A028307 Form a triangle with n numbers in top row; all other numbers are the sum of their parents. E.g.: 4 1 2 7; 5 3 9; 8 12; 20. The numbers must be positive and distinct and the final number is to be minimized. Sequence gives final number.

1, 3, 8, 20, 43, 98, 212, 465, 1000, 2144, 4497, 9504, 19872, 41455, 85356, 178630, 363467, 757085, 1541998, 3183600, 6515066, 13357593, 27432649, 55914902, 114683858, 233517515, 478061719, 972479046, 1986013932

Offset: 1

Author

Mauro Fiorentini

Keywords

• nonn
• nice

Comments

Suggested by Problem 401 of the All-Soviet-Union Mathematical Competitions 1961-1986. Two different links are available for this collection.

Examples

			Solutions for n = 1, 2, ... are:
  1;
  1, 2;
  2, 1, 4;
  4, 1, 2, 7;
  7, 2, 1, 4, 6;
  8, 6, 1, 3, 2, 10;
  ...
		

Links

• Mauro Fiorentini, Further comments
• Vladimir A. Pertsel, Problems of the All-Soviet-Union Mathematical Competitions 1961-1986

Crossrefs

Cf. A028308, A062896, A340389.

Formula

From A.H.M. Smeets, Feb 25 2022: (Start)

a(n) > 2*a(n-1). Proof: Let x, y be the numbers in the second last row, then x >= a(n-1), y >= a(n-1) and x != y, so a(n) = x + y > 2*a(n-1).

It seems that a(n) > (4/3)*(2*a(n-1)-a(n-2)). (End)

Extensions

More terms from the author, Jul 03 2001