A128229 -id:A128229 - cp's OEIS frontend

A056108 Fourth spoke of a hexagonal spiral.

1, 5, 15, 31, 53, 81, 115, 155, 201, 253, 311, 375, 445, 521, 603, 691, 785, 885, 991, 1103, 1221, 1345, 1475, 1611, 1753, 1901, 2055, 2215, 2381, 2553, 2731, 2915, 3105, 3301, 3503, 3711, 3925, 4145, 4371, 4603, 4841, 5085, 5335, 5591, 5853, 6121, 6395

Offset: 0

Henry Bottomley, Jun 09 2000

a(n) = sum of (n+1)-th row terms of triangle A134234. - Gary W. Adamson, Oct 14 2007
If Y is a 4-subset of an n-set X then, for n >= 4, a(n-4) is the number of 4-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 08 2007
Equals binomial transform of [1, 4, 6, 0, 0, 0, ...] - Gary W. Adamson, Apr 30 2008
From A.K. Devaraj, Sep 18 2009: (Start)
Let f(x) be a polynomial in x. Then f(x + n*f(x)) is congruent to 0 (mod f(x)); here n belongs to N.
There is nothing interesting in the quotients f(x + n*f(x))/f(x) when x belongs to Z.
However, when x is irrational these quotients consist of two parts, a) rational integers and b) integer multiples of x.
The present sequence is the integer part when the polynomial is x^2 + x + 1 and x = sqrt(2),
f(x+n*f(x))/f(x) = a(n) + A005563(n)*sqrt(2).
Equals triangle A128229 as an infinite lower triangular matrix * A016777 as a vector, where A016777(n) = (3*n+1). (End)
Numbers of the form ((-h^2+h+1)^2+(h^2-h+1)^2+(h^2+h-1)^2)/(h^2+h+1) for h=n+1. - Bruno Berselli, Mar 13 2013

G. C. Greubel, Table of n, a(n) for n = 0..5000
Henry Bottomley, Illustration of initial terms
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A134234, A000217.
Cf. A005563, A016777, A128229.
Other spokes: A003215, A056105, A056106, A056107, A056109.
Other spirals: A054552.

[3*n^2+n+1: n in [0..50]]; // Bruno Berselli, Mar 13 2013

Table[3 n^2 + n + 1, {n, 0, 50}] (* Bruno Berselli, Mar 13 2013 *)
LinearRecurrence[{3,-3,1},{1,5,15},50] (* Harvey P. Dale, Dec 26 2023 *)

a(n)=3*n^2+n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*n^2 + n + 1.
a(n) = a(n-1) + 6*n - 2 = 2*a(n-1) - a(n-2) + 6
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A056105(n) + 3*n = A056106(n) + 2*n = A056107(n) + n = A056109(n) - n = A003215(n) - 2*n.
a(n) = A096777(3n+1) . - Reinhard Zumkeller, Dec 29 2007
a(n) = 6*n+a(n-1)-2 with n>0, a(0)=1. - Vincenzo Librandi, Aug 07 2010
G.f.: (1+2*x+3*x^2)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 04 2012
a(-n) = A056106(n). - Bruno Berselli, Mar 13 2013
E.g.f.: (3*x^2 + 4*x + 1)*exp(x). - G. C. Greubel, Jul 19 2017

A103450 A figurate number triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 12, 7, 1, 1, 9, 22, 22, 9, 1, 1, 11, 35, 50, 35, 11, 1, 1, 13, 51, 95, 95, 51, 13, 1, 1, 15, 70, 161, 210, 161, 70, 15, 1, 1, 17, 92, 252, 406, 406, 252, 92, 17, 1, 1, 19, 117, 372, 714, 882, 714, 372, 117, 19, 1, 1, 21, 145, 525, 1170, 1722, 1722, 1170, 525, 145, 21, 1

Offset: 0

Paul Barry, Feb 06 2005

Row coefficients are the absolute values of the coefficients of the characteristic polynomials of the n X n matrices A(n) with A(n){i,i} = 2, i>0, A(n){i,j} = 1, otherwise (starts with (0,0) position).

