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.

User: Frans J. Faase

Frans J. Faase's wiki page.

Frans J. Faase has authored 67 sequences. Here are the ten most recent ones:

A022542 Minimum number of possible solutions for all irreducible stick-cutting problems.

Original entry on oeis.org

0, 0, 0, 2, 3, 6, 8, 18, 26, 78, 94, 342, 388, 1660, 1728, 8592, 8616, 45068, 46848, 267536, 282816
Offset: 1

Views

Author

Frans J. Faase

Keywords

Links

Extensions

Changed beginning, added one more term, May 15 1997
Offset corrected, a(1)-a(5) inserted, and a(21) added by Sean A. Irvine, May 19 2019

A022541 Related to number of irreducible stick-cutting problems.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 4, 7, 9, 21, 41, 73, 147, 288, 557, 1111, 2193, 4343, 8728, 17483, 35063, 70828, 143267, 290193, 589705, 1200646, 2448904, 5005001, 10245216, 21005238, 43134355, 88696073, 182621943, 376496023, 777098691, 1605731742, 3321492918, 6877489184
Offset: 1

Views

Author

Frans J. Faase

Keywords

Comments

Number of partitions of n(n+1)/2 with all elements greater than n and less than 2n-1. - David Bevan, Sep 19 2011

Examples

			a(4)=1: 10 can be partitioned as (5,5). - _David Bevan_, Sep 19 2011

Links

Programs

  • Mathematica
    Table[Length[IntegerPartitions[n(n+1)/2, All, Range[n+1,2n-2]]], {n, 20}] (* David Bevan, Sep 19 2011 *)

Formula

a(n) = [x^(n*(n+1)/2)] Product_{k=n+1..2*n-2} 1/(1-x^k). - Sean A. Irvine, May 18 2019

Extensions

a(4) and a(5) corrected by David Bevan, Sep 19 2011
More terms from Alois P. Heinz, Sep 20 2012

A003729 Number of perfect matchings (or domino tilings) in graph C_{5} X P_{2n}.

Original entry on oeis.org

11, 176, 2911, 48301, 801701, 13307111, 220880176, 3666315811, 60855946601, 1010127453401, 16766766924211, 278305942640176, 4619507031938711, 76677648402694901, 1272746577484955101, 21125893715367851311
Offset: 1

Views

Author

Frans J. Faase

Keywords

References

  • F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
  • Per Hakan Lundow, "Computation of matching polynomials and the number of 1-factors in polygraphs", Research report, No 12, 1996, Department of Math., Umea University, Sweden.

Links

Programs

  • Mathematica
    Rest[CoefficientList[Series[x (11-33x+18x^2-x^3)/(1-19x+41x^2- 19x^3+ x^4), {x,0,20}],x]] (* or *) LinearRecurrence[{19,-41,19,-1},{11,176,2911,48301},20] (* Harvey P. Dale, Jul 16 2011 *)

Formula

a(n) = 19a(n-1) - 41a(n-2) + 19a(n-3) - a(n-4), n>4.
G.f. x*(11-33*x+18*x^2-x^3)/(1-19*x+41*x^2-19*x^3+x^4) . [From R. J. Mathar, Mar 11 2010]

A003732 Number of Hamiltonian paths in C_5 X P_n.

Original entry on oeis.org

5, 130, 1660, 16820, 152230, 1275680, 10154290, 77897010, 581452680, 4250594690, 30572999140, 217099260110, 1525905283670, 10636695448300, 73649615037480, 507171127397480, 3476871213780220, 23747634842538120
Offset: 1

Views

Author

Frans J. Faase

Keywords

References

  • F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Links

Formula

Faase gives a 12-term linear recurrence on his web page:
a(1) = 5,
a(2) = 130,
a(3) = 1660,
a(4) = 16820,
a(5) = 152230,
a(6) = 1275680,
a(7) = 10154290,
a(8) = 77897010,
a(9) = 581452680,
a(10) = 4250594690,
a(11) = 30572999140,
a(12) = 217099260110,
a(13) = 1525905283670,
a(14) = 10636695448300 and
a(n) = 19a(n-1) - 127a(n-2) + 328a(n-3) - 117a(n-4) - 675a(n-5)
+ 1127a(n-6) - 1016a(n-7) + 380a(n-8) + 12a(n-9) - 140a(n-10)
+ 68a(n-11) - 20a(n-12), n>14.
G.f. 5*x+130*x^2 -10*x^3*(-166 +1472*x -4347*x^2 +2503*x^3 +7316*x^4 -13386*x^5 +12513*x^6 -4715*x^7 -215*x^8 +1824*x^9 -856*x^10 +252*x^11) / ( (1-7*x-x^2+20*x^3-3*x^4+3*x^5+5*x^6) *(-1+6*x-4*x^2+2*x^3)^2 ). - R. J. Mathar, Aug 21 2012

Extensions

Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009

A003733 Number of spanning trees in C_5 X P_n.

