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 1-10 of 11 results. Next

A074363 Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(3,1).

Original entry on oeis.org

0, 0, 0, 0, 0, 36, 246, 1293, 6057, 26592, 111934, 457353, 1827529, 7176636, 27789976, 106371588, 403204880, 1515647250, 5656172420, 20974163475, 77339044883, 283743384228, 1036296662574, 3769287797151, 13658724680991, 49325767966842, 177570110818794
Offset: 0

Views

Author

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Keywords

Comments

Coefficient of q^0 is A006190(n+1).

Examples

			The first 6 nu polynomials are nu(0)=1, nu(1)=3, nu(2)=10, nu(3)=33+3q, nu(4)=109+19q+10q^2, nu(5)=360+93q+66q^2+36q^3+3q^4, so the coefficients of q^1 are 0,0,0,0,0,36.

Links

Crossrefs

Coefficients of q^0, q^1 and q^2 are in A006190, A074361 and A074362. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074360.

Programs

  • PARI
    concat(vector(4), Vec(x^4*(3 + x)*(12 - 66*x + 69*x^2 + 60*x^3 + 10*x^4) / (1 - 3*x - x^2)^4 + O(x^40))) \\ Colin Barker, Nov 18 2017

Formula

From Colin Barker, Nov 18 2017: (Start)
G.f.: x^4*(3 + x)*(12 - 66*x + 69*x^2 + 60*x^3 + 10*x^4) / (1 - 3*x - x^2)^4.
a(n) = 12*a(n-1) - 50*a(n-2) + 72*a(n-3) + 21*a(n-4) - 72*a(n-5) - 50*a(n-6) - 12*a(n-7) - a(n-8) for n>9.
(End)

Extensions

More terms from Brent Lehman (mailbjl(AT)yahoo.com), Aug 25 2002
Missing a(0)=0 inserted by Sean A. Irvine, Jan 20 2025
Missing a(0)=0 inserted in b-file by David Radcliffe, Aug 01 2025

A074357 Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(1,3).

Original entry on oeis.org

0, 0, 0, 0, 0, 30, 168, 639, 2415, 7872, 25542, 77727, 233547, 679410, 1949862, 5490132, 15276456, 41963844, 114153990, 307595853, 822263313, 2181777252, 5751280350, 15069310365, 39269077809, 101817186264, 262776963360
Offset: 0

Views

Author

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Keywords

Comments

Coefficient of q^0 is A006130.

Examples

			The first 6 nu polynomials are nu(0)=1, nu(1)=1, nu(2)=4, nu(3)=7+3q, nu(4)=19+15q+12q^2, nu(5)=40+45q+42q^2+30q^3+9q^4, so the coefficients of q^3 are 0,0,0,0,0,30.

Links

Crossrefs

Coefficient of q^0, q^1 and q^2 are in A006130, A074355 and A074356. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074354, A074358-A074363.

Programs

  • Maple
    nu := proc(b,lambda,n) global q; local qp,i ; if n = 0 then RETURN(1) ; elif n =1 then RETURN(b) ; fi ; qp:=0 ; for i from 0 to n-2 do qp := qp + q^i ; od ; RETURN( b*nu(b,lambda,n-1)+lambda*qp*nu(b,lambda,n-2)) ; end: A074357 := proc(n) RETURN( coeftayl(nu(1,3,n),q=0,3) ) ; end: for n from 0 to 30 do printf("%d,", A074357(n)) ; od ; # R. J. Mathar, Sep 20 2006
  • Mathematica
    Join[{0, 0, 0}, LinearRecurrence[{4, 6, -32, -19, 96, 54, -108, -81}, {0, 0, 30, 168, 639, 2415, 7872, 25542}, 24]] (* Jean-François Alcover, Sep 22 2017 *)

Formula

Conjecture: O.g.f.: 3*x^5*(3*x+1)*(36*x^4+24*x^3-29*x^2-14*x+10)/(3*x^2+x-1)^4. - R. J. Mathar, Jul 22 2009

Extensions

More terms from R. J. Mathar, Sep 20 2006

A074358 Coefficient of q^1 in nu(n), where nu(0)=1, nu(1)=b and, for n >= 2, nu(n) = b*nu(n-1) + lambda*(1 + q + q^2 + ... + q^(n-2))*nu(n-2) with (b,lambda)=(2,2).