The triangle can be generated by the matrix multiplication A007318 * A114219s, where A114219s = 0; 0,1; 0,1,1; 0,-1,2,1; 0,1,-2,3,1; 0,-1,2,-3,4,1; ... = A097807 * A128229 is a signed variant of A114219. - Gary W. Adamson, Feb 20 2007

			From _Roger L. Bagula_, Oct 21 2008: (Start)
The triangle begins:
  1;
  1,  1;
  1,  3,   1;
  1,  5,   5,   1;
  1,  7,  12,   7,   1;
  1,  9,  22,  22,   9,   1;
  1, 11,  35,  50,  35,  11,   1;
  1, 13,  51,  95,  95,  51,  13,   1;
  1, 15,  70, 161, 210, 161,  70,  15,   1;
  1, 17,  92, 252, 406, 406, 252,  92,  17,  1;
  1, 19, 117, 372, 714, 882, 714, 372, 117, 19, 1; ... (End)

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
F. S. Al-Kharousi, A. Umar, and M. M. Zubairu, On injective partial Catalan monoids, arXiv:2501.00285 [math.GR], 2024. See p. 9.
G. Chiaselotti, W. Keith, and P. A. Oliverio, Two Self-Dual Lattices of Signed Integer Partitions, Appl. Math. Inf. Sci. 8, No. 6, 3191-3199 (2014), via ResearchGate.

Row sums are A045623.
Columns include: A000326, A002412, A002418, A005408.
Cf. A000045, A208354.

A103450:= func< n,k | k eq 0 select 1 else Binomial(n, k)*(k*(n-k) + n)/n >;
[A103450(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 17 2021

(* First program *)
p[x_, n_]:= p[x, n]= If[n==0, 1, (-1+x)^(n-2)*(1 -(n+1)*x +x^2)];
T[n_, k_]:= T[n,k]= (-1)^(n+k)*SeriesCoefficient[p[x, n], {x, 0, k}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* Roger L. Bagula and Gary W. Adamson, Oct 21 2008 *)(* corrected by G. C. Greubel, Jun 17 2021 *)
(* Second program *)
T[n_, k_]:= If[k==0, 1, Binomial[n, k]*(n*(k+1) -k^2)/n];
Table[T[n, k], {n,0,16}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 17 2021 *)

def A103450(n, k): return 1 if (k==0) else binomial(n, k)*(k*(n-k) + n)/n
flatten([[A103450(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 17 2021

T(n, k) = binomial(n-1, k-1)*(k*(n-k) + n)/k with T(n, 0) = 1.
T(n, k) = T(n-1, k-1) + T(n-1, k) + binomial(n-2, k-1) with T(n, 0) = 1.
Column k is generated by (1+k*x)*x^k/(1-x)^(k+1).
Rows are coefficients of the polynomials P(0, x) = 1, P(n, x) = (1+x)^(n-2)*(1 +(n+1)*x + x^2) for n>0.
T(n,k) = Sum_{j=0..n} binomial(k, k-j)*binomial(n-k, j)*(j+1). - Paul Barry, Oct 28 2006
A signed version arises from the coefficients of the polynomials defined by: p(x, 0) = 1, p(x, 1) = (-1 +x), p(x, 2) = (1 -3*x +x^2), p(x,n) = (-1 +x)^(n-2)*(1 - (n + 1)*x + x^2); T(n, k) = (-1)^(n+k)*coefficient of x^k of ( p(x,n) ). - Roger L. Bagula and Gary W. Adamson, Oct 21 2008
T(2*n+1, n) = A141222(n). - Emanuele Munarini, Jun 01 2012 [corrected by Werner Schulte, Nov 27 2021]
G.f.: is 1 / ( (1-q*x/(1-x)) * (1-x/(1-q*x)) ). - Joerg Arndt, Aug 27 2013
Sum_{k=0..floor(n/2)} T(n-k, k) = (1/5)*((-n+5)*Fibonacci(n+1) + (3*n- 2)*Fibonacci(n)) = A208354(n). - G. C. Greubel, Jun 17 2021
T(2*n, n) = A000984(n) * (n + 2) / 2 for n >= 0. - Werner Schulte, Nov 27 2021

A145677 Triangle T(n, k) read by rows: T(n, 0) = 1, T(n, n) = n, n>0, T(n,k) = 0, 0 < k < n-1.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 0, 3, 1, 0, 0, 0, 4, 1, 0, 0, 0, 0, 5, 1, 0, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 0, 8, 1, 0, 0, 0, 0, 0, 0, 0, 0, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11

Offset: 0

Gary W. Adamson and Roger L. Bagula, Mar 28 2009

The first entry in each row is 1, the last entry in each of the rows consist of the positive integers (starting 1,1,2,3,...), and all other entries in the triangle are 0's (see example).

