cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A300998 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.

Original entry on oeis.org

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
Offset: 0

Views

Author

Bryan T. Ek, Mar 20 2018

Keywords

Comments

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]}.

Examples

			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.

Links

Crossrefs

Cf. A301379, A301380, A301381.

Programs

  • Maple
    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);

Formula

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).

A301379 Number of close American football games: number of ways for the game to have n scoring plays and never be separated by more than one score after each play.

Original entry on oeis.org

1, 14, 128, 1378, 13932, 144300, 1480376, 15245184, 156756896, 1612836306, 16589928984, 170664508406, 1755592926518, 18059752212038, 185779058543356, 1911097952732140, 19659326724616886, 202234169412143472, 2080368880383488938, 21400612097499844490, 220146623069820835050
Offset: 0

Views

Author

Bryan T. Ek, Mar 19 2018

Keywords

Comments

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 {[1,8],..,[1,2],[1,-2],..,[1,-8]}.

Examples

			For n=1, any step is valid. For n=2, any walk with steps of opposite direction is valid while [[1,3],[1,6]] is an example of an invalid walk.

Links

Crossrefs

Cf. A300998, A301380, A301381.

Programs

  • Maple
    taylor((1+10*t+13*t^2-37*t^3-40*t^4+28*t^5+26*t^6-2*t^7)/(1-4*t-59*t^2-77*t^3+170*t^4+234*t^5-92*t^6-142*t^7-4*t^8+6*t^9),t=0,N);

Formula

G.f.: (1+10*t+13*t^2-37*t^3-40*t^4+28*t^5+26*t^6-2*t^7)/(1-4*t-59*t^2-77*t^3+170*t^4+234*t^5-92*t^6-142*t^7-4*t^8+6*t^9).

A301381 Number of tied close American football games: number of ways for the game to end at the score of n to n and never be separated by more than one score after each play.

Original entry on oeis.org

1, 0, 2, 2, 6, 24, 80, 208, 922, 2310, 8794, 26000, 86632, 274120, 893552, 2837882, 9254642, 29470852, 95567342, 306155908, 987994256, 3174707284, 10228816628, 32893256236, 105937526030, 340778467916, 1097194416030, 3530389210580, 11364292475448, 36571646955122, 117713073900332
Offset: 0

Views

Author

Bryan T. Ek, Mar 20 2018

Keywords

Comments

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 that return to the x-axis 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]}.

Examples

			There is no way to score 1 point so a(1)=0.
The number of ways to be tied at 4-4 is 6: there must be 2 safeties scored by each team which could be ordered in 4 choose 2 ways.
a(5)=24 since there must be 1 safety and 1 field goal for each team and there are 4! ways to order them.
a(n<=8) is fairly easy to compute since the bounds do not come into effect.

Links

Crossrefs

Cf. A300998, A301379, A301380.

Programs

  • Maple
    taylor((16*t^29-16*t^28-56*t^27+52*t^26+100*t^25-52*t^24-136*t^23+108*t^22+66*t^21-71*t^20+134*t^19-5*t^18-320*t^17+50*t^16+78*t^15-47*t^14+60*t^13+78*t^12-158*t^11-8*t^10+31*t^8+t^7+37*t^6-t^5-10*t^4+2*t^3+6*t^2+t-1)/(32*t^33-112*t^32+24*t^31+324*t^30-300*t^29-40*t^28+52*t^27-542*t^26+784*t^25+766*t^24-1610*t^23+166*t^22+792*t^21-563*t^20+420*t^19+681*t^18-1320*t^17+190*t^16+246*t^15-87*t^14+74*t^13+304*t^12-380*t^11+6*t^10-10*t^9+25*t^8-25*t^7+85*t^6-3*t^5-22*t^4+2*t^3+8*t^2+t-1),t=0,N);

Formula

G.f.: (16*t^29-16*t^28-56*t^27+52*t^26+100*t^25-52*t^24-136*t^23+108*t^22+66*t^21-71*t^20+134*t^19-5*t^18-320*t^17+50*t^16+78*t^15-47*t^14+60*t^13+78*t^12-158*t^11-8*t^10+31*t^8+t^7+37*t^6-t^5-10*t^4+2*t^3+6*t^2+t-1)/(32*t^33-112*t^32+24*t^31+324*t^30-300*t^29-40*t^28+52*t^27-542*t^26+784*t^25+766*t^24-1610*t^23+166*t^22+792*t^21-563*t^20+420*t^19+681*t^18-1320*t^17+190*t^16+246*t^15-87*t^14+74*t^13+304*t^12-380*t^11+6*t^10-10*t^9+25*t^8-25*t^7+85*t^6-3*t^5-22*t^4+2*t^3+8*t^2+t-1).
Showing 1-3 of 3 results.

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