Original entry on oeis.org

0, 0, 0, 4, 20, 80, 288, 976, 3184, 10112, 31488, 96576, 292672, 878336, 2614784, 7731456, 22728448, 66482176, 193617920, 561718272, 1624101888, 4681535488, 13457924096, 38592008192, 110419341312, 315287830528, 898583560192
Offset: 0

Views

Author

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Keywords

Comments

Coefficient of q^0 is A002605.

Examples

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=6, nu(3) = 16 + 4q, nu(4) = 44 + 20q + 12q^2, nu(5) = 120 + 80q + 64q^2 + 40q^3 + 8q^4, so the coefficients of q^1 are 0,0,0,4,20,80.

Links

Crossrefs

Coefficient of q^0, q^2 and q^3 are in A002605, A074359 and A074360. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074357, A074361-A074363.

Programs

  • Maple
    nu := proc(b,lambda,n) global q; local qp,i ; if n = 0 then RETURN(1) ; elif n =1 then RETURN(b) ; fi ; qp:=0 ; for i from 0 to n-2 do qp := qp + q^i ; od ; RETURN( b*nu(b,lambda,n-1)+lambda*qp*nu(b,lambda,n-2)) ; end: A074358 := proc(n) RETURN( coeftayl(nu(2,2,n),q=0,1) ) ; end: for n from 0 to 30 do printf("%d,", A074358(n)) ; od ; # R. J. Mathar, Sep 20 2006
  • Mathematica
    nu[0] = 1; nu[1] = 2; nu[n_] := nu[n] = 2*nu[n-1] + 2*Total[q^Range[0, n-2] ]*nu[n-2] // Expand;
a[n_] := Coefficient[nu[n], q, 1];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Nov 17 2017 *)

Formula

G.f.: 4*x^3*(x + 1)/(2*x^2 + 2*x - 1)^2 (conjectured). - Chai Wah Wu, May 30 2016
a(n) = (1/18)*((1 + sqrt(3))^n*(-9 + 2*sqrt(3)) - (1 - sqrt(3))^n*(9 + 2*sqrt(3)) + 3*((1 - sqrt(3))^n + (1 + sqrt(3))^n)*n) for n > 0 (conjectured). - Colin Barker, Nov 17 2017
a(n) = 4*a(n-1) - 8*a(n-3) - 4*a(n-4) for n > 4 (conjectured). - Colin Barker, Nov 17 2017

Extensions

More terms from R. J. Mathar, Sep 20 2006

A074360 Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(2,2).

Original entry on oeis.org

0, 0, 0, 0, 0, 40, 232, 1072, 4400, 16864, 61728, 218496, 753792, 2547840, 8468608, 27755776, 89886976, 288101888, 915089920, 2883416064, 9021001728, 28042881024, 86672025600, 266472878080, 815347462144, 2483820617728
Offset: 0

Views

Author

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Keywords

Comments

Coefficient of q^0 is A002605.

Examples

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=6, nu(3)=16+4q, nu(4)=44+20q+12q^2, nu(5)=120+80q+64q^2+40q^3+8q^4, so the coefficients of q^1 are 0,0,0,0,0,40.

Links

Crossrefs

Coefficient of q^0, q^1 and q^2 are in A002605, A074358 and A074359. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074357, A074361-A074363.

Programs

  • Maple
    nu := proc(b,lambda,n) global q; local qp,i ; if n = 0 then RETURN(1) ; elif n =1 then RETURN(b) ; fi ; qp:=0 ; for i from 0 to n-2 do qp := qp + q^i ; od ; RETURN( b*nu(b,lambda,n-1)+lambda*qp*nu(b,lambda,n-2)) ; end: A074360 := proc(n) RETURN( coeftayl(nu(2,2,n),q=0,3) ) ; end: for n from 0 to 30 do printf("%d,", A074360(n)) ; od ; # R. J. Mathar, Sep 20 2006
  • Mathematica
    nu[0] = 1; nu[1] = 2; nu[n_] := nu[n] = 2*nu[n - 1] + 2*Total[q^Range[0, n - 2]]*nu[n - 2] // Expand;
a[n_] := Coefficient[nu[n], q, 3];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 18 2017 *)

Formula

Conjecture: O.g.f: 8*x^5*(1+x)*(12*x^4+24*x^3-2*x^2-16*x+5)/(2*x^2+2*x-1)^4. - R. J. Mathar, Jul 22 2009

Extensions

More terms from R. J. Mathar, Sep 20 2006

A074354 Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(1,2).

