A006007 -id:A006007 - cp's OEIS frontend

A081282 Generalized centered polygonal numbers.

0, 1, 6, 22, 62, 147, 308, 588, 1044, 1749, 2794, 4290, 6370, 9191, 12936, 17816, 24072, 31977, 41838, 53998, 68838, 86779, 108284, 133860, 164060, 199485, 240786, 288666, 343882, 407247, 479632, 561968, 655248, 760529, 878934, 1011654

Offset: 0

Paul Barry, Mar 17 2003

Partial sums of A006007.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A081284.

[(n*(n+1)*(n+2)*(n^2+2*n+7))/60: n in [0..35]]; // Vincenzo Librandi, Aug 07 2013

CoefficientList[Series[x (1 + x^2) / (1 - x)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,6,22,62,147},40] (* Harvey P. Dale, Jan 09 2014 *)

a(n) = 2*C(n+3, 5) + C(n+2, 3).
G.f.: x*(1+x^2)/(1-x)^6.
a(n) = n*(n+1)*(n+2)*(n^2+2*n+7)/60. - Carl Najafi, Nov 18 2012
a(0)=0, a(1)=1, a(2)=6, a(3)=22, a(4)=62, a(5)=147, a(n)=6*a(n-1)- 15*a(n-2)+ 20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6). - Harvey P. Dale, Jan 09 2014

A081283 An interleaved sequence of pyramidal and polygonal numbers.

Original entry on oeis.org

0, 1, 1, 5, 6, 16, 20, 40, 50, 85, 105, 161, 196, 280, 336, 456, 540, 705, 825, 1045, 1210, 1496, 1716, 2080, 2366, 2821, 3185, 3745, 4200, 4880, 5440, 6256, 6936, 7905, 8721, 9861, 10830, 12160, 13300, 14840, 16170, 17941, 19481, 21505, 23276, 25576, 27600

Offset: 0

Paul Barry, Mar 17 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A081282, A081284.

A081283:=n->(2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(2*n^2+2*(5+(-1)^n)*n+27-11*(-1)^n)/1536: seq(A081283(n), n=0..80); # Wesley Ivan Hurt, Apr 18 2017

CoefficientList[Series[x (1 + x^3) / ((1 - x) (1 - x^2)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 07 2013 *)

G.f.: x*(1+x^3)/((1-x)*(1-x^2)^4).
a(2*n) = A002415(n);  a(2*n+1) = A006007(n+1).
a(n) = (2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(2*n^2+2*(5+(-1)^n)*n+27-11*(-1)^n)/1536. - Luce ETIENNE, Mar 11 2015

A264854 a(n) = n(n + 1)(11n^2 + 11n - 10)/24.

Original entry on oeis.org

0, 1, 14, 61, 175, 400, 791, 1414, 2346, 3675, 5500, 7931, 11089, 15106, 20125, 26300, 33796, 42789, 53466, 66025, 80675, 97636, 117139, 139426, 164750, 193375, 225576, 261639, 301861, 346550, 396025, 450616, 510664, 576521, 648550, 727125, 812631, 905464, 1006031

Offset: 0

Ilya Gutkovskiy, Nov 26 2015

Partial sums of centered 11-gonal (or hendecagonal) pyramidal numbers.

OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Pyramidal Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A004467.

Cf. similar sequences provided by the partial sums of centered k-gonal pyramidal numbers: A006522 (k=1), A006007 (k=2), A002817 (k=3), A006325 (k=4), A006322 (k=5), A000537 (k=6), A006323 (k=7), A006324 (k=8), A236770 (k=9), A264853 (k=10), this sequence (k=11), A062392 (k=12), A264888 (k=13).

