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 19 results. Next

A074082 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)=(1,1).

Original entry on oeis.org

0, 0, 0, 0, 2, 6, 16, 37, 81, 169, 342, 675, 1307, 2491, 4686, 8718, 16066, 29364, 53282, 96065, 172215, 307151, 545286, 963993, 1697701, 2979383, 5211852, 9090060, 15810530, 27429426, 47473828, 81983773, 141286221, 243011173
Offset: 0

Views

Author

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002

Keywords

Comments

The coefficient of q^0 in nu(n) is the Fibonacci number F(n+1). The coefficient of q^1 is A023610(n-3).

Examples

			The first 6 nu polynomials are nu(0)=1, nu(1)=1, nu(2)=2, nu(3)=3+q, nu(4)=5+3q+2q^2, nu(5)=8+7q+6q^2+4q^3+q^4, so the coefficients of q^2 are 0,0,0,0,2,6.

Links

Crossrefs

Coefficients of q^0, q^1 and q^3 are in A000045, A023610 and A074083. Related sequences with different values of b and lambda are in A074084-A074089.

Programs

  • Mathematica
    b=1; lambda=1; expon=2; nu[0]=1; nu[1]=b; nu[n_] := nu[n]=Together[b*nu[n-1]+lambda(1-q^(n-1))/(1-q)nu[n-2]]; a[n_] := Coefficient[nu[n], q, expon]
(* Second program: *)
Join[{0, 0}, LinearRecurrence[{3, 0, -5, 0, 3, 1}, {0, 0, 2, 6, 16, 37}, 32]] (* Jean-François Alcover, Sep 23 2017 *)

Formula

G.f.: (2*x^4-2*x^6-x^7)/(1-x-x^2)^3.
a(n) = 3*a(n-1)-5*a(n-3)+3*a(n-5)+a(n-6) for n>=8.

Extensions

Edited by Dean Hickerson, Aug 21 2002

A074352 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)=(1,2).

Original entry on oeis.org

0, 0, 0, 2, 8, 22, 60, 146, 352, 814, 1860, 4170, 9256, 20326, 44300, 95874, 206320, 441758, 941780, 2000058, 4233144, 8932310, 18796700, 39457522, 82643328, 172743182, 360399460, 750625066, 1560902472, 3241109574, 6720828460, 13918875490, 28792188176
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 0,0,0,2,8,22.

Links

Crossrefs

Coefficient of q^0, q^2 and q^3 are in A001045, A074353 and A074354. Related sequences with other values of b and lambda are in A074082-A074089, A074353-A074363.

Programs

  • Mathematica
    LinearRecurrence[{2, 3, -4, -4}, {0, 0, 0, 2, 8}, 50] (* Paolo Xausa, Jan 28 2025 *)
  • PARI
    a(n)=if(n<1,0,(1/27)*(2^n*(6*n-11)+(-1)^n*(6*n-16)))
  • PARI
    a(n)=if(n<1,0,(1/81)*(2^(n-1)*(6*n^2-43)+ (-1)^n*(6*n^2-24*n+62)))
  • PARI
    concat(vector(3), Vec(2*x^3*(1 + 2*x) / ((1 + x)^2*(1 - 2*x)^2) + O(x^40))) \\ Colin Barker, Nov 18 2017

Formula

a(0) = 0; for n>0, a(n) = (1/27)*(2^n*(6*n-11) + (-1)^n*(6*n-16)).
From Colin Barker, Nov 18 2017: (Start)
G.f.: 2*x^3*(1 + 2*x) / ((1 + x)^2*(1 - 2*x)^2).
a(n) = 2*a(n-1) + 3*a(n-2) - 4*a(n-3) - 4*a(n-4) for n>4. (End)

Extensions

More terms and formula from Benoit Cloitre, Jan 12 2003
Corrected by Franklin T. Adams-Watters, Oct 25 2006
Corrected by T. D. Noe, Oct 25 2006

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

A074087 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)=(2,3).

Original entry on oeis.org

0, 0, 0, 6, 33, 144, 570, 2118, 7587, 26448, 90420, 304470, 1013061, 3338112, 10911150, 35423862, 114342855, 367242336, 1174368360, 3741029094, 11876859369, 37591894320, 118659631650, 373630740966, 1173847761003
Offset: 0

Views

Author

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002

Keywords

Comments

The coefficient of q^0 is A014983(n+1).

Examples

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=7, nu(3)=20+6q, nu(4)=61+33q+21q^2, nu(5)=182+144q+120q^2+78q^3+18q^4, so the coefficients of q^1 are 0,0,0,6,33,144.

Links

Crossrefs

Coefficients of q^0, q^2 and q^3 are in A014983, A074088 and A074089. Related sequences with other values of b and lambda are in A074082-A074086.

