A005915 -id:A005915 - cp's OEIS frontend

A245556 Irregular triangle read by rows: T(n,k) (n>=0, 0 <= k <= 2n) = number of triples (u,v,w) with entries in the range 0 to n which have some pair adding up to k.

1, 4, 6, 4, 7, 12, 19, 12, 7, 10, 18, 28, 36, 28, 18, 10, 13, 24, 37, 48, 61, 48, 37, 24, 13, 16, 30, 46, 60, 76, 90, 76, 60, 46, 30, 16, 19, 36, 55, 72, 91, 108, 127, 108, 91, 72, 55, 36, 19, 22, 42, 64, 84, 106, 126, 148, 168, 148, 126, 106, 84, 64, 42, 22

Offset: 0

N. J. A. Sloane, Aug 04 2014

			Triangle begins:
[1]
[4, 6, 4]
[7, 12, 19, 12, 7]
[10, 18, 28, 36, 28, 18, 10]
[13, 24, 37, 48, 61, 48, 37, 24, 13]
[16, 30, 46, 60, 76, 90, 76, 60, 46, 30, 16]
[19, 36, 55, 72, 91, 108, 127, 108, 91, 72, 55, 36, 19]
[22, 42, 64, 84, 106, 126, 148, 168, 148, 126, 106, 84, 64, 42, 22]
...
See A245557 for specific examples; also the Example section of A090381 for some of the T(10,10)= 331 triples with n=k=10.

Rows are the partial sums of the rows of A245557.
Main "spine" of triangle is A090381.
Row sums are A005915.

with(LinearAlgebra);
M:=10; A:=Array(0..M,0..2*M); B:=Array(0..M,0..2*M);
for n from 0 to M do
for i from 0 to n do for j from 0 to n do for k from 0 to n do
  s1:={i+j,i+k,j+k}; s1:=convert(s1,list); m1:=max(i,j,k);
    for r1 from 1 to nops(s1) do
       s:=s1[r1]; A[n,s] := A[n,s]+1;
       if (m1=n) then B[n,s] := B[n,s]+1; fi;
                              od:
od: od: od: od:
lprint("A245556");
for i from 0 to M do lprint([seq(A[i,j],j=0..2*i)]); od:
lprint("A245557");
for i from 0 to M do lprint([seq(B[i,j],j=0..2*i)]); od:

A143804 Triangle read by rows, thrice the Connell numbers (A001614) - 2.

Original entry on oeis.org

1, 4, 10, 13, 19, 25, 28, 34, 40, 46, 49, 55, 61, 67, 73, 76, 82, 88, 94, 100, 106, 109, 115, 121, 127, 133, 139, 145, 148, 154, 160, 166, 172, 178, 184, 190, 193, 199, 205, 211, 217, 223, 229, 235, 241, 244, 250, 256, 262, 268, 274, 280, 286, 292, 298

Offset: 1

Gary W. Adamson, Sep 01 2008

Right border of the triangle = A100536: (1, 10, 25, 46, 73,...).
Left border = A056107: (1, 4, 13, 28, 49,...).
Row sums = A005915: (1, 14, 57, 148, 305,...).
n-th row = (right border then going to the left): (n-th term of A100536 followed by (n-1) operations of (-6), (-6), (-6),... As a Connell-like triangle, odd row terms are in the subset 6n-5; even row terms are in the set 6n-2.
Row 3 = (13, 19, 25) beginning with A100536(3) = 25 at the right then following the trajectory (-6), (-6).
Using the modular rules, the triangle begins (1; 4, 10; 13, 19, 25;...) since 1 == 6n-5, while 4 is the next higher term in the set 6n-2, then 10 also in the set 6n-2, being an even row.

			First few rows of the triangle:
  1;
  4, 10;
  13, 19, 25;
  28, 34, 40, 46;
  49, 55, 61, 67, 73;
  76, 82, 88, 94, 100, 106;
  ...

Cf. A001614, A100536, A056107, A005915.

from math import isqrt
def A143804(n): return 3*((m:=n<<1)-(k:=isqrt(m))-int(m>=k*(k+1)+1))-2 # Chai Wah Wu, Aug 01 2022

a(n) = 3*A001614(n) - 2.

A095266 A sequence generated from the Narayana triangle considered as a matrix, or from Pascal's triangle.