[n*(n+1)*(11*n^2+11*n-10)/24: n in [0..50]]; // Vincenzo Librandi, Nov 27 2015

Table[n (n + 1) (11 n^2 + 11 n - 10)/24, {n, 0, 50}]

a(n)=n*(n+1)*(11*n^2+11*n-10)/24 \\ Charles R Greathouse IV, Jul 26 2016

G.f.: x*(1 + 9*x + x^2)/(1 - x)^5.
a(n) = Sum_{k = 0..n} A004467(k).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Nov 27 2015

A129687 A129686 * A007318.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 4, 3, 1, 2, 6, 7, 4, 1, 2, 8, 13, 11, 5, 1, 2, 10, 21, 24, 16, 6, 1, 2, 12, 31, 45, 40, 22, 7, 1, 2, 14, 43, 76, 85, 62, 29, 8, 1, 2, 16, 57, 119, 161, 147, 91, 37, 9, 1, 2, 18, 73, 176, 280, 308, 238, 128, 46, 10, 1, 2, 20, 91, 249, 456

Offset: 0

Gary W. Adamson, Apr 28 2007

Row sums = A084215: (1, 2, 5, 10, 20, 40, 80, ...). A007318 * A129686 = A124725.
From Philippe Deléham, Feb 12 2014: (Start)
Riordan array ((1+x^2)/(1-x), x/(1-x)).
Diagonal sums are A000032(n) - 0^n (cf. A000204).
T(n,0) = A046698(n+1).
T(n+1,1) = A004277(n).
T(n+2,2) = A002061(n+1).
T(n+3,3) = A006527(n+1) = A167875(n).
T(n+4,4) = A006007(n+1).
T(n+5,5) = A081282(n+1). (End)

			First few rows of the triangle:
  1;
  1,   1;
  2,   2,   1;
  2,   4,   3,   1;
  2,   6,   7,   4,   1;
  2,   8,  13,  11,   5,   1;
  2,  10,  21,  24,  16,   6,   1;
  2,  12,  31,  45,  40,  22,   7,   1;
  2,  14,  43,  76,  85,  62,  29,   8,   1;
  2,  16,  57, 119, 161, 147,  91,  37,   9,   1;
  ...

Cf. A129686, A007318, A124725.

Cf. Columns: A046698, A004277, A002061, A006527, A006007, A081282

A129686 * A007318 (Pascal's Triangle), as infinite lower triangular matrices.

T(n,k) = T(n-1,k) + T(n-1,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = T(2,1) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Feb 12 2014

More terms from Philippe Deléham, Feb 12 2014

A153977 One-fourth of partial sums of A153976.

Original entry on oeis.org

2, 9, 27, 65, 135, 252, 434, 702, 1080, 1595, 2277, 3159, 4277, 5670, 7380, 9452, 11934, 14877, 18335, 22365, 27027, 32384, 38502, 45450, 53300, 62127, 72009, 83027, 95265, 108810, 123752, 140184, 158202, 177905, 199395, 222777, 248159, 275652, 305370, 337430

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

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

Cf. A000217, A000537, A000578, A003215, A005898, A006007, A027602, A153976.

A153977:=n->(1/4)*sum(i^3 + (i+2)^3, i=0..n): seq(A153977(n), n=0..50); # Wesley Ivan Hurt, Feb 04 2017

a[n_]:=n^3;lst={};s=0;Do[s+=(a[n]+a[n+2]);AppendTo[lst,s/4],{n,0,6!}];lst
Accumulate[Array[#^3+(#+2)^3&,40,0]]/4 (* or *) LinearRecurrence[ {5,-10,10,-5,1},{2,9,27,65,135},40] (* Harvey P. Dale, Aug 02 2011 *)