Original entry on oeis.org

0, 0, 0, 0, 0, 14, 64, 218, 692, 1982, 5496, 14562, 37692, 95142, 236032, 576074, 1387780, 3304078, 7787656, 18190386, 42151116, 96972534, 221651472, 503650970, 1138286740, 2559944414, 5731095704, 12776843138, 28374100572
Offset: 0

Views

Author

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Keywords

Comments

Coefficient of q^0 is A001045(n+1).

Examples

			The first 6 nu polynomials are nu(0)=1, nu(1)=1, nu(2)=3, nu(3)=5+2q, nu(4)=11+8q+6q^2, nu(5)=21+22q+20q^2+14q^3+4q^4, so the coefficients of q^1 are 0,0,0,0,0,14.

Links

Crossrefs

Coefficients of q^0, q^1 and q^2 are in A001045, A074352 and A074353. Related sequences with other values of b and lambda are in A074082-A074089, A074355-A074363.

Formula

Conjectures from Colin Barker, Nov 18 2017: (Start)
G.f.: 2*x^5*(1 + 2*x)*(7 - 10*x - 13*x^2 + 12*x^3 + 12*x^4) / ((1 + x)^4*(1 - 2*x)^4).
a(n) = 4*a(n-1) + 2*a(n-2) - 20*a(n-3) - a(n-4) + 40*a(n-5) + 8*a(n-6) - 32*a(n-7) - 16*a(n-8) for n>10.
(End)

Extensions

More terms from Benoit Cloitre, Jan 16 2003
Corrected by T. D. Noe, Oct 25 2006

A074355 Coefficient of q^1 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(1,3).

Original entry on oeis.org

0, 0, 0, 3, 15, 45, 147, 402, 1134, 2991, 7917, 20367, 52167, 131748, 330876, 824187, 2042763, 5035473, 12361755, 30226614, 73664298, 178971879, 433649769, 1048133619, 2527706127, 6083434824, 14613750648, 35045236083, 83909261319
Offset: 0

Views

Author

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Keywords

Comments

Coefficient of q^0 is A006130.

Examples

			The first 6 nu polynomials are nu(0)=1, nu(1)=1, nu(2)=4, nu(3)=7+3q, nu(4)=19+15q+12q^2, nu(5)=40+45q+42q^2+30q^3+9q^4, so the coefficients of q^1 are 0,0,0,3,15,45.

Links

Crossrefs

Coefficient of q^0, q^2 and q^3 are in A006130, A074356 and A074357. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074354, A074358-A074363.

Programs

  • Maple
    nu := proc(n,b,lambda) option remember ; if n = 0 then 1 ; elif n = 1 then b ; else b*nu(n-1,b,lambda)+lambda*nu(n-2,b,lambda)*add(q^i,i=0..n-2) ; fi ; end:
A074355 := proc(n) local b,lambda,thisnu ; b := 1 ; lambda := 3 ; thisnu := nu(n,b,lambda) ; RETURN( coeftayl(thisnu,q=0,1) ) ; end: # R. J. Mathar, Mar 20 2007
  • Mathematica
    nu[n_, b_, lambda_] := nu[n, b, lambda] = Which[ n == 0, 1, n == 1, b, True, b*nu[n - 1, b, lambda] + lambda*nu[n - 2, b, lambda]*Sum[q^i, {i, 0, n - 2}]];
a[n_] := a[n] = Coefficient[nu[n, 1, 3], q, 1];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 30}] (* Jean-François Alcover, Nov 23 2017, from 1st Maple program *)

Formula

G.f.: (9x^4+3x^3)/(1-3x-3x^2)^2 (conjectured). - Ralf Stephan, May 09 2004

Extensions

More terms from R. J. Mathar, Mar 20 2007

A074361 Coefficient of q^1 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(3,1).

