new sequences

[Peter Pein]

I am very sorry. I stared 30 minutes or longer at the message and nevertheless
overlooked that I had the same example in all four sequences. I noticed it
immediately after sending the text :-(


%I A138796
%S A138796 2,3,4,3,6,4,8,4,10,6,5,7,5,6,16,9,6,10,6,8,7,12,9,7,8,7,28,15,8,16,32,
%T A138796 8,10,8,13,19,11,9,10,21,9,22,9,10,13,24,17,10,12,11,10,27,10,13,11,12,
%U A138796 16,30,11,31,17,11,64,11,18,34,12,14,13,36,12,37,20,12,13,12,21,40,18
%N A138796 Least possible k>0 with T(k)-T(j)=n, where T(i)>0 are the
triangular numbers A000217
%H A138796 There is a <a
href="http://freenet-homepage.de/Peter_Berlin/triadiff.nb">Mathematica
notebook containing a faster algorithm</a>.
%Y A138796 Cf. A000217,A109814,A118235,A136107,A138797,A138798,A138799.
%O A138796 2,1
%e A138796 a(30)=8, because 30 = T(30)-T(29)=T(11)-T(8)=T(9)-T(5)=T(8)-T(3)
and 8 is the least index of the minuends
%t A138796
T=#(#+1)/2&;Min[k/.{ToRules[Reduce[{T[k]-T[j]\[Equal]#,0<j<k},{j,k},Integers]]}]&/@Range[2,100]
%K A138796 nonn
%C A138796 for T(k) see A138797, for j see A138798 and for T(j) see A138799.
%C A138796 The number of ways n can be written as difference of two triangular
numbers is sequence A136107
%A A138796 Peter Pein (petsie(AT)dordos.net), Mar 30, 2008


%I A138797
%S A138797 3,6,10,6,21,10,36,10,55,21,15,28,15,21,136,45,21,55,21,36,28,78,45,28,
%T A138797 36,28,406,120,36,136,528,36,55,36,91,190,66,45,55,231,45,253,45,55,91,
%U A138797 300,153,55,78,66,55,378,55,91,66,78,136,465,66,496,153,66,2080,66,171
%N A138797 Least possible T(k) with T(k)-T(j)=n, where T(i)>0 are the
triangular numbers A000217
%Y A138797 Cf. A000217,A109814,A118235,A136107,A138796,A138798,A138799.
%O A138797 2,1
%e A138797 a(4)=10, because T(A138796(4))=10
%t A138797
T=#(#+1)/2&;T[Min[k/.{ToRules[Reduce[{T[k]-T[j]\[Equal]#,0<j<k},{j,k},Integers]]}]]&/@Range[2,100]
%K A138797 nonn
%C A138797 for k see A138796, for j see A138798 and for T(j) see A138799.
%C A138797 The number of ways n can be written as difference of two triangular
numbers is sequence A136107
%A A138797 Peter Pein (petsie(AT)dordos.net), Mar 30, 2008


%I A138798
%S A138798 1,2,3,1,5,2,7,1,9,4,2,5,1,3,15,7,2,8,1,5,3,10,6,2,4,1,27,13,3,14,31,2,
%T A138798 6,1,10,17,7,3,5,19,2,20,1,4,9,22,14,3,7,5,2,25,1,8,4,6,12,28,3,29,13,
%U A138798 2,63,1,14,32,4,8,6,34,3,35,16,2,5,1,17,38,13,4,18,40,6,3,19,11,2,43,1
%N A138798 Corresponding j to least possible k>0 with T(k)-T(j)=n, where
T(i)>0 are the triangular numbers A000217
%Y A138798 Cf. A000217,A109814,A118235,A136107,A138796,A138797,A138799.
%O A138798 2,2
%e A138798 a(30)=3, because 30 = T(30)-T(29)=T(11)-T(8)=T(9)-T(5)=T(8)-T(3)
and 3 is the least index of the subtrahends
%t A138798
T=#(#+1)/2&;Sort[{k,j}/.{ToRules[Reduce[{T[k]-T[j]\[Equal]#,0<j<k},{j,k},Integers]]}][[1,2]]&/@Range[2,100]
%K A138798 nonn
%C A138798 for k see A138796, for T(k) see A138797 and for T(j) see A138799.
%C A138798 The number of ways n can be written as difference of two triangular
numbers is sequence A136107
%A A138798 Peter Pein (petsie(AT)dordos.net), Mar 30, 2008


%I A138799
%S A138799 1,3,6,1,15,3,28,1,45,10,3,15,1,6,120,28,3,36,1,15,6,55,21,3,10,1,378,
%T A138799 91,6,105,496,3,21,1,55,153,28,6,15,190,3,210,1,10,45,253,105,6,28,15,
%U A138799 3,325,1,36,10,21,78,406,6,435,91,3,2016,1,105,528,10,36,21,595,6,630
%N A138799 Corresponding T(j) to least possible T(k) with T(k)-T(j)=n, where
T(i)>0 are the triangular numbers A000217
%Y A138799 Cf. A000217,A109814,A118235,A136107,A138796,A138797,A138798.
%O A138799 2,2
%e A138799 a(30)=6, because 30 = T(30)-T(29)=T(11)-T(8)=T(9)-T(5)=T(8)-T(3)
and T(3)=6 is the least minuend
%t A138799
T=#(#+1)/2&;T[Sort[{k,j}/.{ToRules[Reduce[{T[k]-T[j]\[Equal]#,0<j<k},{j,k},Integers]]}][[1,2]]]&/@Range[2,100]
%K A138799 nonn
%C A138799 for k see A138796,  for T(k) see A138797 and for j see A138798.
%C A138799 The number of ways n can be written as difference of two triangular
numbers is sequence A136107
%A A138799 Peter Pein (petsie(AT)dordos.net), Mar 30, 2008