a(n)=(n^4 + 2*n^3 + 7*n^2 + 6*n)/8 \\ Charles R Greathouse IV, Feb 06 2017

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5, a(1)=2, a(2)=9, a(3)=27, a(4)=65, a(5)=135. - Harvey P. Dale, Aug 02 2011
a(n) = (A000217(n-1)^2 + A000217(n+1)^2 - 1)/4. - Richard R. Forberg, Dec 25 2013
Recurrence: (n-1)*(n^2 - n + 6)*a(n) = (n+1)*(n^2 + n + 6)*a(n-1). - Vaclav Kotesovec, Dec 26 2013
a(n) = A000217(A000217(n)) + A000217(n). - Bruno Berselli, May 28 2015
a(n) = (A000217(n)^2 + 3*A000217(n))/2 where A000217(n) is the n-th triangular number. - Frederic Isenmann, Feb 04 2017
Sum_{n>=1} 1/a(n) = 14/9 - 4*tanh(sqrt(23)*Pi/2)*Pi/(3*sqrt(23)). - Amiram Eldar, Aug 23 2022
From Elmo R. Oliveira, Aug 28 2025: (Start)
G.f.: x*(2 - x + 2*x^2)/(1-x)^5.
E.g.f.: x*(2 + x)^2*(4 + x)*exp(x)/8. (End)

A067956 Number of nodes in virtual, "optimal", chordal graphs of diameter 4 and degree n+1.

Original entry on oeis.org

9, 16, 41, 66, 129, 192, 321, 450, 681, 912, 1289, 1666, 2241, 2816, 3649, 4482, 5641, 6800, 8361, 9922, 11969, 14016, 16641, 19266, 22569, 25872, 29961, 34050, 39041, 44032, 50049, 56066, 63241, 70416, 78889

Offset: 1

S. Bujnowski & B. Dubalski (slawb(AT)atr.bydgoszcz.pl), Mar 08 2002

			For n=5, n is odd; t=3; a(5) = (2*3*(3+1)*(3^2+3+4)/3)+1 = ((6*4*16)/3)+1 = 129.
For n=6, n is even; t=3; a(6) = a(5) + ((2*3+1)*(2*t^2+2*t+3))/3 = 129 + (7*27)/3 = 192.

Concrete Mathematics, R. L. Graham, D. E. Knuth, O. Patashnik, 1994, Addison-Wesley Company, Inc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Ronald Cools, Ian H. Sloan, Minimial cubature formulae of trigonometric degree, Math. Comp. 65 (216) (1996) 1583-1600. Table 1 dimension 4.
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A006007.

for n from 1 to k do if ((n mod 2 ) = 1) then t := (n+1)/2; a[n] := ((2*(t*(t+1)*(t^2+t+4))/3)+1); else t := (n)/2; a[n] := ((2*(t*(t+1)*(t^2+t+4)/3)+1)+(2*t+1)*(2*t^2+2*t+3)/3); fi; print(a[n]); od;

Array[((2 #2 (#2 + 1) (#2^2 + #2 + 4))/3) + 1 + (Boole[EvenQ[#1]]*((2 #2 + 1) (2 #2^2 + 2 #2 + 3))/3) & @@ {#, (# + Boole[OddQ[#]])/2} &, 35] (* Michael De Vlieger, Jul 29 2022 *)

For n odd, t = (n+1)/2, a(n) = ((2*t*(t+1)*(t^2+t+4))/3)+1;
for n even, t = n/2, a(n) = (((2*t*(t+1)*(t^2+t+4))/3)+1)+((2*t+1)*(2*t^2+2*t+3))/3.
G.f.: -x*(9-2*x-9*x^2+6*x^3+11*x^4-6*x^5-3*x^6+2*x^7)/(1+x)^3/(x-1)^5 . - R. J. Mathar, Apr 07 2025

A355010 Array read by ascending antidiagonals: T(n, k) is the number of n-core partitions with k corners.

