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 1-5 of 5 results.

A047923 Main diagonal of array in A038150.

Original entry on oeis.org

1, 6, 29, 97, 343, 1131, 3338, 10336, 29644, 88555, 260497, 728358, 2103284, 6020698, 16594432, 46969365, 128670281, 361020986, 1008411198, 2742388946, 7613161908, 20632925370, 56988914979, 156977658446, 423559114311
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A026351, A038150.

Programs

  • Mathematica
    max = 24; t[0, 0] = 1; t[n_, 1] := t[n, 1] = 2*t[n, 0]+n+1; t[n_, 0] := t[n, 0] = Catch[For[ u = Table[t[m, k], {m, 0, n-1}, {k, 0, max - m}] // Flatten // Union; k = 1, k <= n*(n+1)/2+1 , k++, If[u[[k]] != k, Throw[k]]]]; t[n_, k_] := t[n, k] = 3*t[n, k-1] - t[n, k-2] ; a[n_] := t[n, n]; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Jan 02 2013 *)

Formula

a(n) = F(2n)*n + F(2n+1)*A026351(n). - Charlie Neder, Feb 07 2019

Extensions

More terms from Naohiro Nomoto, Jun 07 2001

A047925 3rd column of array in A038150.

Original entry on oeis.org

8, 16, 29, 37, 50, 63, 71, 84, 92, 105, 118, 126, 139, 152, 160, 173, 181, 194, 207, 215, 228, 236, 249, 262, 270, 283, 296, 304, 317, 325, 338, 351, 359, 372, 385, 393, 406, 414, 427, 440, 448, 461, 469, 482, 495, 503, 516, 529, 537, 550, 558, 571, 584, 592
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A038150.

Programs

  • Mathematica
    max = 53; Clear[t]; t[0, 0] = 1; t[n_, 1] := t[n, 1] = 2*t[n, 0] + n + 1; t[n_, 0] := t[n, 0] = For[u = Table[t[m, k], {m, 0, n-1}, {k, 0, max-m}] // Flatten // Union; k = 1, k <= n*(n+1)/2 + 1, k++, If[u[[k]] != k, Return[k]]]; t[n_, k_] := t[n, k] = 3*t[n, k-1] - t[n, k-2]; a[n_] := t[n, 2]; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Jul 16 2015 *)

Extensions

More terms from Naohiro Nomoto, Jun 08 2001

A120874 Fractal sequence of the Fraenkel array (A038150).

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 5, 1, 6, 7, 3, 8, 9, 4, 10, 2, 11, 12, 5, 13, 1, 14, 15, 6, 16, 17, 7, 18, 3, 19, 20, 8, 21, 22, 9, 23, 4, 24, 25, 10, 26, 2, 27, 28, 11, 29, 30, 12, 31, 5, 32, 33, 13, 34, 1, 35, 36, 14, 37, 38, 15, 39, 6, 40, 41, 16, 42, 43, 17, 44, 7, 45, 46, 18, 47, 3, 48, 49, 19
Offset: 1

Views

Author

Clark Kimberling, Jul 10 2006

Keywords

Comments

A fractal sequence f contains itself as a proper subsequence; e.g., if you delete the first occurrence of each positive integer, the remaining sequence is f; thus f properly contains itself infinitely many times.

Examples

			The fractal sequence f(n) of a dispersion D={d(g,h,)} is defined as follows. For each positive integer n there is a unique (g,h) such that n=d(g,h) and f(n)=g. So f(6)=2 because the row of the Fraenkel array in which 6 occurs is row 2.

References

  • Clark Kimberling, The equation (j+k+1)^2-4*k=Q*n^2 and related dispersions, Journal of Integer Sequences 10 (2007, Article 07.2.7) 1-17.

Links

Crossrefs

Cf. A038150.

Programs

  • Mathematica
    num[n_, b_] := Last[NestWhile[{Mod[#[[1]], Last[#[[2]]]], Drop[#[[2]], -1], Append[#[[3]], Quotient[#[[1]], Last[#[[2]]]]]} &, {n, b, {}}, #[[2]] =!= {} &]];
left[n_, b_] := If[Last[num[n, b]] == 0, Dot[num[n, b], Rest[Append[Reverse[b], 0]]], n];
fractal[n_, b_] := # - Count[Last[num[Range[#], b]], 0] &@ FixedPoint[left[#, b] &, n];
Table[fractal[n, Table[Fibonacci[2 i], {i, 12}]], {n, 30}] (* Birkas Gyorgy, Apr 13 2011 *)
Table[Ceiling[NestWhile[Ceiling[#/GoldenRatio^2] - 1 &, n, Ceiling[#/GoldenRatio] == Ceiling[(# - 1)/GoldenRatio]&]/ GoldenRatio], {n, 30}] (* Birkas Gyorgy, Apr 15 2011 *)

A047924 a(n) = B_{A_n+1}+1, where A_n = floor(n*phi) = A000201(n), B_n = floor(n*phi^2) = A001950(n) and phi is the golden ratio.

