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

1,2

Comments

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

Links

Table of n, a(n) for n=1..29.

Mauro Fiorentini, Further comments

Vladimir A. Pertsel, Problems of the All-Soviet-Union Mathematical Competitions 1961-1986

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)

Example

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;
  ...

Crossrefs

Cf. A028308, A062896, A340389.

Sequence in context: A182735 A135565 A139488 * A027298 A000236 A109327

Adjacent sequences: A028304 A028305 A028306 * A028308 A028309 A028310

Keyword

nonn,nice

Author

Mauro Fiorentini

Extensions

More terms from the author, Jul 03 2001

Status

approved