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 1-3 of 3 results.

A192951 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, 3, 9, 20, 40, 74, 131, 225, 379, 630, 1038, 1700, 2773, 4511, 7325, 11880, 19252, 31182, 50487, 81725, 132271, 214058, 346394, 560520, 906985, 1467579, 2374641, 3842300, 6217024, 10059410, 16276523, 26336025, 42612643, 68948766
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) + 3n - 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.
...
The list of examples at A192744 is extended here; the recurrence is given by p(n,x) = x*p(n-1,x) + v(n), with p(0,x)=1, and the reduction of p(n,x) by x^2 -> x+1 is represented by u1 + u2*x:
...
If v(n)= n, then u1=A001595, u2=A104161.
If v(n)= n-1, then u1=A001610, u2=A066982.
If v(n)= 3n-1, then u1=A171516, u2=A192951.
If v(n)= 3n-2, then u1=A192746, u2=A192952.
If v(n)= 2n-1, then u1=A111314, u2=A192953.
If v(n)= n^2, then u1=A192954, u2=A192955.
If v(n)= -1+n^2, then u1=A192956, u2=A192957.
If v(n)= 1+n^2, then u1=A192953, u2=A192389.
If v(n)= -2+n^2, then u1=A192958, u2=A192959.
If v(n)= 2+n^2, then u1=A192960, u2=A192961.
If v(n)= n+n^2, then u1=A192962, u2=A192963.
If v(n)= -n+n^2, then u1=A192964, u2=A192965.
If v(n)= n(n+1)/2, then u1=A030119, u2=A192966.
If v(n)= n(n-1)/2, then u1=A192967, u2=A192968.
If v(n)= n(n+3)/2, then u1=A192969, u2=A192970.
If v(n)= 2n^2, then u1=A192971, u2=A192972.
If v(n)= 1+2n^2, then u1=A192973, u2=A192974.
If v(n)= -1+2n^2, then u1=A192975, u2=A192976.
If v(n)= 1+n+n^2, then u1=A027181, u2=A192978.
If v(n)= 1-n+n^2, then u1=A192979, u2=A192980.
If v(n)= (n+1)^2, then u1=A001891, u2=A053808.
If v(n)= (n-1)^2, then u1=A192981, u2=A192982.

Links

Crossrefs

Cf. A000045, A192232, A192744.

Programs

  • GAP
    F:=Fibonacci;; List([0..40], n-> F(n+4)+2*F(n+2)-(3*n+5)); # G. C. Greubel, Jul 12 2019
  • Magma
    I:=[0, 1, 3, 9]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2)-1*Self(n-3)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Nov 16 2011
  • Magma
    F:=Fibonacci; [F(n+4)+2*F(n+2)-(3*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] + 3n - 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}] (* A171516 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192951 *)
(* Additional programs *)
LinearRecurrence[{3,-2,-1,1},{0,1,3,9},40] (* Vincenzo Librandi, Nov 16 2011 *)
With[{F=Fibonacci}, Table[F[n+4]+2*F[n+2]-(3*n+5), {n,0,40}]] (* G. C. Greubel, Jul 12 2019 *)
  • PARI
    a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 1,-1,-2,3]^n*[0;1;3;9])[1,1] \\ Charles R Greathouse IV, Mar 22 2016
  • PARI
    vector(40, n, n--; f=fibonacci; f(n+4)+2*f(n+2)-(3*n+5)) \\ G. C. Greubel, Jul 12 2019
  • Sage
    f=fibonacci; [f(n+4)+2*f(n+2)-(3*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 Bruno Berselli, Nov 16 2011: (Start)
G.f.: x*(1+2*x^2)/((1-x)^2*(1 - x - x^2)).
a(n) = ((25+13*t)*(1+t)^n + (25-13*t)*(1-t)^n)/(10*2^n) - 3*n - 5 = A000285(n+2) - 3*n - 5 where t=sqrt(5). (End)
a(n) = Fibonacci(n+4) + 2*Fibonacci(n+2) - (3*n+5). - G. C. Greubel, Jul 12 2019

A263519 T(n,k) = Number of (n+1) X (k+1) arrays of permutations of 0..(n+1)*(k+1)-1 filled by rows with each element moved a city block distance of 0 or 1, and rows and columns in increasing lexicographic order.

Original entry on oeis.org

3, 7, 8, 15, 35, 23, 29, 160, 208, 66, 53, 660, 2076, 1198, 190, 93, 2651, 18369, 25968, 7022, 547, 159, 10350, 158109, 489294, 331130, 41035, 1575, 267, 39807, 1317780, 9051857, 13332096, 4213002, 240237, 4535, 443, 151463, 10791350, 162207955
Offset: 1

Views

Author

R. H. Hardin, Oct 19 2015

Keywords

Comments

Table starts
.....3.......7.........15............29...............53..................93
.....8......35........160...........660.............2651...............10350
....23.....208.......2076.........18369...........158109.............1317780
....66....1198......25968........489294..........9051857...........162207955
...190....7022.....331130......13332096........529329240.........20339400914
...547...41035....4213002.....362159570......30867389241.......2543460828164
..1575..240237...53712998....9866744449....1805523575884.....319022980139204
..4535.1406038..684799391..268827612021..105637731091773...40028581755172441
.13058.8230727.8732881192.7327820172316.6184312882582853.5025951440933512579

Examples

			Some solutions for n=3 k=4
..0..1..7..8..9....0..1..7..8..9....0..1..2..3..4....0..1..2..4..9
..6..5..2..3..4...10..5..2..3..4....5..6..8..7..9....5..7..6..3..8
.10.12.11.13.19...11..6.12.13.14...15.10.13.12.14...10.12.11.14.13
.15.17.16.18.14...15.16.17.19.18...16.11.18.17.19...15.16.17.18.19

Links

Crossrefs

Column 1 is A147704(n+1).
Row 1 is A192960.

Formula

Empirical for column k:
k=1: a(n) = 3*a(n-1) -a(n-3)
k=2: [order 10]
k=3: [order 35]
Empirical for row n:
n=1: a(n) = 3*a(n-1) -2*a(n-2) -a(n-3) +a(n-4)
n=2: [order 10]
n=3: [order 29]
n=4: [order 92]

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
Showing 1-3 of 3 results.

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