

A114143


Possible sums of the final scores of completed American football games where both teams score.


0



4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40
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OFFSET

1,1


COMMENTS

One point only is an impossible score in American football. But with the safety 2 and the field goal 3, we can construct the set of integers greater than 1.
We can prove this by noting that if a score is even, we can build it with a series of safeties. Of course the other allowed scorings of 3, 6, and 1 after a touchdown could also be used. Now if a score is odd it is of the form 2k + 3. So for any odd number 2m + 1, we subtract 3 (or 1 field goal) from it to make it even and divide by 2 to get the number of safeties we need to add back to the field goal. Symbolically, let the odd number be 2m + 1. Then, (2m+1  3)/2 = m  1 safeties are needed. Add this to 3 and you will have the number.
For example, say we want a score of 99. 99 = 2m + 1 and m = 49. So m  1 = 48 safeties + 1 field goal = 99 points. From the first statement that 1 is an impossible score, it follows that there are infinitely many impossible scores in football, i.e., (1, 0), (1, 1), (1, 2), (1, 3), ...
Moreover, this restriction greatly influences the outcome of the ending digits for match up to the 100 square game cards that are floating around throughout the world before the Super Bowl game.


LINKS

Table of n, a(n) for n=1..37.
Tanya Khovanova, Non Recursions


EXAMPLE

A safety or 2 points is the minimum possible score at the end of a game. So if two teams score one safety each, the sum of their scores will be 4. The next number, 5, is a safety for one team and a field goal for the other.


CROSSREFS

Sequence in context: A178052 A030543 A214084 * A171526 A171463 A020705
Adjacent sequences: A114140 A114141 A114142 * A114144 A114145 A114146


KEYWORD

easy,nonn


AUTHOR

Cino Hilliard, Feb 03 2006


EXTENSIONS

Comments and example edited by Jon E. Schoenfield, Sep 20 2013


STATUS

approved



