A076110 Triangle (read by rows) in which the n-th row contains first n terms of an arithmetic progression with first term 1 and common difference (n-1).

1, 1, 2, 1, 3, 5, 1, 4, 7, 10, 1, 5, 9, 13, 17, 1, 6, 11, 16, 21, 26, 1, 7, 13, 19, 25, 31, 37, 1, 8, 15, 22, 29, 36, 43, 50, 1, 9, 17, 25, 33, 41, 49, 57, 65, 1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 101, 1, 12, 23, 34, 45, 56, 67, 78, 89, 100, 111, 122

Offset: 1

Author

Amarnath Murthy, Oct 09 2002

Keywords

• nonn
• tabl
• easy

Comments

Leading diagonal contains n^2 + 1 (A002522).

Sum of the n-th row is (n+1)(n^2+2)/2 (A064808).

Examples

			1;
1, 2;
1, 3, 5;
1, 4, 7, 10;
1, 5, 9, 13, 17;
1, 6, 11, 16, 21, 26;
1, 7, 13, 19, 25, 31, 37; ...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10011(rows 1 to 141, flattened)

Crossrefs

Cf. A002522, A064808, A076111 (row products), A079904.

Programs

GAP

Flat(List([1..12],n->List([1..n],k->1+(n-1)*(k-1)))); # Muniru A Asiru, Dec 05 2018

Magma

/* As triangle */ [[1+(n-1)*(k-1): k in [1..n]]: n in [1.. 12]]; // Vincenzo Librandi, Dec 05 2018

Maple

T:= (n,k) -> 1+(n-1)*(k-1):for n from 1 to 10 do seq(T(n,k),k=1..n) od; # Robert Israel, Dec 04 2018

Mathematica

T[n_, k_] := 1 + (n-1) * (k-1); Table[T[n, k], {n,1,10}, {k,1,n}] // Flatten (* Amiram Eldar, Dec 04 2018 *)

Formula

A076110(n) = L(n) with L=seq(seq(n*k+1, k = 0..n), n = 0..+inf). - Yalcin Aktar, Jul 14 2009

From Robert Israel, Dec 04 2018: (Start)

T(n,k) = 1 + (n-1)*(k-1).

G.f. as triangle: (1-x-x*y+2*x^2*y+2*x^2*y^2-3*x^3*y^2)*x*y/((1-x)^2*(1-x*y)^3).

G.f. as sequence: x/(1-x) + Sum_{m>=0} (-m*(m+1)*x^((m^2+3*m+4)/2) + (1+m*(m+1))*x^((m^2+3*m+6)/2))/(1-x)^2.

(End)

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003