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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A192313 Constant term of the reduction of n-th polynomial at A157751 by x^2->x+1.

Original entry on oeis.org

1, 2, 5, 13, 34, 91, 247, 680, 1893, 5319, 15056, 42867, 122605, 351898, 1012729, 2920521, 8435362, 24392655, 70599403, 204472264
Offset: 1

Views

Author

Clark Kimberling, Jun 28 2011

Keywords

Comments

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

Examples

			The first five polynomials at A157751 and their reductions are as follows:
p0(x)=1 -> 1
p1(x)=2+x -> 2+x
p2(x)=4+2x+x^2 -> 5+3x
p3(x)=8+4x+4x^2+x^3 -> 13+10x
p4(x)=16+8x+12x^2+4x^3+x^4 -> 34+31x.
From these, we read
A192313=(1,2,5,13,34,...) and A192314=(0,1,3,19,31,...)

Crossrefs

Cf. A192232, A192337.

Programs

  • Mathematica
    q[x_] := x + 1;
p[0, x_] := 1;
p[n_, x_] := (x + 1)*p[n - 1, x] + p[n - 1, -x] /; n > 0  (* A157751 *)
Table[Simplify[p[n, x]], {n, 0, 5}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 20}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 20}]
  (* A192313 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 20}]
  (* A192337 *)

Formula

Empirical G.f.: x*(x+1)*(x^2-3*x+1)/(x^4+6*x^3+x^2-4*x+1). [Colin Barker, Nov 13 2012]

A192373 Constant term in the reduction of the polynomial p(n,x) defined at A162517 and below in Comments.

Original entry on oeis.org

1, 0, 7, 8, 77, 192, 1043, 3472, 15529, 57792, 240655, 934808, 3789653, 14963328, 60048443, 238578976, 953755537, 3798340224, 15162325975, 60438310184, 241126038941, 961476161856, 3835121918243, 15294304429744, 61000836720313, 243280700771904
Offset: 1

Views

Author

Clark Kimberling, Jun 29 2011

Keywords

Comments

The polynomials are given by p(n,x)=((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+4).
For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

Examples

			The first five polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=2x -> 2x
p(2,x)=4+x+3x^2 -> 7+4x
p(3,x)=16x+4x^2+4x^3 -> 8+28x
p(4,x)=16+8x+41x^2+10x^3+5x^4 -> 77+84x.
From these, read A192352=(1,0,7,8,77,...) and A049602=(0,2,4,28,84,...).

Crossrefs

Cf. A192232, A192374, A192375, A162517.

Programs

  • Mathematica
    q[x_] := x + 1; d = Sqrt[x + 4];
p[n_, x_] := ((x + d)^n - (x - d)^n )/(2 d) (* A162517 *)
Table[Expand[p[n, x]], {n, 1, 6}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)}; t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]  (* A192373 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]  (* A192374 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}] (* A192375 *)

Formula

Conjecture: a(n) = 2*a(n-1)+10*a(n-2)-6*a(n-3)-9*a(n-4). G.f.: -x*(x+1)*(3*x-1) / (9*x^4+6*x^3-10*x^2-2*x+1). - Colin Barker, May 09 2014

A192374 Coefficient of x in the reduction of the polynomial p(n,x) defined at A162517 and below in Comments.

Original entry on oeis.org

0, 2, 4, 28, 84, 406, 1448, 6200, 23688, 97034, 380716, 1533844, 6079452, 24339742, 96844496, 386805104, 1541301648, 6150529682, 24521644756, 97819530508, 390080615652, 1555871900710, 6204937972088, 24747735482792, 98698893741336
Offset: 1

Views

Author

Clark Kimberling, Jun 29 2011

Keywords

Comments

The polynomials p(n,x) is defined by ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+4). A192374=2*A192375. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

Examples

			(See A192373.)

Crossrefs

Cf. A192232, A192373, A192375.

Programs

Formula

Empirical G.f.: 2*x^2/(9*x^4+6*x^3-10*x^2-2*x+1). [Colin Barker, Nov 23 2012]

A192376 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

1, 0, 7, 16, 73, 256, 975, 3616, 13521, 50432, 188247, 702512, 2621849, 9784832, 36517535, 136285248, 508623521, 1898208768, 7084211623, 26438637648, 98670339049, 368242718464, 1374300534895, 5128959421024, 19141537149297, 71437189176064
Offset: 1

Views

Author

Clark Kimberling, Jun 29 2011

Keywords

Comments

The polynomial p(n,x) is defined by ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+1). For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

Examples

			The first five polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=2x -> 2x
p(2,x)=4+x+3x^2 -> 7+4x
p(3,x)=16x+4x^2+4x^3 -> 16+20x
p(4,x)=16+8x+41x^2+10x^3+5x^4 -> 73+68x.
From these, read A192376=(1,0,7,16,73,...) and A192377=(0,2,4,20,68,...).

