A076110 - OEIS

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

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OFFSET

1,3

COMMENTS

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

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

LINKS

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

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)

EXAMPLE

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; ...

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A368563 A093412 A119355 * A117584 A199847 A047997

Adjacent sequences: A076107 A076108 A076109 * A076111 A076112 A076113

KEYWORD

nonn,tabl,easy

AUTHOR

Amarnath Murthy, Oct 09 2002

EXTENSIONS

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

STATUS

approved