The vector of (1, 1, 2, 5, 16, 65, 326,...), which is 1 followed by A000522, is an eigenvector of the matrix: 1 + Sum_{k=1..n} T(n,k)*A000522(k-1) = A000522(n).

			First few rows of the triangle:
  1;
  1, 1;
  1, 0, 2;
  1, 0, 0, 3;
  1, 0, 0, 0, 4;
  1, 0, 0, 0, 0, 5;
  1, 0, 0, 0, 0, 0, 6;
  1, 0, 0, 0, 0, 0, 0, 7;
  1, 0, 0, 0, 0, 0, 0, 0, 8;
  1, 0, 0, 0, 0, 0, 0, 0, 0, 9;
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A128229, A002061, A000522, A152271, A158416.

T[n_, k_]:= If[k==0, 1, If[k==n, n, 0]];
Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 23 2021 *)

def A145677(n,k):
    if (k==0): return 1
    elif (k==n): return n
    else: return 0
flatten([[A145677(n,k) for k in (0..n)] for n in (0..14)]) # G. C. Greubel, Dec 23 2021

T(n, k) = A158821(n,n-k).
1 + Sum_{k= 1..n} T(n,k) *(k-1) = A002061(n).
From G. C. Greubel, Dec 23 2021: (Start)
Sum_{k=0..n} T(n, k) = A000027(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A158416(n) = A152271(n+1). (End)

Edited by R. J. Mathar, Oct 02 2009

A128255 A114219(signed) * A007318.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 2, 6, 5, 1, 3, 10, 13, 7, 1, 3, 15, 27, 23, 9, 1, 4, 21, 48, 57, 36, 1, 4, 28, 78, 118, 104, 52, 13, 1, 5, 36, 118, 218, 246, 172, 71, 15, 1, 5, 45, 170, 370, 510, 458, 265, 93, 17, 1

Offset: 1

Gary W. Adamson, Feb 20 2007

Row sums = A059570: (1, 2, 6, 14, 34, 78, 178,...).

			First few rows of the triangle are:
1;
1, 1;
2, 3, 1;
2, 6, 5, 1;
3, 10, 13, 7, 1;
3, 15, 27, 23, 9, 1;
4, 21, 48, 57, 36, 11, 1;
...

Cf. A103450, A114219, A007318, A059570, A097807, A128229.

Let the signed version of A114219 {1; 0,1; 0,1,1; 0,-1,2,1; 0,1,-2,-3,4;...} = M; and P = Pascal's triangle, A007318. Then A128255 = A114219(signed) * A007318.

A128261 a(n) = tau(n) + (n-1)*tau(n-1).

Original entry on oeis.org

1, 3, 6, 9, 14, 14, 26, 18, 35, 31, 42, 28, 74, 30, 60, 65, 82, 40, 110, 44, 124, 88, 90, 54, 195, 79, 108, 114, 170, 66, 242, 68, 196, 136, 140, 149, 326, 78, 156, 164, 322, 90, 338, 92, 270, 274, 186, 104, 483, 153, 304, 210, 314, 114, 436, 228, 452, 232

Offset: 1

Gary W. Adamson, Feb 21 2007

nonn

			a(5) = 14 = (5 + 4 + 0 + 4 + 1), where (5, 4, 0, 4, 1) = row 5 of A128260.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Row sums of A128260.

Cf. A000005 (tau), A128229, A051731.

Table[DivisorSigma[0,n]+(n-1)DivisorSigma[0,n-1],{n,60}] (* Harvey P. Dale, Sep 17 2020 *)

a(n)={numdiv(n) + if(n > 1, (n-1)*numdiv(n-1))} \\ Andrew Howroyd, Aug 09 2018

Name changed and terms a(11) and beyond from Andrew Howroyd, Aug 09 2018

cp's OEIS Frontend

A056108 Fourth spoke of a hexagonal spiral.

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A103450 A figurate number triangle read by rows.

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A145677 Triangle T(n, k) read by rows: T(n, 0) = 1, T(n, n) = n, n>0, T(n,k) = 0, 0 < k < n-1.

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A128255 A114219(signed) * A007318.

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A128261 a(n) = tau(n) + (n-1)*tau(n-1).

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