A007584 -id:A007584 - cp's OEIS frontend

A218329 Even 9-gonal (nonagonal) pyramidal numbers.

10, 34, 80, 266, 420, 624, 1210, 1606, 2080, 3290, 4040, 4896, 6954, 8170, 9520, 12650, 14444, 16400, 20826, 23310, 25984, 31930, 35216, 38720, 46410, 50610, 55056, 64714, 69940, 75440, 87290, 93654, 100320, 114586, 122200, 130144, 147050, 156026, 165360

Offset: 1

Ant King, Oct 28 2012

nonn

			The sequence of 9-gonal (nonagonal) pyramidal numbers A007584 begins 1, 10, 34, 80, 155, 266, 420, 624, 885, 1210,.... As the third even term is 80, then a(3) = 80.

Index entries for linear recurrences with constant coefficients, signature (1, 0, 3, -3, 0, -3, 3, 0, 1, -1).

Cf. A007584, A218328.

LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1},{10,34,80,266,420,624,1210,1606,2080,3290},39]

a(n) = a(n-1) + 3*a(n-3) - 3*a(n-4) - 3*a(n-6) + 3*a(n-7) + a(n-9) - a(n-10).
a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9) + 448.
a(n) = phi(n)*(phi(n)+9)*(7*phi(n)-36)/4374, where phi(n) = 3 + 12*n - 3*cos(2*n*Pi/3) + sqrt(3)*sin(2*n*Pi/3).
G.f.: 2*x*(5+12*x+23*x^2+78*x^3+41*x^4+33*x^5+29*x^6+3*x^7)/((1-x)^4*(1+x+x^2)^3).

A255687 a(n) = n(n + 1)(7*n + 11)/6.

Original entry on oeis.org

0, 6, 25, 64, 130, 230, 371, 560, 804, 1110, 1485, 1936, 2470, 3094, 3815, 4640, 5576, 6630, 7809, 9120, 10570, 12166, 13915, 15824, 17900, 20150, 22581, 25200, 28014, 31030, 34255, 37696, 41360, 45254, 49385, 53760, 58386, 63270, 68419, 73840, 79540, 85526

Offset: 0

Luce ETIENNE, Mar 02 2015

This sequence gives the number of triangles of all sizes in (3*n^2+2*n)-polyiamonds in a pentagonal or heptagonal configuration.

Also sum of 2*n*(n+1)*(n+2)/3 triangles oriented in one direction and n*(n+1)^2/2 oriented in the opposite direction.

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

First bisection of A212977.
Partial sums of A179986.
Cf. A000292, A006002, A007584, A255211.

[n*(n+1)*(7*n+11)/6: n in [0..50]]; // Bruno Berselli, Mar 02 2015

A255687:=n->n*(n+1)*(7*n+11)/6: seq(A255687(n), n=0..50); # Wesley Ivan Hurt, Mar 03 2015

Table[n (n + 1) (7 n + 11)/6, {n, 0, 50}] (* Bruno Berselli, Mar 02 2015 *)
LinearRecurrence[{4,-6,4,-1},{0,6,25,64},50] (* Harvey P. Dale, Jul 17 2015 *)

PARI
```
vector(50, n, n--; n*(n+1)*(7*n+11)/6)
```

concat(0, Vec(x*(x+6)/(x-1)^4 + O(x^100))) \\ Colin Barker, Mar 02 2015

[n*(n+1)*(7*n+11)/6 for n in (0..50)] # Bruno Berselli, Mar 02 2015

a(n) = (1/2)*(Sum_{j=0..n} (n+1-j)*(3*n-j) + Sum_{j=0..n-1} (n-j)*(3*n+1-3*j)).
From Colin Barker, Mar 02 2015: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x*(x + 6)/(x - 1)^4. (End)
a(n) = -A007584(-n-1). - Bruno Berselli, Mar 02 2015
From Elmo R. Oliveira, Aug 18 2025: (Start)
E.g.f.: exp(x)*x*(36 + 39*x + 7*x^2)/6.
a(n) = A212977(2*n). (End)

A266086 Alternating sum of 9-gonal (or nonagonal) numbers.

Original entry on oeis.org

0, -1, 8, -16, 30, -45, 66, -88, 116, -145, 180, -216, 258, -301, 350, -400, 456, -513, 576, -640, 710, -781, 858, -936, 1020, -1105, 1196, -1288, 1386, -1485, 1590, -1696, 1808, -1921, 2040, -2160, 2286, -2413, 2546, -2680, 2820, -2961, 3108, -3256, 3410

Offset: 0

Ilya Gutkovskiy, Dec 21 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Nonagonal Number
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A001106, A006578, A007584, A035608, A083392, A089594.