Original entry on oeis.org

1, 42, 303, 1144, 3105, 6906, 13447, 23808, 39249, 61210, 91311, 131352, 183313, 249354, 331815, 433216, 556257, 703818, 878959, 1084920, 1325121, 1603162, 1922823, 2288064, 2703025, 3172026, 3699567, 4290328, 4949169, 5681130

Offset: 1

Gary W. Adamson, May 31 2004

A095267 has the same recursion rule but is derived from the matrix derived from A056939 (a type of generalized Narayana triangle).

			a(7) = 23808 = 5*a(6) - 10*a(5) + 10*a(4) - 5*a(3) + a(2) = 5*13447 - 10*6906 + 10*3105 - 5*1144 + 303.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A095267, A001263.

a[n_] := (MatrixPower[{{1, 0, 0, 0, 0}, {1, 1, 0, 0, 0}, {1, 3, 1, 0, 0}, {1, 6, 6, 1, 0}, {1, 10, 20, 10, 1}}, n].{{1}, {0}, {0}, {0}, {0}})[[5, 1]]; Table[ a[n], {n, 30}] (* Robert G. Wilson v, Jun 05 2004 *)

a(n+6) = 5*a(n+5) - 10*a(n+4) + 10*a(n+3) - 5*a(n+2) + a(n), where the multipliers with changed signs are found in the characteristic polynomial of the generating matrix M: x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1. Let M be the 5th-order Matrix M, having Narayana triangle (A001263) rows (fill in with zeros): [1 0 0 0 0 / 1 1 0 0 0 / 1 3 1 0 0 / 1 6 6 1 0 / 1 10 20 10 1]. Then M^n *[1 0 0 0 0] = [1 n A000326(n) A005915(n) a(n)] where A000326 = the pentagonal numbers and A005915 = the hex prism numbers.
From Colin Barker, Oct 21 2012: (Start)
a(n) = (n*(-8 + 25*n - 30*n^2 + 15*n^3))/2.
G.f.: -x*(39*x^3 + 103*x^2 + 37*x + 1)/(x-1)^5. (End)

Edited and corrected by Robert G. Wilson v, Jun 05 2004

Typo in recurrence fixed by Colin Barker, Oct 21 2012

A112338 Triangle read by rows, generated from A001263.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 12, 14, 1, 1, 5, 22, 57, 42, 1, 1, 6, 35, 148, 303, 132, 1, 1, 7, 51, 305, 1144, 1743, 429, 1, 8, 70, 546, 3105, 9784, 10629, 1430, 1

Offset: 0

Gary W. Adamson, Sep 04 2005

Rows of the array are row sums of n-th powers of the Narayana triangle; e.g., row 1 = A000108: (1, 2, 5, 14, 42, ...); row 2 = row sums of the Narayana triangle squared (A103370): (1, 3, 12, 57, 303, ...), etc.

			In the array, antidiagonal terms (1, 3, 5, 1) become row 3 of the triangle.
First few rows of the array:
  1, 1,  1,   1,    1,     1, ...
  1, 2,  5,  14,   42,   132, ...
  1, 3, 12,  57,  303,  1743, ...
  1, 4, 22, 148, 1144,  9784, ...
  1, 5, 35, 305, 3105, 35505, ...
First few rows of the triangle:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  5,   1;
  1, 4, 12,  14,   1;
  1, 5, 22,  57,  42,   1;
  1, 6, 35, 148, 303, 132, 1;

Cf. A001263, A000326, A005915, A095266, A000108, A103370.

Let M be the infinite lower triangular Narayana triangle (A001263). Perform M^n * [1 0 0 0 ...] getting an array. Take antidiagonals of the array which become rows of the triangle A112338.

A329530 a(n) = n * (7*binomial(n, 2) + 1).

Original entry on oeis.org

0, 1, 16, 66, 172, 355, 636, 1036, 1576, 2277, 3160, 4246, 5556, 7111, 8932, 11040, 13456, 16201, 19296, 22762, 26620, 30891, 35596, 40756, 46392, 52525, 59176, 66366, 74116, 82447, 91380, 100936, 111136, 122001, 133552, 145810, 158796, 172531, 187036, 202332, 218440

Offset: 0

Ilya Gutkovskiy, Nov 15 2019

Centered heptagonal prism numbers.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), 144.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Centered m-gonal prism numbers: A100175 (m = 3), A059722 (m = 4), A006564 (m = 5), A005915 (m = 6), this sequence (m = 7), A139757 (m = 8), A006566 (m = 9).

Cf. A000217, A002413, A006597, A024966, A069099.

Table[n (7 Binomial[n, 2] + 1), {n, 0, 40}]
nmax = 40; CoefficientList[Series[x (1 + 12 x + 8 x^2)/(1 - x)^4, {x, 0, nmax}], x]
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 16, 66}, 41]

G.f.: x * (1 + 12*x + 8*x^2) / (1 - x)^4.
E.g.f.: exp(x) * x * (2 + 14*x + 7*x^2) / 2.
a(n) = n * (7*n^2 - 7*n + 2) / 2.
a(n) = n * (7*A000217(n-1) + 1).
a(n) = n * A069099(n).

cp's OEIS Frontend

A245556 Irregular triangle read by rows: T(n,k) (n>=0, 0 <= k <= 2n) = number of triples (u,v,w) with entries in the range 0 to n which have some pair adding up to k.

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A143804 Triangle read by rows, thrice the Connell numbers (A001614) - 2.

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A095266 A sequence generated from the Narayana triangle considered as a matrix, or from Pascal's triangle.

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A112338 Triangle read by rows, generated from A001263.

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A329530 a(n) = n * (7*binomial(n, 2) + 1).

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