Original entry on oeis.org

5, 1805, 508805, 140503005, 38720000000, 10668237057005, 2939274449134805, 809816405722655805, 223117116976138566005, 61472262298219520000000, 16936571572967914651674005, 4666290873812984282155907805, 1285636259054921313298518442805
Offset: 1

Views

Author

Frans J. Faase

Keywords

References

  • F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Links

Crossrefs

Cf. A143699.

Programs

  • Maple
    a:= n-> (Matrix(1,9, (i,j)-> [0, 5, 1805, 508805, 140503005][1+abs(j-5)]). Matrix(9, (i,j)-> if (i=j-1) then 1 elif j=1 then -[408001, 128319, 12441, 319, 1][1/2+abs(i-9/2)] *(-1)^i else 0 fi)^n)[1, 5]: seq(a(n), n=1..20); # Alois P. Heinz, Mar 28 2009
  • Mathematica
    a[n_] := (16/41)*Sinh[n*ArcCosh[(-9 - Sqrt[5])/4]]^2*Sinh[n*ArcCosh[(-9 + Sqrt[5])/4]]^2 // Round; Array[a, 20] (* Jean-François Alcover, Jan 31 2016, after Peter Bala in A143699 *)

Formula

a(n) = 319*a(n-1) - 12441*a(n-2) + 128319*a(n-3) - 408001*a(n-4) + 408001*a(n-5) - 128319*a(n-6) + 12441*a(n-7) - 319*a(n-8) + a(n-9). [Modified by Paul Raff, Oct 30 2009]
G.f.: -5*x *(1+x) *(x^6+41*x^5-998*x^4+2722*x^3-998*x^2+41*x+1) / ( (x-1)*(x^4-279*x^3+961*x^2-279*x+1) *(x^4-39*x^3+281*x^2-39*x+1) ).
a(n) = 5 * (A143699(n))^2. - R. K. Guy, Mar 11 2010

Extensions

Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009

A003734 Number of spanning trees with degrees 1 and 3 in C_5 X P_2n.

Original entry on oeis.org

0, 260, 27420, 2504560, 223723080, 19923617840, 1773563554900, 157870122686600, 14052371971981100, 1250831588811052320, 111339169110472830220, 9910535055491682625400, 882157695038695625086700, 78522722964255506997330800, 6989473714324564174042717340
Offset: 1

Views

Author

Frans J. Faase

Keywords

References

  • F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Links

Formula

Faase gives a 12-term linear recurrence on his web page:
If b(n) denotes the number of spanning trees with degrees 1 and 3 in C_5 X P_n we have:
b(1) = 0,
b(2) = 0,
b(3) = 0,
b(4) = 260,
b(5) = 0,
b(6) = 27420,
b(7) = 0,
b(8) = 2504560,
b(9) = 0,
b(10) = 223723080,
b(11) = 0,
b(12) = 19923617840,
b(13) = 0,
b(14) = 1773563554900,
b(15) = 0,
b(16) = 157870122686600,
b(17) = 0,
b(18) = 14052371971981100,
b(19) = 0,
b(20) = 1250831588811052320,
b(21) = 0,
b(22) = 111339169110472830220,
b(23) = 0,
b(24) = 9910535055491682625400,
b(25) = 0,
b(26) = 882157695038695625086700, and
b(n) = 98b(n-2) - 745b(n-4) - 4916b(n-6) - 234b(n-8) + 160624b(n-10)
- 26648b(n-12) + 338976b(n-14) - 1265216b(n-16) - 2291392b(n-18) - 1695488b(n-20)
- 307200b(n-22) + 32768b(n-24).

Extensions

Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009
More terms from Sean A. Irvine, Jul 29 2015

A003740 Number of spanning trees with degrees 1 and 3 in W_5 X P_2n.