Crossrefs

Cf. A192232, A192377, A192378.

Programs

  • Mathematica
    q[x_] := x + 2; d = Sqrt[x + 1];
p[n_, x_] := ((x + d)^n - (x - d)^ n )/(2 d)   (* Cf. A162517 *)
Table[Expand[p[n, x]], {n, 1, 6}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 1,  30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192376 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192377 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}] (* A192378 *)

Formula

Conjecture: a(n) = 2*a(n-1)+6*a(n-2)+2*a(n-3)-a(n-4). G.f.: x*(x-1)^2 / ((x+1)^2*(x^2-4*x+1)). - Colin Barker, May 11 2014

A192377 Coefficient of x in the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

0, 2, 4, 20, 68, 262, 968, 3624, 13512, 50442, 188236, 702524, 2621836, 9784846, 36517520, 136285264, 508623504, 1898208786, 7084211604, 26438637668, 98670339028, 368242718486, 1374300534872, 5128959421048, 19141537149272, 71437189176090
Offset: 1

Views

Author

Clark Kimberling, Jun 29 2011

Keywords

Comments

The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2d), where d=sqrt(x+1). A192377=2*A192378. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

Examples

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0,x)=1 -> 1
  p(1,x)=2x -> 2x
  p(2,x)=4+x+3x^2 -> 7+4x
  p(3,x)=16x+4x^2+4x^3 -> 16+20x
  p(4,x)=16+8x+41x^2+10x^3+5x^4 -> 73+68x.
From these, read (0,2,4,20,68,...)

Links

Crossrefs

Cf. A192376, A192378.

Programs

Formula

From Colin Barker, Dec 09 2012: (Start)
a(n) = 2*a(n-1) + 6*a(n-2) + 2*a(n-3) - a(n-4).
G.f.: 2*x^2 / ((x+1)^2*(x^2-4*x+1)). (End)

A192379 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

1, 0, 5, 8, 45, 128, 505, 1680, 6089, 21120, 74909, 262680, 926485, 3258112, 11474865, 40382752, 142171985, 500432640, 1761656821, 6201182760, 21829269181, 76841888640, 270495370025, 952182350768, 3351823875225, 11798909226368
Offset: 1

Views

Author

Clark Kimberling, Jun 29 2011

Keywords

Comments

The polynomial p(n,x) is defined by ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+2). For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

Examples

			The first five polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=2x -> 2x
p(2,x)=2+x+3x^2 -> 5+4x
p(3,x)=8x+4x^2+4x^3 -> 8+20x
p(4,x)=4+4x+21x^2+10x^3+5x^4 -> 45+60x.
From these, read A192379=(1,0,5,8,45,...) and A192380=(0,2,4,20,60,...).

Crossrefs

Cf. A192232, A192380, A192381.

Programs

  • Mathematica
    q[x_] := x + 1; d = Sqrt[x + 2];
p[n_, x_] := ((x + d)^n - (x - d)^n )/(2 d)   (* Cf. A162517 *)
Table[Expand[p[n, x]], {n, 1, 6}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]   (* A192379 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]   (* A192380 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]   (* A192381 *)

Formula

Conjecture: a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: -x*(x^2+2*x-1) / (x^4+2*x^3-6*x^2-2*x+1). - Colin Barker, May 11 2014

A192383 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

1, 0, 6, 8, 60, 160, 744, 2496, 10064, 36480, 140512, 522624, 1983168, 7439360, 28091520, 105674752, 398391552, 1500057600, 5652182528, 21288560640, 80200784896, 302101094400, 1138045495296, 4286942363648, 16149041172480, 60833034895360
Offset: 1

Views

Author

Clark Kimberling, Jun 30 2011

Keywords

Comments

The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d=sqrt(x+3). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.