Programs

  • Magma
    I:=[0,0,6,33]; [0] cat [n le 4 select I[n] else 4*Self(n-1) + 2*Self(n-2) -12*Self(n-3) -9*Self(n-4): n in [1..30]]; // G. C. Greubel, May 26 2018
  • Mathematica
    b=2; lambda=3; expon=1; nu[0]=1; nu[1]=b; nu[n_] := nu[n]=Together[b*nu[n-1]+lambda(1-q^(n-1))/(1-q)nu[n-2]]; a[n_] := Coefficient[nu[n], q, expon]
(* Second program: *)
Join[{0}, LinearRecurrence[{4,2,-12,-9}, {0,0,6,33}, 50]] (* G. C. Greubel, May 26 2018 *)
  • PARI
    x='x+O('x^30); concat([0,0,0], Vec((6*x^3 +9*x^4)/(1-2*x-3*x^2)^2)) \\ G. C. Greubel, May 26 2018

Formula

G.f.: (6*x^3 +9*x^4)/(1-2*x-3*x^2)^2.
a(n) = 4*a(n-1) +2*a(n-2) -12*a(n-3) -9*a(n-4) for n>=5.

Extensions

Edited by Dean Hickerson, Aug 21 2002

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

A074084 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)=(2,1).

Original entry on oeis.org

0, 0, 0, 2, 9, 32, 102, 306, 883, 2480, 6828, 18514, 49597, 131568, 346194, 904738, 2350695, 6076960, 15641304, 40103778, 102473969, 261046144, 663180222, 1680628946, 4249496795, 10722962256, 27007159428, 67904097074
Offset: 0

Views

Author

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002

Keywords

Comments

The coefficient of q^0 is the Pell number A000129(n+1).

Examples

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=5, nu(3)=12+2q, nu(4)=29+9q+5q^2, nu(5)=70+32q+24q^2+14q^3+2q^4, so the coefficients of q^1 are 0,0,0,2,9,32.

Links

Crossrefs

Coefficients of q^0, q^2 and q^3 are in A000129, A074085 and A074086. Related sequences with other values of b and lambda are in A074082-A074083 and A074087-A074089.

Programs

  • Mathematica
    b=2; lambda=1; expon=1; nu[0]=1; nu[1]=b; nu[n_] := nu[n]=Together[b*nu[n-1]+lambda(1-q^(n-1))/(1-q)nu[n-2]]; a[n_] := Coefficient[nu[n], q, expon]
(* Second program: *)
Join[{0},LinearRecurrence[{4,-2,-4,-1},{0,0,2,9},30]] (* Harvey P. Dale, Apr 18 2012 *)

Formula

G.f.: (2x^3+x^4)/(1-2x-x^2)^2.
a(n) = 4a(n-1)-2a(n-2)-4a(n-3)-a(n-4) for n>=5.

Extensions

Edited by Dean Hickerson, Aug 21 2002

A074086 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)=(2,1).

Original entry on oeis.org

0, 0, 0, 0, 0, 14, 71, 282, 997, 3298, 10439, 32012, 95834, 281494, 814131, 2324422, 6564135, 18362810, 50947395, 140329400, 384031508, 1044880222, 2828084399, 7618214354, 20432838121, 54585196818, 145287466799, 385397215108
Offset: 0

Views

Author

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002

Keywords

Comments

The coefficient of q^0 is the Pell number A000129(n+1).

Examples

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=5, nu(3)=12+2q, nu(4)=29+9q+5q^2, nu(5)=70+32q+24q^2+14q^3+2q^4, so the coefficients of q^3 are 0,0,0,0,0,14.

Links

Crossrefs

Coefficients of q^0, q^1 and q^2 are in A000129, A074084 and A074085. Related sequences with other values of b and lambda are in A074082-A074083 and A074087-A074089.

Programs

  • Mathematica
    b=2; lambda=1; expon=3; nu[0]=1; nu[1]=b; nu[n_] := nu[n]=Together[b*nu[n-1]+lambda(1-q^(n-1))/(1-q)nu[n-2]]; a[n_] := Coefficient[nu[n], q, expon]
(* Second program: *)
Join[{0, 0, 0}, LinearRecurrence[{8, -20, 8, 26, -8, -20, -8, -1}, {0, 0, 14, 71, 282, 997, 3298, 10439}, 25]] (* Jean-François Alcover, Jan 27 2019 *)

Formula

G.f.: (14x^5-41x^6-6x^7+49x^8+30x^9+5x^10)/(1-2x-x^2)^4.
a(n) = 8a(n-1)-20a(n-2)+8a(n-3)+26a(n-4)-8a(n-5)-20a(n-6)-8a(n-7)-a(n-8) for n>=11.

Extensions

Edited by Dean Hickerson, Aug 21 2002

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
Showing 1-10 of 19 results. Next

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

Content is available under The OEIS End-User License Agreement.