Original entry on oeis.org

0, 0, 0, 3, 19, 93, 407, 1674, 6618, 25455, 95953, 356151, 1305887, 4741092, 17072484, 61055787, 217074895, 767882865, 2704365719, 9487509102, 33170122494, 115614094071, 401864286637, 1393378817259, 4820368210175
Offset: 0

Views

Author

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Keywords

Comments

Coefficient of q^0 is A006190(n+1).

Examples

			The first 6 nu polynomials are nu(0)=1, nu(1)=3, nu(2)=10, nu(3)=33+3q, nu(4)=109+19q+10q^2, nu(5)=360+93q+66q^2+36q^3+3q^4, so the coefficients of q^1 are 0,0,0,3,19,93.

Links

Crossrefs

Coefficient of q^0, q^2 and q^3 are in A006190, A074362 and A074363. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074360.

Programs

  • Mathematica
    CoefficientList[Series[(x^4+3x^3)/(1-3x-x^2)^2,{x,0,30}],x] (* or *) Join[{0},LinearRecurrence[{6,-7,-6,-1},{0,0,3,19},30]] (* Harvey P. Dale, Jan 16 2012 *)

Formula

G.f.: (x^4+3x^3)/(1-3x-x^2)^2.
a(0)=0, a(1)=0, a(2)=0, a(3)=3, a(4)=19, a(n)=6*a(n-1)-7*a(n-2)- 6*a(n-3)- a(n-4). - Harvey P. Dale, Jan 16 2012

Extensions

More terms from Brent Lehman (mailbjl(AT)yahoo.com), Aug 25 2002

A074353 Coefficient of q^2 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(1,2).

Original entry on oeis.org

0, 0, 0, 0, 6, 20, 70, 196, 542, 1396, 3526, 8628, 20766, 49092, 114598, 264356, 603998, 1368148, 3076166, 6870740, 15256158, 33696804, 74073510, 162127940, 353460766, 767816500, 1662394310, 3588252916, 7723318942, 16580031876, 35506388646, 75864499428
Offset: 0

Views

Author

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Keywords

Comments

Coefficient of q^0 is A001045(n+1).

Examples

			The first 6 nu polynomials are nu(0)=1, nu(1)=1, nu(2)=3, nu(3)=5+2q, nu(4)=11+8q+6q^2, nu(5)=21+22q+20q^2+14q^3+4q^4, so the coefficients of q^2 are 0,0,0,0,6,20.

Links

Crossrefs

Coefficients of q^0, q^1 and q^3 are in A001045, A074352 and A074354. Related sequences with other values of b and lambda are in A074082-A074089, A074355-A074363.

Programs

  • Mathematica
    LinearRecurrence[{3, 3, -11, -6, 12, 8}, {0, 0, 0, 0, 6, 20, 70, 196}, 50] (* Paolo Xausa, Jan 28 2025 *)
  • PARI
    concat(vector(4), Vec(2*x^4*(3 + x - 4*x^2 - 4*x^3) / ((1 + x)^3*(1 - 2*x)^3) + O(x^40))) \\ Colin Barker, Nov 18 2017

Formula

a(0)=0 for n>0, a(n) = (1/81)*(2^(n-1)*(6*n^2-43) + (-1)^n*(6*n^2-24*n+62)). - Benoit Cloitre, Jan 16 2003
From Colin Barker, Nov 18 2017: (Start)
G.f.: 2*x^4*(3 + x - 4*x^2 - 4*x^3) / ((1 + x)^3*(1 - 2*x)^3).
a(n) = 3*a(n-1) + 3*a(n-2) - 11*a(n-3) - 6*a(n-4) + 12*a(n-5) + 8*a(n-6) for n>7.
(End)

Extensions

More terms from Benoit Cloitre, Jan 16 2003
Corrected by T. D. Noe, Oct 25 2006

A074356 Coefficient of q^2 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(1,3).

Original entry on oeis.org

0, 0, 0, 0, 12, 42, 180, 561, 1833, 5373, 15798, 44367, 123561, 336243, 906054, 2408094, 6344832, 16561824, 42922602, 110472933, 282678423, 719404803, 1822117962, 4594816221, 11540742615, 28880919975, 72033463644, 179107709004
Offset: 0

Views

Author

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Keywords

Comments

Coefficient of q^0 is A006130.

