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.

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A192942 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x)=(2x+1)(2x+2)...(2x+n).

Original entry on oeis.org

0, 2, 10, 62, 448, 3688, 34056, 348568, 3916352, 47919520, 634256480, 9029234720, 137569217280, 2233574372480, 38497936301440, 702049663399040, 13504656880506880, 273280886412413440, 5803407252377602560, 129044887279907315200
Offset: 0

Views

Author

Clark Kimberling, Jul 13 2011

Keywords

Comments

For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.

Examples

			The first four polynomials p(n,x) and their reductions are as follows:
p(0,x) = 1
p(1,x) = 2x+1 -> 1+2x
p(2,x) = (2x+1)(2x+2) -> 6+10x
p(3,x) = (2x+1)(2x+2)(2x+3) -> 38+62x
From these, read
A192941=(1,2,6,38,...) and A192942=(0,2,10,62,...)

Links

Crossrefs

Cf. A192232, A192744, A192941, A192950.

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); s:=Sqrt(5); [Round(s*Gamma(n+2+s)/Gamma(s+2) - Sin(Pi(R)*(s+3))*Gamma(s+1) *Gamma(n+2-s)/(Pi(R)*(s-1)))/5: n in [0..20]]; // G. C. Greubel, Jul 25 2019
  • Mathematica
    (* First program *)
q = x^2; s = x + 1; z = 26;
p[0, x]:= 1;
p[n_, x_]:= (2x+n)*p[n-1, x];
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192941 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192942 *)
u2/2 (* A192950 *)
(* Second program *)
With[{s = Sqrt[5]}, Table[FullSimplify[(s*Gamma[n+2+s]/Gamma[s+2] - Sin[Pi*(s+3)]*Gamma[s+1]*Gamma[n+2-s]/(Pi*(s-1)))/5], {n, 0, 20}]] (* G. C. Greubel, Jul 26 2019 *)
  • PARI
    default(realprecision, 100); vector(20, n, n--; s=sqrt(5); round(s*gamma(n+2+s)/gamma(s+2) - sin(Pi*(s+3))*gamma(s+1)*gamma(n+2-s)/(Pi*(s-1)))/5 ) \\ G. C. Greubel, Jul 25 2019
  • Sage
    s=sqrt(5); [round(s*gamma(n+2+s)/gamma(s+2) - sin(pi*(s+3))* gamma(s+1)*gamma(n+2-s)/(pi*(s-1)))/5 for n in (0..20)] # G. C. Greubel, Jul 25 2019

Formula

Conjecture: a(n) +(-2*n-1)*a(n-1) +(n^2-5)*a(n-2)=0. - R. J. Mathar, May 08 2014

A192953 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

0, 1, 2, 6, 13, 26, 48, 85, 146, 246, 409, 674, 1104, 1801, 2930, 4758, 7717, 12506, 20256, 32797, 53090, 85926, 139057, 225026, 364128, 589201, 953378, 1542630, 2496061, 4038746, 6534864, 10573669, 17108594, 27682326, 44790985, 72473378
Offset: 0

Views

Author

Clark Kimberling, Jul 13 2011

Keywords

Comments

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 2n - 1, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.

Links

Crossrefs

Cf. A000045, A192232, A192744, A192951.

Programs

  • GAP
    F:=Fibonacci;; List([0..40], n-> 3*F(n+2)-(2*n+3)); # G. C. Greubel, Jul 12 2019
  • Magma
    F:=Fibonacci; [3*F(n+2)-(2*n+3): n in [0..40]]; // G. C. Greubel, Jul 12 2019
  • Mathematica
    (* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + 2n - 1;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A111314 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192953 *)
(* Second program *)
With[{F=Fibonacci}, Table[3*F[n+2]-(2*n+3), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)
  • PARI
    vector(40, n, n--; f=fibonacci; 3*f(n+2)-(2*n+3)) \\ G. C. Greubel, Jul 12 2019
  • Sage
    f=fibonacci; [3*f(n+2)-(2*n+3) for n in (0..40)] # G. C. Greubel, Jul 12 2019