Original entry on oeis.org

1, 3, 1, 6, 5, 1, 10, 16, 7, 1, 15, 40, 31, 9, 1, 21, 85, 105, 51, 11, 1, 28, 161, 295, 219, 76, 13, 1, 36, 280, 721, 771, 396, 106, 15, 1, 45, 456, 1582, 2331, 1681, 650, 141, 17, 1, 55, 705, 3186, 6244, 6083, 3235, 995, 181, 19, 1, 66, 1045, 5985, 15156, 19348, 13663, 5685, 1445, 226, 21, 1

Offset: 2

Stefano Spezia, Jun 15 2022

T(n, k) is also equal to the number of cornerless Motzkin paths of length 2*k + n - 1 with n - 1 flat steps (see Theorem 3.3 and Proposition 3.4 at pp. 13 - 14 in Cho et al.).

In proposition 3.4 in Cho et al., the Narayana number is defined as N(k, i) = binomial(k, i)*binomial(k, i-1)/k, unlike A001263.

			The array begins:
    1,  1,   1,   1,    1,    1,    1,    1, ...
    3,  5,   7,   9,   11,   13,   15,   17, ...
    6, 16,  31,  51,   76,  106,  141,  181, ...
   10, 40, 105, 219,  396,  650,  995, 1445, ...
   15, 85, 295, 771, 1681, 3235, 5685, 9325, ...
   ...

Hyunsoo Cho, JiSun Huh, Hayan Nam, and Jaebum Sohn, Combinatorics on bounded free Motzkin paths and its applications, arXiv:2205.15554 [math.CO], 2022.

Cf. A000012 (n = 2), A001263, A005408 (n = 3), A005891 (n = 4), A006007, A063490 (n = 5), A160747 (n = 6), A161680 (k = 1), A355011.

T[n_,k_]:=Sum[Binomial[k,i]Binomial[k,i-1]Binomial[n+2(k-i),2k]/k,{i,Min[k,Floor[n/2]]}]; Flatten[Table[T[n-k+1,k],{n,2,12},{k,1,n-1}]]

T(n, k) = Sum_{i=1..min(k,floor(n/2))} N(k, i)*binomial(n+2*(k-i), 2*k), where N(k, i) = binomial(k, i)*binomial(k, i-1)/k. (See proposition 3.4 in Cho et al.).

T(n, 2) = A006007(n-1).

A383957 Sum of the legs of the unique primitive Pythagorean triple whose inradius is A000108(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

7, 7, 17, 71, 449, 3697, 35377, 369799, 4095521, 47297537, 564278417, 6911822737, 86538816337, 1103803791601, 14305269324961, 187980077927431, 2500329797088481, 33615543666867361, 456277457385934801, 6246438372527004961, 86175353802778434481, 1197196443885744428881, 16738118900659230353761

Offset: 0

Miguel-Ángel Pérez García-Ortega, May 16 2025

			For n=1, the short leg is A383251(1,1) = 3 and the long leg is A383251(1,2) = 4 so the sum of the legs is then a(1) = 3 + 4 = 7.

José Miguel Blanco Casado and Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

Cf. A000108, A383251, A381483, A382114, A006007, A058919, A336535.

a=Table[(2n)!/(n!(n+1)!),{n,0,23}];Apply[Join,Map[{2#^2+4#+1}&,a]]

a(n) = A383251(n,1) + A383251(n,2).

a(n) = 2*(A000108(n))^2 + 4*A000108(n) + 1.

cp's OEIS Frontend

A081282 Generalized centered polygonal numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A081283 An interleaved sequence of pyramidal and polygonal numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A264854 a(n) = n*(n + 1)*(11*n^2 + 11*n - 10)/24.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A129687 A129686 * A007318.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A153977 One-fourth of partial sums of A153976.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A067956 Number of nodes in virtual, "optimal", chordal graphs of diameter 4 and degree n+1.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A355010 Array read by ascending antidiagonals: T(n, k) is the number of n-core partitions with k corners.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A383957 Sum of the legs of the unique primitive Pythagorean triple whose inradius is A000108(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Views

Author

A264854 a(n) = n(n + 1)(11n^2 + 11n - 10)/24.