Original entry on oeis.org

208, 335344, 503672968, 757005488704, 1137734095903816, 1709944335224262352, 2569941155563565968488, 3862463470575397280285088, 5805045002479537990606632936
Offset: 1

Views

Author

Frans J. Faase

Keywords

References

  • F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Links

Formula

If b(n) denotes the number of spanning trees with degrees 1 and 3 in W_5 X P_n we have:
b(1) = 0,
b(2) = 208,
b(3) = 0,
b(4) = 335344,
b(5) = 0,
b(6) = 503672968,
b(7) = 0,
b(8) = 757005488704,
b(9) = 0,
b(10) = 1137734095903816,
b(11) = 0,
b(12) = 1709944335224262352,
b(13) = 0,
b(14) = 2569941155563565968488,
b(15) = 0,
b(16) = 3862463470575397280285088,
b(17) = 0,
b(18) = 5805045002479537990606632936,
b(19) = 0,
b(20) = 8724625549856078166453269723376,
b(21) = 0,
b(22) = 13112575518826856642901203139743240,
b(23) = 0,
b(24) = 19707394403851935411114869745719526144,
b(25) = 0,
b(26) = 29619001517386258600018494299567252781896,
b(27) = 0,
b(28) = 44515537310983054901068606912734277302893072,
b(29) = 0,
b(30) = 66904114270101652083096747543361961556161338280,
b(31) = 0,
b(32) = 100552768239022085083137539569611934600600485769824,
b(33) = 0,
b(34) = 151124625306471850563573728012268031905685321872309416,
b(35) = 0,
b(36) = 227131015624872535892492790329036203871753015873169846576,
b(37) = 0,
b(38) = 341363944851262010688127945467040823127463725134532755058760,
b(39) = 0,
b(40) = 513049010606610528824074852666729120665123598849369486838352320,
b(41) = 0,
b(42) = 771081103480659083177648561305159418338110532879217116850112505608,
b(43) = 0,
b(44) = 1158887466602766746036049127283646002598030062997458201209529788050000, and
b(n) = 1498b(n-2) + 9727b(n-4) - 3430420b(n-6) - 51780334b(n-8) + 2175631056b(n-10)
- 3049771912b(n-12) + 20785260864b(n-14) - 885420351008b(n-16) + 2723994857536b(n-18) + 5274700679360b(n-20)
+ 125883661338368b(n-22) + 354089303896576b(n-24) - 880465464686592b(n-26) - 28529345908736b(n-28) + 3938132497694720b(n-30)
- 1757770863747072b(n-32) - 1334108047147008b(n-34) - 337906312937472b(n-36) - 49853396680704b(n-38) - 3371549327360b(n-40).

Extensions

Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009

A003744 Number of Hamiltonian paths in O_5 X P_n.

Original entry on oeis.org

36, 3960, 197172, 8372376, 313590732, 10961493288, 364496212992, 11715923002644, 367218115613412, 11297962590845364, 342721436917704060, 10284809936813182116, 306078425919342660924
Offset: 1

Views

Author

Frans J. Faase

Keywords

References

  • F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Links

Formula

a(1) = 36,
a(2) = 3960,
a(3) = 197172,
a(4) = 8372376,
a(5) = 313590732,
a(6) = 10961493288,
a(7) = 364496212992,
a(8) = 11715923002644,
a(9) = 367218115613412,
a(10) = 11297962590845364,
a(11) = 342721436917704060,
a(12) = 10284809936813182116,
a(13) = 306078425919342660924,
a(14) = 9050314137435866812308,
a(15) = 266262758895847900204044,
a(16) = 7802857128214786920966468,
a(17) = 227964188131745757879553596,
a(18) = 6644168196971243295712163700,
a(19) = 193287318120848681996183075244,
a(20) = 5614785173559337471057013732388,
a(21) = 162918194408431653609336890189340,
a(22) = 4723043996602440520832973512325972,
a(23) = 136828273928341927052870400623002380, and
a(n) = 59a(n-1) - 731a(n-2) - 11403a(n-3) + 204688a(n-4) + 697232a(n-5)
- 13575824a(n-6) + 15466532a(n-7) + 288258520a(n-8) - 1327022000a(n-9) + 1631290560a(n-10)
+ 3212771840a(n-11) - 12023726208a(n-12) + 9649896000a(n-13) + 11298643072a(n-14) - 24109594624a(n-15)
+ 6239014400a(n-16) + 14028280832a(n-17) - 8564428800a(n-18) - 2763866112a(n-19) + 2175729664a(n-20)
+ 199229440a(n-21) - 150994944a(n-22).

Extensions

Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009

A003746 Number of spanning trees with degrees 1 and 3 in O_5 X P_2n.