Formula

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
G.f.: x*(1 -x +2*x^2)/((1-x-x^2)*(1-x)^2). - R. J. Mathar, Aug 01 2011
a(n) = -2*n - 3 + 3*A000045(n+2). - R. J. Mathar, Aug 01 2011
a(n) = A131300(n) - 1. - R. J. Mathar, Mar 24 2018
a(n) = 3*Fibonacci(n+2) - (2*n+3). - G. C. Greubel, Jul 12 2019

A192954 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

1, 1, 5, 11, 23, 43, 77, 133, 225, 375, 619, 1015, 1657, 2697, 4381, 7107, 11519, 18659, 30213, 48909, 79161, 128111, 207315, 335471, 542833, 878353, 1421237, 2299643, 3720935, 6020635, 9741629, 15762325, 25504017, 41266407, 66770491
Offset: 0

Views

Author

Clark Kimberling, Jul 13 2011

Keywords

Comments

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + n^2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.

Links

Crossrefs

Cf. A000032, A000045, A192232, A192744, A192951, A192955.

Programs

  • GAP
    List([0..40], n-> 2*Lucas(1,-1,n+2)[2]-(2*n+5)); # G. C. Greubel, Jul 12 2019
  • Magma
    [2*Lucas(n+2)-(2*n+5): n in [0..40]]; // G. C. Greubel, Jul 12 2019
  • Mathematica
    (* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + n^2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192954 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192955 *)
(* Second program *)
Table[2*LucasL[n+2]-(2*n+5), {n,0,40}] (* G. C. Greubel, Jul 12 2019 *)
LinearRecurrence[{3,-2,-1,1},{1,1,5,11},40] (* Harvey P. Dale, Jan 13 2022 *)
  • PARI
    vector(40, n, n--; f=fibonacci; 2*(f(n+3)+f(n+1))-(2*n+5)) \\ G. C. Greubel, Jul 12 2019
  • Sage
    [2*lucas_number2(n+2,1,-1)-(2*n+5) for n in (0..40)] # G. C. Greubel, Jul 12 2019

Formula

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
From R. J. Mathar, May 08 2014: (Start)
G.f.: (1 -2*x +4*x^2 -x^3)/((1-x-x^2)*(1-x)^2).
a(n) - a(n-1) = A168674(n-1). (End)
a(n) = 2*Lucas(n+2) - (2*n+5). - G. C. Greubel, Jul 12 2019

A192955 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

0, 1, 2, 7, 18, 41, 84, 161, 294, 519, 894, 1513, 2528, 4185, 6882, 11263, 18370, 29889, 48548, 78761, 127670, 206831, 334942, 542257, 877728, 1420561, 2298914, 3720151, 6019794, 9740729, 15761364, 25502993, 41265318, 66769335, 108035742
Offset: 0

Views

Author

Clark Kimberling, Jul 13 2011

Keywords

Comments

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + n^2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.

Links

Crossrefs

Cf. A000032, A000045, A192232, A192744, A192951, A192955.

Programs

  • GAP
    List([0..40], n-> 2*Lucas(1,-1,n+3)[2]-(n^2+4*n+8)); # G. C. Greubel, Jul 12 2019
  • Magma
    [2*Lucas(n+3)-(n^2+4*n+8): n in [0..40]]; // G. C. Greubel, Jul 12 2019
  • Mathematica
    (* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + n^2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192954 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192955 *)
(* Second program *)Table[2*LucasL[n+3]-(n^2+4*n+8), {n,0,40}] (* G. C. Greubel, Jul 12 2019 *)
  • PARI
    vector(40, n, n--; f=fibonacci; 2*(f(n+4)+f(n+2))-(n^2+4*n+8)) \\ G. C. Greubel, Jul 12 2019
  • Sage
    [2*lucas_number2(n+3,1,-1)-(n^2+4*n+8) for n in (0..40)] # G. C. Greubel, Jul 12 2019

