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A304780 Consider a triangle whose first row is {1,2} and, for n > 1, has as its n-th row the integers k through 2k where k is the sum of the numbers in the (n-1)th row. Then a(n) is the first number in the n-th row.

%I A304780 #47 Sep 07 2018 03:48:28
%S A304780 1,3,18,513,395523,234658258578,82596747478641253260993,
%T A304780 10233334041075645341729789249315281196742910563,
%U A304780 157081688394356396673208173772909833928515988895188885472258972148661958252271815996039831298
%N A304780 Consider a triangle whose first row is {1,2} and, for n > 1, has as its n-th row the integers k through 2k where k is the sum of the numbers in the (n-1)th row. Then a(n) is the first number in the n-th row.
%F A304780 a(1) = 1, a(n) = (3*a(n-1)*(a(n-1) + 1))/2 for n > 1.
%F A304780 a(n) ~ (2/3) * c^(2^n), where c = 1.515006464529590220430714781603262955960312205695360166833... - _Vaclav Kotesovec_, Jul 23 2018
%e A304780 Triangle begins:
%e A304780   1, 2; (row sum = 3)
%e A304780   3, 4, 5, 6; (row sum = 18)
%e A304780   18, 19, 20, 21, ... 33, 34, 35, 36; (row sum = 513)
%e A304780   513, 514, 515, 516, ..., 1023, 1024, 1025, 1026;
%e A304780   ...
%t A304780 RecurrenceTable[{a[1] == 1, a[n] == (3*a[n-1]*(a[n-1] + 1))/2}, a, {n, 1, 10}] (* _Vaclav Kotesovec_, Jul 23 2018 *)
%t A304780 Nest[Append[#, Range[#, 2 #] &@ Total@ Last@ #] &, {{1, 2}}, 3] // Flatten (* _Michael De Vlieger_, Jul 26 2018 *)
%o A304780 (PARI) a(n) = if (n==1, 1, (3*a(n - 1)*(a(n - 1) + 1))/2); \\ _Michel Marcus_, May 24 2018
%Y A304780 Cf. A000217.
%K A304780 nonn
%O A304780 1,2
%A A304780 _Nathaniel J. Strout_, May 18 2018