Examples

			The first 6 nu polynomials are nu(0)=1, nu(1)=1, nu(2)=4, nu(3)=7+3q, nu(4)=19+15q+12q^2, nu(5)=40+45q+42q^2+30q^3+9q^4, so the coefficients of q^2 are 0,0,0,0,12,42.

Links

Crossrefs

Coefficient of q^0, q^1 and q^3 are in A006130, A074355 and A074357. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074354, A074358-A074363.

Programs

  • Maple
    nu := proc(n,b,lambda) if n = 0 then 1 ; elif n = 1 then b ; else b*nu(n-1,b,lambda)+lambda*nu(n-2,b,lambda)*add(q^i,i=0..n-2) ; fi ; end: A074356 := proc(n) local b,lambda,thisnu ; b := 1 ; lambda := 3 ; thisnu := nu(n,b,lambda) ; RETURN( coeftayl(thisnu,q=0,2) ) ; end: for n from 0 to 40 do printf("%d, ",A074356(n) ) ; od ; # R. J. Mathar, Mar 20 2007
  • Mathematica
    nu[n_, b_, lambda_] := nu[n, b, lambda] = Which[n == 0, 1, n == 1, b, True, b*nu[n - 1, b, lambda] + lambda*nu[n - 2, b, lambda]*Sum[q^i, {i, 0, n - 2}]];
a[n_] := a[n] = Coefficient[nu[n, 1, 3], q, 2];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 30}] (* Jean-François Alcover, Nov 23 2017, from Maple *)

Formula

Conjectures from Colin Barker, Nov 18 2017: (Start)
G.f.: 3*x^4*(2 - 3*x)*(2 + 4*x + 3*x^2) / (1 - x - 3*x^2)^3.
a(n) = 3*a(n-1) + 6*a(n-2) - 17*a(n-3) - 18*a(n-4) + 27*a(n-5) + 27*a(n-6) for n>7.
(End)

Extensions

More terms from R. J. Mathar, Mar 20 2007

A074359 Coefficient of q^2 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(2,2).

Original entry on oeis.org

0, 0, 0, 0, 12, 64, 280, 1088, 3968, 13856, 46912, 155136, 503616, 1610496, 5086336, 15895552, 49229312, 151275008, 461662208, 1400356864, 4224703488, 12683452416, 37911164928, 112865394688, 334788444160, 989756825600
Offset: 0

Views

Author

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Keywords

Comments

Coefficient of q^0 is A002605.

Examples

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=6, nu(3)=16+4q, nu(4)=44+20q+12q^2, nu(5)=120+80q+64q^2+40q^3+8q^4, so the coefficients of q^2 are 0,0,0,0,12,64.

Links

Crossrefs

Coefficient of q^0, q^1 and q^3 are in A002605, A074358 and A074360. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074357, A074361-A074363.

Programs

  • Maple
    nu := proc(n,b,lambda) if n = 0 then 1 ; elif n = 1 then b ; else b*nu(n-1,b,lambda)+lambda*nu(n-2,b,lambda)*add(q^i,i=0..n-2) ; fi ; end: A074359 := proc(n) local b,lambda,thisnu ; b := 2 ; lambda := 2 ; thisnu := nu(n,b,lambda) ; RETURN( coeftayl(thisnu,q=0,2) ) ; end: for n from 0 to 40 do printf("%d, ",A074359(n) ) ; od ; # R. J. Mathar, Mar 20 2007
  • Mathematica
    Join[{0, 0}, LinearRecurrence[{6, -6, -16, 12, 24, 8}, {0, 0, 12, 64, 280, 1088}, 24]] (* Jean-François Alcover, Sep 23 2017 *)

Formula

Conjecture: O.g.f: 4*x^4*(-3+2*x+8*x^2+4*x^3)/(2*x^2+2*x-1)^3. - R. J. Mathar, Jul 22 2009

Extensions

More terms from R. J. Mathar, Mar 20 2007
Showing 1-10 of 11 results. Next

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