Examples

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0, x) = 1 -> 1
  p(1, x) =     2*x -> 2*x
  p(2, x) = 3 +   x +  3*x^2 -> 6 + 4*x
  p(3, x) =    12*x +  4*x^2 +  4*x^3 -> 8 + 24*x
  p(4, x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 60 + 72*x.
From these, read A192383 = (1, 0, 6, 8, 60, ...) and A192384 = (0, 2, 4, 24, 72, ...).

Links

Crossrefs

Cf. A083084, A192232.
Cf. A192384, A192385.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 41);
Coefficients(R!( x*(1-2*x-2*x^2)/(1-2*x-8*x^2+4*x^3+4*x^4) )) // G. C. Greubel, Jul 10 2023
  • Mathematica
    q[x_]:= x+1; d= Sqrt[x+3];
p[n_, x_]:= ((x+d)^n - (x-d)^n)/(2*d); (* suggested by A162517 *)
Table[Expand[p[n, x]], {n,6}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}];
Table[Coefficient[Part[t, n], x, 0], {n,30}] (* A192383 *)
Table[Coefficient[Part[t, n], x, 1], {n,30}] (* A192384 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n,30}] (* A192385 *)
LinearRecurrence[{2,8,-4,-4}, {1,0,6,8}, 40] (* G. C. Greubel, Jul 10 2023 *)
  • SageMath
    @CachedFunction
def a(n): # a = A192383
    if (n<5): return (0,1,0,6,8)[n]
    else: return 2*a(n-1) +8*a(n-2) -4*a(n-3) -4*a(n-4)
[a(n) for n in range(1,41)] # G. C. Greubel, Jul 10 2023

Formula

From Colin Barker, May 11 2014: (Start)
a(n) = 2*a(n-1) + 8*a(n-2) - 4*a(n-3) - 4*a(n-4).
G.f.: x*(1-2*x-2*x^2)/(1-2*x-8*x^2+4*x^3+4*x^4). (End)
From G. C. Greubel, Jul 10 2023: (Start)
T(n, k) = [x^k] ((x+sqrt(x+3))^n - (x-sqrt(x+3))^n)/(2*sqrt(x+3)).
a(n) = Sum_{k=0..n-1} T(n, k)*Fibonacci(k-1). (End)

A192384 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

0, 2, 4, 24, 72, 312, 1088, 4288, 15744, 60192, 224832, 851072, 3197056, 12062592, 45398016, 171104256, 644354048, 2427699712, 9144222720, 34448209920, 129761986560, 488821962752, 1841370087424, 6936475090944, 26129575084032
Offset: 1

Views

Author

Clark Kimberling, Jun 30 2011

Keywords

Comments

Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d = sqrt(x+3). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.

Examples

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0, x) = 1 -> 1
  p(1, x) =     2*x -> 2*x
  p(2, x) = 3 +   x +  3*x^2 -> 6 + 4*x
  p(3, x) =    12*x +  4*x^2 +  4*x^3 -> 8 + 24*x
  p(4, x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 60 + 72*x.
From these, read A192383 = (1, 0, 6, 8, 60, ...) and A192384 = (0, 2, 4, 24, 72, ...).

Links

Crossrefs

Cf. A192232, A192383, A192385.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 41);
[0] cat Coefficients(R!( 2*x^2/(1-2*x-8*x^2+4*x^3+4*x^4) )) // G. C. Greubel, Jul 10 2023
  • Mathematica
    (See A192383.)
LinearRecurrence[{2,8,-4,-4}, {0,2,4,24}, 40] (* G. C. Greubel, Jul 10 2023 *)
  • SageMath
    @CachedFunction
def a(n): # a = A192384
    if (n<5): return (0,0,2,4,24)[n]
    else: return 2*a(n-1) +8*a(n-2) -4*a(n-3) -4*a(n-4)
[a(n) for n in range(1,41)] # G. C. Greubel, Jul 10 2023

Formula

From Colin Barker, Dec 09 2012: (Start)
a(n) = 2*a(n-1) + 8*a(n-2) - 4*a(n-3) - 4*a(n-4).
G.f.: 2*x^2/(1-2*x-8*x^2+4*x^3+4*x^4). (End)
From G. C. Greubel, Jul 10 2023: (Start)
T(n, k) = [x^k] ((x+sqrt(x+3))^n - (x-sqrt(x+3))^n)/(2*sqrt(x+3)).
a(n) = Sum_{k=0..n-1} T(n, k)*Fibonacci(k). (End)

A192385 a(n) = A192384(n)/2.

