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A300998
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Number of close American football games: number of ways for the game to end after n points have been scored and never be separated by more than one score after each play.
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3
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1, 0, 2, 2, 4, 8, 14, 28, 52, 78, 156, 272, 520, 832, 1616, 2734, 5224, 8756, 16798, 28192, 54118, 90644, 173876, 292816, 561574, 938748, 1802188, 3031400, 5812998, 9734470, 18684588, 31367492, 60172174, 100893834, 193598664, 324824728, 623209036, 1045201398, 2005438304, 3364638978
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OFFSET
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0,3
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COMMENTS
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Each play (counting untimed downs as part of the previous play) can score at most 8 points for one team.
The same as counting walks of x-length n from the origin bounded above by y=8, below by y=-8, and using the steps {[2,2],[3,3],[8,4],[7,5],[6,6],[7,7],[8,8],[2,-2],[3,-3],[8,-4],[7,-5],[6,-6],[7,-7],[8,-8]}.
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LINKS
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FORMULA
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G.f.: -(16*t^58-16*t^57-48*t^56+56*t^55-20*t^54-8*t^53+168*t^52-164*t^51-32*t^50+104*t^49-128*t^48+96*t^47-64*t^46+52*t^45-188*t^44+66*t^43+350*t^42-352*t^41+421*t^40-160*t^39-606*t^38+540*t^37-145*t^36-54*t^35+234*t^34-26*t^33-56*t^32-162*t^31+334*t^30-200*t^29+107*t^28-18*t^27-388*t^26+352*t^25-94*t^24-34*t^23+136*t^22-54*t^21+48*t^20-112*t^19+64*t^18-8*t^17+7*t^16+40*t^15-81*t^14+62*t^13-71*t^12-2*t^11+31*t^10-18*t^9+24*t^8+4*t^7-8*t^6+6*t^5-6*t^4+2*t^3+t^2+1)/(32*t^66-112*t^64+24*t^62+324*t^60-300*t^58-40*t^56+52*t^54-542*t^52+784*t^50+766*t^48-1610*t^46+166*t^44+792*t^42-563*t^40+420*t^38+681*t^36-1320*t^34+190*t^32+246*t^30-87*t^28+74*t^26+304*t^24-380*t^22+6*t^20-10*t^18+25*t^16-25*t^14+85*t^12-3*t^10-22*t^8+2*t^6+8*t^4+t^2-1).
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EXAMPLE
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There is no way to score 1 point so a(1)=0.
There are 2 ways to score 2 or 3 points.
a(n<=8) is fairly easy to compute since the bounds do not come into effect.
a(9)=78. The unallowable walks are those with 9 points all of the same magnitude: [2,2,2,3],[3,3,3],[2,7],[3,6] (and the negatives and reorderings). A total of 18 unallowable walks. The total walks of length 9 are 2*4*2 (2 and 7 points and ordering) + 2*2*2 (3 and 6) + 2*2*2 (3 and 3 and 3) + 2*2*2*2*4 (2 and 2 and 2 and 3). The total is then 16+8+8+64-18=78.
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MAPLE
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taylor(-(16*t^58-16*t^57-48*t^56+56*t^55-20*t^54-8*t^53+168*t^52-164*t^51-32*t^50+104*t^49-128*t^48+96*t^47-64*t^46+52*t^45-188*t^44+66*t^43+350*t^42-352*t^41+421*t^40-160*t^39-606*t^38+540*t^37-145*t^36-54*t^35+234*t^34-26*t^33-56*t^32-162*t^31+334*t^30-200*t^29+107*t^28-18*t^27-388*t^26+352*t^25-94*t^24-34*t^23+136*t^22-54*t^21+48*t^20-112*t^19+64*t^18-8*t^17+7*t^16+40*t^15-81*t^14+62*t^13-71*t^12-2*t^11+31*t^10-18*t^9+24*t^8+4*t^7-8*t^6+6*t^5-6*t^4+2*t^3+t^2+1)/(32*t^66-112*t^64+24*t^62+324*t^60-300*t^58-40*t^56+52*t^54-542*t^52+784*t^50+766*t^48-1610*t^46+166*t^44+792*t^42-563*t^40+420*t^38+681*t^36-1320*t^34+190*t^32+246*t^30-87*t^28+74*t^26+304*t^24-380*t^22+6*t^20-10*t^18+25*t^16-25*t^14+85*t^12-3*t^10-22*t^8+2*t^6+8*t^4+t^2-1), t=0, N);
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CROSSREFS
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KEYWORD
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nonn,walk
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AUTHOR
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STATUS
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approved
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