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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).

%I A076110 #24 Mar 30 2023 12:33:11
%S A076110 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,
%T A076110 1,8,15,22,29,36,43,50,1,9,17,25,33,41,49,57,65,1,10,19,28,37,46,55,
%U A076110 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
%N 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).
%C A076110 Leading diagonal contains n^2 + 1 (A002522).
%C A076110 Sum of the n-th row is (n+1)(n^2+2)/2 (A064808).
%H A076110 Robert Israel, <a href="/A076110/b076110.txt">Table of n, a(n) for n = 1..10011</a>(rows 1 to 141, flattened)
%F A076110 A076110(n) = L(n) with L=seq(seq(n*k+1, k = 0..n), n = 0..+inf). - _Yalcin Aktar_, Jul 14 2009
%F A076110 From _Robert Israel_, Dec 04 2018: (Start)
%F A076110 T(n,k) = 1 + (n-1)*(k-1).
%F A076110 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).
%F A076110 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.
%F A076110 (End)
%e A076110 1;
%e A076110 1, 2;
%e A076110 1, 3, 5;
%e A076110 1, 4, 7, 10;
%e A076110 1, 5, 9, 13, 17;
%e A076110 1, 6, 11, 16, 21, 26;
%e A076110 1, 7, 13, 19, 25, 31, 37; ...
%p A076110 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
%t A076110 T[n_, k_] := 1 + (n-1) * (k-1); Table[T[n, k], {n,1,10}, {k,1,n}] // Flatten (* _Amiram Eldar_, Dec 04 2018 *)
%o A076110 (GAP) Flat(List([1..12],n->List([1..n],k->1+(n-1)*(k-1)))); # _Muniru A Asiru_, Dec 05 2018
%o A076110 (Magma) /* As triangle */ [[1+(n-1)*(k-1): k in [1..n]]: n in [1.. 12]]; // _Vincenzo Librandi_, Dec 05 2018
%Y A076110 Cf. A002522, A064808, A076111 (row products), A079904.
%K A076110 nonn,tabl,easy
%O A076110 1,3
%A A076110 _Amarnath Murthy_, Oct 09 2002
%E A076110 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003