Original entry on oeis.org

3, 6, 11, 14, 19, 24, 27, 32, 35, 40, 45, 48, 53, 58, 61, 66, 69, 74, 79, 82, 87, 90, 95, 100, 103, 108, 113, 116, 121, 124, 129, 134, 137, 142, 147, 150, 155, 158, 163, 168, 171, 176, 179, 184, 189, 192, 197, 202, 205, 210, 213, 218, 223, 226, 231, 234, 239
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

2nd column of array in A038150.
Apart from the first term also the second column of A126714; see also A223025. - Casey Mongoven, Mar 11 2013

References

  • Clark Kimberling, Stolarsky interspersions, Ars Combinatoria 39 (1995), 129-138.

Links

Crossrefs

Cf. A007066.

Programs

  • Maple
    A001950 := proc(n)
        local phi;
        phi := (1+sqrt(5))/2 ;
        floor(n*phi^2) ;
end proc:
A000201 := proc(n)
        local phi;
        phi := (1+sqrt(5))/2 ;
        floor(n*phi) ;
end proc:
A047924 := proc(n)
        1+A001950(1+A000201(n)) ;
end proc: # R. J. Mathar, Mar 20 2013
  • Mathematica
    A[n_] := Floor[n*GoldenRatio]; B[n_] := Floor[n*GoldenRatio^2]; a[n_] := B[A[n]+1]+1; Table[a[n], {n, 0, 56}] (* Jean-François Alcover, Feb 11 2014 *)
  • Python
    from mpmath import *
mp.dps=100
import math
def A(n): return int(math.floor(n*phi))
def B(n): return int(math.floor(n*phi**2))
def a(n): return B(A(n) + 1) + 1 # Indranil Ghosh, Apr 25 2017
  • Python
    from math import isqrt
def A047924(n): return ((m:=(n+isqrt(5*n**2)>>1)+1)+isqrt(5*m**2)>>1)+m+1 # Chai Wah Wu, Aug 25 2022

Extensions

More terms from Naohiro Nomoto, Jun 08 2001
New description from Aviezri S. Fraenkel, Aug 03 2007

A136451 Triangle T(n,k) with the coefficient [x^k] of the characteristic polynomial of the following n X n matrix: 2 on the main antidiagonal, -1 on the adjacent sub-antidiagonals and 0 otherwise.

Original entry on oeis.org

1, 2, -1, -3, 2, 1, -4, 6, 2, -1, 5, -10, -9, 2, 1, 6, -19, -16, 12, 2, -1, -7, 28, 42, -22, -15, 2, 1, -8, 44, 68, -74, -28, 18, 2, -1, 9, -60, -138, 126, 115, -34, -21, 2, 1, 10, -85, -208, 316, 202, -165, -40, 24, 2, -1, -11, 110, 363, -506, -605, 296, 224, -46, -27, 2, 1
Offset: 0

Views

Author

Roger L. Bagula, Mar 19 2008

Keywords

Comments

We start from tri-antidiagonal variants of the Cartan A-n group matrix. For n=1 this is {2}, for n=2 this is {{-1,2},{2,-1}}, for n=3 {{0,-1,2},{-1,2,-1},{2,-1,0}}, for n =4 {{0,0,-1,2},{0,-1,2,-1},{-1,2,-1,0},{2,-1,0,0}} etc. The n-th row of the triangle are the expansion coefficients of the characteristic polynomial.
For n=0, the empty product of the empty matrix is assigned the value T(0,0)=1.
Row sums (characteristic polynomials evaluated at x=0) are 1, 1, 0, 3, -11, -16, 29, 21, 0, 55, -199, -288, 521, 377, 0, 987, -3571, -5168, 9349, 6765, 0, ... (see A038150).

Examples

			1;
2, -1;
-3,2, 1;
-4, 6, 2, -1;
5, -10, -9, 2, 1;
6, -19, -16, 12, 2, -1;
-7,28, 42, -22, -15, 2, 1;
-8, 44, 68, -74, -28,18, 2, -1;
9, -60, -138, 126, 115, -34, -21, 2, 1;
10, -85, -208,316, 202, -165, -40, 24, 2, -1;
-11, 110, 363, -506, -605, 296, 224, -46, -27, 2, 1;

Crossrefs

Cf. A124018 (variant), A005993 (column k=1), A061927 (bisection column k=2).

Programs

  • Maple
    A136451x := proc(n,x)
    local A,r,c ;
    A := Matrix(1..n,1..n) ;
    for r from 1 to n do
    for c from 1 to n do
            A[r,c] :=0 ;
        if r+c = 1+n then
            A[r,c] := A[r,c]+2 ;
        elif abs(r+c-1-n)= 1 then
            A[r,c] :=  A[r,c]-1 ;
        end if;
    end do:
    end do:
    (-1)^n*LinearAlgebra[CharacteristicPolynomial](A,x) ;
end proc;
A136451 := proc(n,k)
    coeftayl( A136451x(n,x),x=0,k) ;
end proc:
seq(seq(A136451(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Dec 04 2011
  • Mathematica
    H[n_] := Table[Table[If[i + j - 1 == n, 2,If[i + j - 1 == n + 1, -1, If[i + j - 1 == n - 1, -1, 0]]], {i, 1, n}], {j, 1, n}]; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[H[n], x], x], {n, 1, 10}]]; Flatten[a']
Showing 1-5 of 5 results.

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

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