528, 4472, 5572, 6840, 8288, 9928, 11772, 13832, 16120, 18648, 21428, 24472, 27792, 31400, 35308, 39528, 44072, 48952, 54180, 59768, 65728, 72072, 78812, 85960, 93528, 101528, 109972, 118872, 128240

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..765
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000537, A000578, A003215, A005898, A006007, A027602.

[n^3+(n+2)^3: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

f[n_]:=n^3;lst={};Do[AppendTo[lst,(f[n]+f[n+2])],{n,0,6!}];lst
Array[#^3+(#+2)^3&,40,0] (* or *) LinearRecurrence[{4,-6,4,-1},{8,28,72,152},40] (* Harvey P. Dale, Aug 02 2011 *)

def a(n): return n**3 + (n+2)**3
print([a(n) for n in range(40)]) # Michael S. Branicky, Aug 28 2021

For n>3, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Aug 02 2011
G.f.: 4*( 2-x+2*x^2 ) / (x-1)^4 . - R. J. Mathar, Apr 11 2016
a(n) = 4*A229183(n+1). - R. J. Mathar, Apr 11 2016

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 26 2011

A153978 a(n) = n(n-1)(n+1)(3n-2)/12.

Original entry on oeis.org

0, 2, 14, 50, 130, 280, 532, 924, 1500, 2310, 3410, 4862, 6734, 9100, 12040, 15640, 19992, 25194, 31350, 38570, 46970, 56672, 67804, 80500, 94900, 111150, 129402, 149814, 172550, 197780, 225680, 256432, 290224, 327250, 367710, 411810, 459762

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Partial sums of A011379.

Antidiagonal sums of the convolution array A213819. - Clark Kimberling, Jul 04 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A003215, A000537, A000578, A005898, A027602, A006007, A153976, A153977, A011379, A052149, A213819.

With[{r=Range[0,50]},Accumulate[r^2+r^3]] (* Harvey P. Dale, Jan 16 2011 *)
Rest[CoefficientList[Series[-2 x^2 * (2 x + 1)/(x - 1)^5, {x, 0, 40}], x]] (* Vincenzo Librandi, Jun 30 2014 *)
LinearRecurrence[{5,-10,10,-5,1}, {0,2,14,50,130}, 25] (* G. C. Greubel, Sep 01 2016 *)

concat(0, Vec(-2*x^2*(2*x+1)/(x-1)^5 + O(x^100))) \\ Colin Barker, Jun 28 2014

a(n) = n*(n-1)*(n+1)*(3*n-2)/12 \\ Charles R Greathouse IV, Sep 01 2016

a(n) = 2 * A001296(n-1) = (n-1)*n*(n+1)*(3*n-2)/12 (n>0). - Bruno Berselli, Apr 21 2010
a(n) = Sum_{i=1..n-1} binomial(i+1,i)*i^2. - Enrique Pérez Herrero, Jun 28 2014
G.f.: 2*x^2*(2*x+1) / (1 - x)^5. - Colin Barker, Jun 28 2014
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4. - Vincenzo Librandi, Jun 30 2014
a(n) = Sum_{k=1..n-1}k*((n-1)*n/2 + k) for n > 1. - J. M. Bergot, Feb 16 2018
From Amiram Eldar, Aug 23 2022: (Start)
Sum_{n>=2} 1/a(n) = 141/5 - 9*sqrt(3)*Pi/5 - 81*log(3)/5.
Sum_{n>=2} (-1)^n/a(n) = 18*sqrt(3)*Pi/5 + 48*log(2)/5 - 129/5. (End)

Edited by Bruno Berselli, Jun 15 2010

Simpler definition as suggested by Wesley Ivan Hurt, Jun 29 2014

A061315 Array read by antidiagonals: T(n,k)=T(n,k-1)*(T(n,k-1)+k-1)/k with T(n,1)=n.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 5, 6, 4, 1, 10, 16, 10, 5, 1, 28, 76, 40, 15, 6, 1, 154, 1216, 430, 85, 21, 7, 1, 3520, 247456, 37324, 1870, 161, 28, 8, 1, 1551880

Offset: 1

Henry Bottomley, Apr 24 2001

Not always an integer

			1,1,1,1,1,1,1,1,1,1,...
2,3,5,10,28,154,3520,1551880,267593772160,7160642690122633501504,...
3,6,16,76,1216,247456,61235956672/7,468730299066952899064/49,...
4,10,40,430,37324,232211266,7703153350084336,7417321441864447837991470393906,
5,15,85,1870,700876,81871778626,957569733696568731376,...
6,21,161,6601,8719921,12672844307641,22942997549397847673832961,...
7,28,280,19810,78503068,1027122012987994,...

Rows include A000012 and A003504. Columns include A000027, A000217, A006007, A061319 and A061321.

a(n) =A061314(n, k-1)/k

A061318 Column 3 of A061314.

Original entry on oeis.org

0, 4, 40, 304, 1720, 7480, 26404, 79240, 209304, 499140, 1095160, 2242504, 4332640, 7966504, 14036260, 23829040, 39156304, 62512740, 97268904, 147902080, 220270120, 321933304, 462529540, 654208504, 912130600, 1255036900, 1705896504, 2292638040, 3048972304, 4015313320

Offset: 0

Henry Bottomley, Apr 24 2001

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1)

Cf. A006007, A061314, A061316, A061319.

CoefficientList[Series[-4x (1+x+22x^2+22x^3+22x^4+x^5+x^6)/(x-1)^9,{x,0,30}],x] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,4,40,304,1720,7480,26404,79240,209304},30] (* Harvey P. Dale, Jul 04 2023 *)

a(n) = A061316(n) + A006007(n)^2 = 4*A061319(n).

G.f. -4*x*(1+x+22*x^2+22*x^3+22*x^4+x^5+x^6)/(x-1)^9. - R. J. Mathar, Aug 11 2012

a(25)-a(29) from Stefano Spezia, Aug 31 2023

cp's OEIS Frontend

A351153 Triangle read by rows: T(n, k) = n*(k - 1) - k*(k - 3)/2 with 0 < k <= n.

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A132366 Partial sum of centered tetrahedral numbers A005894.

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A236770 a(n) = n*(n + 1)*(3*n^2 + 3*n - 2)/8.

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A383834 Sum of the legs of the unique primitive Pythagorean triple whose inradius is A000217(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

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A047662 Square array a(n,k) read by antidiagonals: a(n,1)=n, a(1,k)=k, a(n,k)=a(n-1,k-1)+a(n-1,k)+a(n,k-1)+1.

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A061316 a(n) = n*(n+1)*(n^2 + n + 4)/4.

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A153976 a(n) = n^3 + (n+2)^3.

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A153978 a(n) = n*(n-1)*(n+1)*(3*n-2)/12.

A351153 Triangle read by rows: T(n, k) = n(k - 1) - k(k - 3)/2 with 0 < k <= n.

A236770 a(n) = n(n + 1)(3n^2 + 3n - 2)/8.

A061316 a(n) = n(n+1)(n^2 + n + 4)/4.

A153978 a(n) = n(n-1)(n+1)(3n-2)/12.