Original entry on oeis.org

540, 1751352, 5386703316, 16582103036544, 51045000577926816, 157132783947988296192, 483704801377335372564480, 1488997578825205151673656448, 4583609224965381313988566950144
Offset: 1

Views

Author

Frans J. Faase

Keywords

References

  • F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Links

Formula

If b(n) denotes the number of spanning trees with degrees 1 and 3 in O_5 X P_n then
b(1) = 0,
b(2) = 540,
b(3) = 0,
b(4) = 1751352,
b(5) = 0,
b(6) = 5386703316,
b(7) = 0,
b(8) = 16582103036544,
b(9) = 0,
b(10) = 51045000577926816,
b(11) = 0,
b(12) = 157132783947988296192,
b(13) = 0,
b(14) = 483704801377335372564480,
b(15) = 0,
b(16) = 1488997578825205151673656448,
b(17) = 0,
b(18) = 4583609224965381313988566950144,
b(19) = 0,
b(20) = 14109810402621649533503234558344704,
b(21) = 0,
b(22) = 43434494483860386599671308650864330496,
b(23) = 0,
b(24) = 133705220498070622788909783421076412386304,
b(25) = 0,
b(26) = 411587292562609297454750726054600269987912704,
b(27) = 0,
b(28) = 1266996896366237649178359003459366628005457649664,
b(29) = 0,
b(30) = 3900220352788196660232362097608501848215326938755072,
b(31) = 0,
b(32) = 12006121596612176283154633057320394687803565435297505280,
b(33) = 0,
b(34) = 36958669704287162536274146164634194441880201040907341168640,
b(35) = 0,
b(36) = 113770567399219775084499535791661980035376168565367523333734400,
b(37) = 0,
b(38) = 350222075358923174025212352063864697242943327666094722900436582400,
b(39) = 0,
b(40) = 1078095195203820521745918151197065855397382661823414208194364252422144,
b(41) = 0,
b(42) = 3318720696661962582358070874565591095886422622888933137425721520537337856, and
b(n) = 2976b(n-2) + 311460b(n-4) + 10745408b(n-6) + 185361600b(n-8) - 11015685472b(n-10)
- 384432909824b(n-12) + 12586530486400b(n-14) - 142686379766272b(n-16) + 471457558327040b(n-18) + 3354655475796480b(n-20)
- 12936942677605376b(n-22) + 29721236628888576b(n-24) - 167487137019375616b(n-26) - 745271272714235904b(n-28) + 1043959728550182912b(n-30)
- 1512329782916284416b(n-32) + 206265260306202624b(n-34) + 59399388450127872b(n-36) + 26359905185169408b(n-38) + 154793410560000b(n-40).

Extensions

Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009

A003753 Number of spanning trees in C_4 X P_n.

Original entry on oeis.org

4, 384, 31500, 2558976, 207746836, 16864848000, 1369080572444, 111141302329344, 9022397309950500, 732433860440996736, 59458627396289740076, 4826822683620921984000, 391839136544897998002484, 31809312044806091140235904, 2582264604005182130741437500
Offset: 1

Views

Author

Frans J. Faase

Keywords

References

  • F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Links

Crossrefs

Column k=4 of A173958. - Alois P. Heinz, Sep 20 2012

Programs

  • Maple
    a:= n-> (Matrix([[4,0,-4,-384,-31500,-2558976]]). Matrix(6, (i,j)-> if (i=j-1) then 1 elif j=1 then [90, -735, 1548, -735, 90, -1][i] else 0 fi)^(n-1))[1,1]; seq(a(n), n=1..20);  # Alois P. Heinz, Aug 01 2008
  • Mathematica
    a[n_] := (Sqrt[2]/3)*Sinh[n*ArcCosh[3]]*Sinh[n*ArcCosh[7]/2]^2 // Round; Array[a, 20] (* Jean-François Alcover, Jan 31 2016 *)

Formula

a(1) = 4,
a(2) = 384,
a(3) = 31500,
a(4) = 2558976,
a(5) = 207746836,
a(6) = 16864848000 and
a(n) = 90a(n-1) - 735a(n-2) + 1548a(n-3) - 735a(n-4) + 90a(n-5) - a(n-6).
G.f.: 4x(x^4+6x^3-30x^2+6x+1)/(x^6-90x^5+735x^4-1548x^3+735x^2-90x+1). [Paul Raff, Mar 06 2009]
a(n) = 4*A001109(n)*A098301(n). [R. K. Guy, seqfan list, Mar 28 2009] [From R. J. Mathar, Jun 03 2009]

Extensions

Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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