Original entry on oeis.org

0, 1, 2, 12, 36, 156, 544, 2144, 7872, 30096, 112416, 425536, 1598528, 6031296, 22699008, 85552128, 322177024, 1213849856, 4572111360, 17224104960, 64880993280, 244410981376, 920685043712, 3468237545472, 13064787542016
Offset: 1

Views

Author

Clark Kimberling, Jun 30 2011

Keywords

Links

Crossrefs

Cf. A192232, A192383, A192384.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 41);
[0] cat Coefficients(R!( x^2/(1-2*x-8*x^2+4*x^3+4*x^4) )); // G. C. Greubel, Jul 10 2023
  • Mathematica
    (See A192384.)
LinearRecurrence[{2,8,-4,-4}, {0,1,2,12}, 40] (* G. C. Greubel, Jul 10 2023 *)
  • SageMath
    @CachedFunction
def a(n): # a = A192385
    if (n<5): return (0,0,1,2,12)[n]
    else: return 2*a(n-1) +8*a(n-2) -4*a(n-3) -4*a(n-4)
[a(n) for n in range(1,41)] # G. C. Greubel, Jul 10 2023

Formula

From G. C. Greubel, Jul 10 2023: (Start)
T(n, k) = [x^k] ((x+sqrt(x+3))^n - (x-sqrt(x+3))^n)/(2*sqrt(x+3)).
a(n) = (1/2)*Sum_{k=0..n-1} T(n, k)*Fibonacci(k).
a(n) = 2*a(n-1) + 8*a(n-2) - 4*a(n-3) - 4*a(n-4).
G.f.: x^2/(1 - 2*x - 8*x^2 + 4*x^3 + 4*x^4). (End)

A192386 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

Original entry on oeis.org

1, 0, 8, 8, 96, 224, 1408, 4608, 22784, 86016, 386048, 1548288, 6676480, 27467776, 116490240, 484409344, 2040135680, 8521777152, 35786063872, 149761818624, 628140015616, 2630784909312, 11028578435072, 46205266558976, 193656954814464
Offset: 1

Views

Author

Clark Kimberling, Jun 30 2011

Keywords

Comments

The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d = sqrt(x+5). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232.

Examples

			The first five polynomials p(n,x) and their reductions are as follows:
  p(0, x) = 1 -> 1
  p(1, x) =     2*x -> 2*x
  p(2, x) = 3 +   x +  3*x^2 -> 8 + 4*x
  p(3, x) =    12*x +  4*x^2 +  4*x^3 -> 8 + 32*x
  p(4, x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 96 + 96*x.
From these, read A192386 = (1, 0, 8, 8, 96, ...) and A192387 = (0, 2, 4, 32, 96, ...).

Links

Crossrefs

Cf. A083087, A192232, A192387, A192388.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 41);
Coefficients(R!( x*(1-2*x-4*x^2)/(1-2*x-12*x^2+8*x^3+16*x^4) )); // G. C. Greubel, Jul 10 2023
  • Mathematica
    q[x_]:= x+1; d= Sqrt[x+5];
p[n_, x_]:= ((x+d)^n - (x-d)^n)/(2*d); (* suggested by A162517 *)
Table[Expand[p[n, x]], {n,6}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[ FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}];
Table[Coefficient[Part[t, n], x, 0], {n,30}] (* A192386 *)
Table[Coefficient[Part[t, n], x, 1], {n,30}] (* A192387 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n,30}] (* A192388 *)
LinearRecurrence[{2,12,-8,-16}, {1,0,8,8}, 40] (* G. C. Greubel, Jul 10 2023 *)
  • SageMath
    @CachedFunction
def a(n): # a = A192386
    if (n<5): return (0,1,0,8,8)[n]
    else: return 2*a(n-1) +12*a(n-2) -8*a(n-3) -16*a(n-4)
[a(n) for n in range(1,41)] # G. C. Greubel, Jul 10 2023

Formula

From Colin Barker, May 11 2014: (Start)
a(n) = 2*a(n-1) + 12*a(n-2) - 8*a(n-3) - 16*a(n-4).
G.f.: x*(1-2*x-4*x^2)/(1-2*x-12*x^2+8*x^3+16*x^4). (End)
From G. C. Greubel, Jul 10 2023: (Start)
T(n, k) = [x^k] ((x+sqrt(x+5))^n - (x-sqrt(x+5))^n)/(2*sqrt(x+5)).
a(n) = Sum_{k=0..n-1} T(n, k)*Fibonacci(k-1). (End)
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