Formula

a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
From R. J. Mathar, May 08 2014: (Start)
G.f.: x*(1 -2*x +4*x^2 -x^3)/((1-x-x^2)*(1-x)^3).
a(n) - a(n-1) = A192954(n-1). (End)
a(n) = 2*Lucas(n+3) - (n^2+4*n+8). - G. C. Greubel, Jul 12 2019

A192956 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

1, 0, 4, 9, 20, 38, 69, 120, 204, 341, 564, 926, 1513, 2464, 4004, 6497, 10532, 17062, 27629, 44728, 72396, 117165, 189604, 306814, 496465, 803328, 1299844, 2103225, 3403124, 5506406, 8909589, 14416056, 23325708, 37741829, 61067604, 98809502
Offset: 0

Views

Author

Clark Kimberling, Jul 13 2011

Keywords

Comments

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) +- 1 + n^2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.

Links

Crossrefs

Cf. A000045, A192232, A192744, A192951, A192957.

Programs

  • GAP
    F:=Fibonacci;; List([0..40], n-> F(n+3)+4*F(n+1)-(2*n+5)); # G. C. Greubel, Jul 12 2019
  • Magma
    F:=Fibonacci; [F(n+3)+4*F(n+1)-(2*n+5): n in [0..40]]; // G. C. Greubel, Jul 12 2019
  • Mathematica
    (* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + n^2 - 1;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192956 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192957 *)
(* Second program *)
With[{F=Fibonacci}, Table[F[n+3]+4*F[n+1]-(2*n+5), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)
  • PARI
    vector(40, n, n--; f=fibonacci; f(n+3)+4*f(n+1)-(2*n+5)) \\ G. C. Greubel, Jul 12 2019
  • Sage
    f=fibonacci; [f(n+3)+4*f(n+1)-(2*n+5) for n in (0..40)] # G. C. Greubel, Jul 12 2019

Formula

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
From R. J. Mathar, May 09 2014: (Start)
G.f.: (1 -3*x +6*x^2 -2*x^3)/((1-x-x^2)*(1-x)^2).
a(n) -2*a(n+1) +a(n+2) = A022096(n-3). (End)
a(n) = Fibonacci(n+3) + 4*Fibonacci(n+1) - (2*n+5). - G. C. Greubel, Jul 12 2019

A192957 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

0, 1, 1, 5, 14, 34, 72, 141, 261, 465, 806, 1370, 2296, 3809, 6273, 10277, 16774, 27306, 44368, 71997, 116725, 189121, 306286, 495890, 802704, 1299169, 2102497, 3402341, 5505566, 8908690, 14415096, 23324685, 37740741, 61066449, 98808278
Offset: 0

Views

Author

Clark Kimberling, Jul 13 2011

Keywords

Comments

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) +- 1 + n^2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.

Links

Crossrefs

Cf. A000045, A192232, A192744, A192951, A192956.

Programs

  • GAP
    F:=Fibonacci;; List([0..40], n-> F(n+4)+4*F(n+2)-(n^2+4*n+7)); # G. C. Greubel, Jul 12 2019
  • Magma
    F:=Fibonacci; [F(n+4)+4*F(n+2)-(n^2+4*n+7): n in [0..40]]; // G. C. Greubel, Jul 12 2019
  • Mathematica
    (* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + n^2 - 1;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192956 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192957 *)
(* Second program *)With[{F=Fibonacci}, Table[F[n+4]+4*F[n+2]-(n^2+4*n+7), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)
  • PARI
    vector(40, n, n--; f=fibonacci; f(n+4)+4*f(n+2)-(n^2+4*n+7)) \\ G. C. Greubel, Jul 12 2019
  • Sage
    f=fibonacci; [f(n+4)+4*f(n+2)-(n^2+4*n+7) for n in (0..40)] # G. C. Greubel, Jul 12 2019

Formula

a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
From R. J. Mathar, May 09 2014: (Start)
G.f.: x*(1 -3*x +6*x^2 -2*x^3)/((1-x-x^2)*(1-x)^3).
a(n) - a(n-1) = A192956(n-1). (End)
a(n) = Fibonacci(n+4) + 4*Fibonacci(n+2) - (n^2+4*n+7). - G. C. Greubel, Jul 12 2019

