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A096032 Take pairs (a, b), sorted on a, such that T(a)+T(b)=concatenation of a and b, where T(k) is the k-th triangular number A000217(k). Sequence gives values of b. 6

%I #13 Jun 15 2020 21:08:51

%S 1,415,1545,1726,2196,910,3676,3846,910,5226,415,6970,7171,8526,9231,

%T 9300,9756,9850,9880,44835,9880,9850,9756,9300,9231,52830,8526,7171,

%U 6970,5226,3846,3676,2196,1726,1545,84906,89386,99580,99580,89386,84906

%N Take pairs (a, b), sorted on a, such that T(a)+T(b)=concatenation of a and b, where T(k) is the k-th triangular number A000217(k). Sequence gives values of b.

%C For values of a see A096031.

%C It is easier to generate the pairs sorted by b. A d-digit number b is a member iff 4*(10^(2*d)-10^d-b^2+b)+1 is a square. All such b occur twice, except for 1, which occurs once. There are no members with 2, 6, 7, or 8 digits. There are six distinct nine-digit members. - _David Wasserman_, May 15 2007

%D J. S. Madachy, Madachy's Mathematical Recreations, pp. 166 Dover NY 1979.

%H Chai Wah Wu, <a href="/A096032/b096032.txt">Table of n, a(n) for n = 1..294</a>

%e 1726 of the sequence forms a pair with 150 and we indeed have T(150)+T(1726)=11325+1490401=1501726.

%t f[n_] := Block[{k = n + 1, t1 = n(n + 1)/2, td = IntegerDigits[n]}, While[k < 15*n && t1 + k(k + 1)/2 != FromDigits[ Join[ td, IntegerDigits[k]]], k++ ]; If[k != 15*n, k, 0]]; Do[ k = f[n]; If[k != 0, Print[n, " & ", k]], {n, 10^6}] (* _Robert G. Wilson v_, Jun 21 2004 *)

%K nonn,base

%O 1,2

%A _Lekraj Beedassy_, Jun 16 2004

%E Two more terms from _Robert G. Wilson v_, Jun 21 2004

%E Terms from a(19) onwards from _David Wasserman_, May 15 2007

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Last modified June 16 14:12 EDT 2024. Contains 373430 sequences. (Running on oeis4.)