[(14*(-1)^n*n^2 + 4*(-1)^n*n - 5*(-1)^n + 5)/8: n in [0..50]]; // Vincenzo Librandi, Dec 21 2015

Table[((14 n^2 + 4 n - 5) (-1)^n + 5)/8, {n, 0, 44}]
CoefficientList[Series[(x - 6 x^2)/(x^4 + 2 x^3 - 2 x - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)

x='x+O('x^100); concat(0, Vec(-x*(1-6*x)/((1-x)*(1+x)^3))) \\ Altug Alkan, Dec 21 2015

G.f.: -x*(1 - 6*x)/((1 - x)*(1 + x)^3).
a(n) = ((14*n^2 + 4*n - 5)*(-1)^n + 5)/8.
a(n) = Sum_{k = 0..n} (-1)^k*A001106(k).
Lim_{n -> infinity} a(n + 1)/a(n) = -1.

A349147 Triangle T(n,m) read by rows: the sum of runs of all sequences arranging n objects of one type and m objects of another type.

Original entry on oeis.org

1, 1, 4, 1, 7, 18, 1, 10, 34, 80, 1, 13, 55, 155, 350, 1, 16, 81, 266, 686, 1512, 1, 19, 112, 420, 1218, 2982, 6468, 1, 22, 148, 624, 2010, 5412, 12804, 27456, 1, 25, 189, 885, 3135, 9207, 23595, 54483, 115830, 1, 28, 235, 1210, 4675, 14872, 41041, 101530, 230230, 486200, 1, 31

Offset: 0

R. J. Mathar, Nov 08 2021

			The triangle starts
  1,
  1,  4,
  1,  7,  18,
  1, 10,  34,   80,
  1, 13,  55,  155,  350,
  1, 16,  81,  266,  686,  1512,
  1, 19, 112,  420, 1218,  2982,  6468,
  1, 22, 148,  624, 2010,  5412, 12804,  27456,
  1, 25, 189,  885, 3135,  9207, 23595,  54483, 115830,
  1, 28, 235, 1210, 4675, 14872, 41041, 101530, 230230, 486200,
  1, 31, 286, 1606, 6721, 23023, 68068, 179608, 432718, 967538, 2032316
For n=m=1 the sequences are ab (2 runs) and ba (2 runs), so T(1,1)=2+2=4.
For n=1, m=2 the sequences are aab (2 runs), aba (3 runs), baa (2 runs), so T(1,2)=2+3+2=7.
For n=m=2 the sequences are aabb (2 runs), abab (4 runs), abba (3 runs), baab (3 runs), baba (4 runs), bbaa (2 runs), so T(2,2) = 2+4+3+3+4+2=18.

R. J. Mathar, The Number of Runs of Words on a 2-letter Alphabet (2021)

Cf. A016777 (row/col 1), A000566 (row/col 2), A007584 (row/col 3), A051798 (row/col 4).

Diagonal gives A037965(n+1).

T(n,m) = T(m,n).
Sum_{m=0..n} T(n,m) = A000917(n-1) + A000984(n) = 1, 5, 26, 125, 574, ... - R. J. Mathar, Nov 09 2021
T(n,m) = binomial(n+m,n)*(2*n*m+n+m)/(n+m) for n+m >= 1.

A269440 Alternating sum of 9-gonal (or decagonal) pyramidal numbers.

Original entry on oeis.org

0, -1, 9, -25, 55, -100, 166, -254, 370, -515, 695, -911, 1169, -1470, 1820, -2220, 2676, -3189, 3765, -4405, 5115, -5896, 6754, -7690, 8710, -9815, 11011, -12299, 13685, -15170, 16760, -18456, 20264, -22185, 24225, -26385, 28671, -31084, 33630, -36310, 39130

Offset: 0

Ilya Gutkovskiy, Feb 26 2016

OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Pyramidal Number
Index entries for linear recurrences with constant coefficients, signature (-3,-2,2,3,1).

Cf. A000292, A002717, A007584, A173196, A266677.

Table[(-1)^n (2 n - 1) ((14 n^2 + 34 n + 15)/48) + 5/16, {n, 0, 40}]
LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 9, -25, 55}, 41]

G.f.: x*(1 - 6*x)/((x - 1)*(x + 1)^4).
a(n) = (-1)^n*(2*n - 1)*(14*n^2 + 34*n + 15)/48 + 5/16.
a(n) = Sum_{k = 0..n} (-1)^k*A007584(k).

cp's OEIS Frontend

A218329 Even 9-gonal (nonagonal) pyramidal numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A255687 a(n) = n*(n + 1)*(7*n + 11)/6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A266086 Alternating sum of 9-gonal (or nonagonal) numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A349147 Triangle T(n,m) read by rows: the sum of runs of all sequences arranging n objects of one type and m objects of another type.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A269440 Alternating sum of 9-gonal (or decagonal) pyramidal numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A255687 a(n) = n(n + 1)(7*n + 11)/6.