A192958 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

1, -1, 3, 7, 17, 33, 61, 107, 183, 307, 509, 837, 1369, 2231, 3627, 5887, 9545, 15465, 25045, 40547, 65631, 106219, 171893, 278157, 450097, 728303, 1178451, 1906807, 3085313, 4992177, 8077549, 13069787, 21147399, 34217251, 55364717, 89582037
Offset: 0

Views

Author

Clark Kimberling, Jul 13 2011

Keywords

Comments

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) - 2 + n^2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.

Links

Crossrefs

Cf. A000045, A192232, A192744, A192951, A192959.

Programs

  • GAP
    F:=Fibonacci;; List([0..40], n-> 6*F(n+1)-(2*n+5)); # G. C. Greubel, Jul 12 2019
  • Magma
    F:=Fibonacci; [6*F(n+1)-(2*n+5): n in [0..40]]; // G. C. Greubel, Jul 12 2019
  • Mathematica
    (* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + n^2 - 2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192958 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192959 *)
(* Second program *)
With[{F=Fibonacci}, Table[6*F[n+1]-(2*n+5), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)
  • PARI
    vector(40, n, n--; f=fibonacci; 6*f(n+1)-(2*n+5)) \\ G. C. Greubel, Jul 12 2019
  • Sage
    f=fibonacci; [6*f(n+1)-(2*n+5) for n in (0..40)] # G. C. Greubel, Jul 12 2019

Formula

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
From R. J. Mathar, May 09 2014: (Start)
G.f.: (1 -4*x +8*x^2 -3*x^3)/((1-x-x^2)*(1-x)^2).
a(n) - 2*a(n-1) +a(n-2) = A022089(n-3). (End)
a(n) = 6*Fibonacci(n+1) - (2*n+5). - G. C. Greubel, Jul 12 2019

A192959 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

0, 1, 0, 3, 10, 27, 60, 121, 228, 411, 718, 1227, 2064, 3433, 5664, 9291, 15178, 24723, 40188, 65233, 105780, 171411, 277630, 449523, 727680, 1177777, 1906080, 3084531, 4991338, 8076651, 13068828, 21146377, 34216164, 55363563, 89580814
Offset: 0

Views

Author

Clark Kimberling, Jul 13 2011

Keywords

Comments

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) - 2 + n^2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.

Links

Crossrefs

Cf. A000045, A192232, A192744, A192951, A192958.

Programs

  • GAP
    F:=Fibonacci;; List([0..40], n-> 6*F(n+2)-(n^2+4*n+6)); # G. C. Greubel, Jul 12 2019
  • Magma
    F:=Fibonacci; [6*F(n+2)-(n^2+4*n+6): n in [0..40]]; // G. C. Greubel, Jul 12 2019
  • Mathematica
    (* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + n^2 - 2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192958 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192959 *)
(* Second program *)
With[{F=Fibonacci}, Table[6*F[n+2]-(n^2+4*n+6), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)
  • PARI
    vector(40, n, n--; f=fibonacci; 6*f(n+2)-(n^2+4*n+6)) \\ G. C. Greubel, Jul 12 2019
  • Sage
    f=fibonacci; [6*f(n+2)-(n^2+4*n+6) for n in (0..40)] # G. C. Greubel, Jul 12 2019

Formula

a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
From R. J. Mathar, May 09 2014: (Start)
G.f.: x*(1 -4*x +8*x^2 -3*x^3)/((1-x-x^2)*(1-x)^3).
a(n) - a(n-1) = A192958(n-1). (End)
a(n) = 6*Fibonacci(n+2) - (n^2 + 4*n + 6). - G. C. Greubel, Jul 12 2019

A192961 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

0, 1, 4, 11, 26, 55, 108, 201, 360, 627, 1070, 1799, 2992, 4937, 8100, 13235, 21562, 35055, 56908, 92289, 149560, 242251, 392254, 634991, 1027776, 1663345, 2691748, 4355771, 7048250, 11404807, 18453900, 29859609, 48314472, 78175107, 126490670
Offset: 0

Views

Author

Clark Kimberling, Jul 13 2011

Keywords

Comments

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 2 + n^2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.

Links

Crossrefs

Cf. A000045, A192232, A192744, A192951, A192960.

Programs

  • GAP
    F:=Fibonacci;; List([0..40], n-> 2*F(n+5)-(n^2+4*n+10)); # G. C. Greubel, Jul 12 2019
  • Magma
    F:=Fibonacci; [2*F(n+5)-(n^2+4*n+10): n in [0..40]]; // G. C. Greubel, Jul 12 2019
  • Mathematica
    (* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + n^2 + 2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192960 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192961 *)
(* Second program *)
With[{F=Fibonacci}, Table[2*F[n+5]-(n^2+4*n+10), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)
LinearRecurrence[{4,-5,1,2,-1},{0,1,4,11,26},40] (* Harvey P. Dale, Dec 30 2024 *)
  • PARI
    vector(40, n, n--; f=fibonacci; 2*f(n+5)-(n^2+4*n+10)) \\ G. C. Greubel, Jul 12 2019
  • Sage
    f=fibonacci; [2*f(n+5)-(n^2+4*n+10) for n in (0..40)] # G. C. Greubel, Jul 12 2019

Formula

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
From R. J. Mathar, May 09 2014: (Start)
G.f.: x*(1+x)*(1-x+x^2)/((1-x-x^2)*(1-x)^3).
a(n) - a(n-1)= A192960(n-1). (End)
a(n) = 2*Fibonacci(n+5) - (n^2 + 4*n + 10). - G. C. Greubel, Jul 12 2019

A192962 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

Original entry on oeis.org

1, 2, 7, 15, 30, 55, 97, 166, 279, 463, 762, 1247, 2033, 3306, 5367, 8703, 14102, 22839, 36977, 59854, 96871, 156767, 253682, 410495, 664225, 1074770, 1739047, 2813871, 4552974, 7366903, 11919937, 19286902, 31206903, 50493871, 81700842
Offset: 1

Views

Author

Clark Kimberling, Jul 13 2011

Keywords

Comments

The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + n + n^2, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.

Links

Crossrefs

Cf. A000045, A192232, A192744, A192951, A192963.

Programs

  • GAP
    F:=Fibonacci;; List([1..40], n-> 3*F(n+1) +4*F(n) -2*(n+2)); # G. C. Greubel, Jul 12 2019
  • Magma
    F:=Fibonacci; [3*F(n+1) +4*F(n) -2*(n+2): n in [1..40]]; // G. C. Greubel, Jul 12 2019
  • Mathematica
    (* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 1;
p[n_, x_]:= x*p[n-1, x] + n(n+1);
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192962 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192963 *)
(* Additional programs *)
CoefficientList[Series[(1-x+3x^2-x^3)/((1-x-x^2)(1-x)^2), {x, 0, 40}], x] (* Vincenzo Librandi, May 09 2014 *)
With[{F=Fibonacci}, Table[3*F[n+1]+4*F[n] -2*(n+2), {n,1,40}]] (* G. C. Greubel, Jul 12 2019 *)
  • PARI
    vector(40, n, f=fibonacci; 3*f(n+1)+4*f(n)-2*(n+2)) \\ G. C. Greubel, Jul 12 2019
  • Sage
    f=fibonacci; [3*f(n+1) +4*f(n) -2*(n+2) for n in (1..40)] # G. C. Greubel, Jul 12 2019

Formula

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
From R. J. Mathar, May 09 2014: (Start)
G.f.: x*(1 -x +3*x^2 -x^3)/((1-x-x^2)*(1-x)^2).
a(n) -2*a(n-1) + a(n-2) = A022120(n-4). (End)
a(n) = 3*Fibonacci(n+1) + 4*Fibonacci(n) - 2*(n+2). - G. C. Greubel, Jul 12 2019
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