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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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A269725 a(n) = row number of extended Wythoff array (see A035513) which contains the sequence n times the Fibonacci numbers 1,2,3,5,8,13,21,... .

Original entry on oeis.org

0, 2, 3, 4, 15, 18, 21, 24, 27, 30, 33, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 630, 651, 672, 693, 714, 735, 756, 777, 798, 819, 840, 861, 882, 903, 924, 945, 966, 987, 1008, 1029, 1050, 1071, 1092, 1113, 1134, 1155, 1176, 1197, 1218, 1239, 1260
Offset: 1

Views

Author

N. J. A. Sloane, Mar 07 2016

Keywords

Examples

			Take n=5: 5 times 1,2,3,5,8,13,... gives 5,10,15,25,40,65,.., which is row 15 of the extended Wythoff array (when extended to the left), so a(5) = 15.

References

  • J. H. Conway, Posting to Math Fun Mailing List, Nov 25 1996.

Links

Crossrefs

Cf. A000045, A022413-A022423, A035513, A269726, A269729.

Programs

  • Maple
    A269725 := proc(n)
    local f,sl,r,c,wrks ;
    f := [seq(n*combinat[fibonacci](i),i=2..30)] ;
    for sl from 0 do
        for r from 1 do
            if A035513(r,1) = op(1+sl,f) then
                wrks := true;
                for c from 2 to 5 do
                    if A035513(r,c) <> op(c+sl,f) then
                        wrks := false;
                    end if;
                end do:
                if wrks then
                    print(n,f,r) ;
                    return r-1 ;
                end if;
            elif A035513(r,1) > op(1+sl,f) then
                break ;
            end if;
        end do:
    end do:
end proc: # R. J. Mathar, May 06 2017
  • Mathematica
    W[n_, k_] := Fibonacci[k+1] Floor[n*GoldenRatio] + (n-1) Fibonacci[k];
a[n_] := Module[{f, sl, r, c, wrks}, f = Table[n*Fibonacci[i], {i, 2, 30}]; For[sl = 0, True, sl++, For[r = 1, True, r++, Which[W[r, 1] == f[[1 + sl]], wrks = True; For[c = 2, c <= 5, c++, If[W[r, c] != f[[c+sl]], wrks = False]]; If[wrks, Return[r-1]], W[r, 1] > f[[1+sl]], Break[]]]]];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Sep 13 2022, after R. J. Mathar *)

Formula

a(n) = A173027(n)-1. - R. J. Mathar, May 06 2017

A269729 a(n) = row number of extended Wythoff array (see A035513) which contains the sequence obtained by reading the n-th row backwards (and adjusting signs).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 10, 5, 8, 11, 6, 9, 12, 20, 28, 15, 23, 31, 18, 26, 13, 21, 29, 16, 24, 32, 19, 27, 14, 22, 30, 17, 25, 33, 54, 75, 41, 62, 83, 49, 70, 36, 57, 78, 44, 65, 86, 52, 73, 39, 60, 81, 47, 68, 34, 55, 76, 42, 63, 84, 50, 71, 37, 58, 79, 45, 66, 87, 53, 74, 40
Offset: 0

Views

Author

N. J. A. Sloane, Mar 08 2016

Keywords

Comments

Conjecture: sequence is its own inverse. - R. J. Mathar, May 08 2019

Examples

			Take n=5: reading row 5 of A035513 backwards gives ... 23, 14, 9, 5, 4, 1, 3, -2, 5, -7, 12, -19, ..., which after adjusting the signs is row 7, so a(5) = 7.

References

  • J. H. Conway, Postings to Math Fun Mailing List, Nov 25 1996 and Dec 02 1996.

Links

Crossrefs

Cf. A000045, A022413-A022423, A035513, A269725, A269726, A269729.
See A269733 for first differences.

Programs

  • Maple
    A035513 := proc(r::integer, c::integer)
    option remember;
    if c = 1 then
        A003622(r) ;
    elif c > 1 then
        A022342(1+procname(r, c-1)) ;
    elif c < 1 then
        procname(r,c+2)-procname(r,c+1) ;
    end if;
end proc:
# search in A035513 for row with consecutive w1,w2
A035513inv := proc(w1::integer,w2::integer)
    local r,c,W1,W2 ;
    for r from 1 do
        if A035513(r,1) > w2 then
            return -1 ;
        end if;
        for c from 1 do
            W1 := A035513(r,c) ;
            W2 := A035513(r,c+1) ;
            if W1=w1 and W2=w2 then
                return r-1 ;
            elif W2 > w2 then
                break;
            end if;
        end do:
    end do:
end proc:
A269729 := proc(n)
    option remember;
    local c,W1,W2,r,n35513;
    n35513 := n+1 ;
    for c from 1 by -1 do
        W1 := A035513(n35513,c) ;
        W2 := A035513(n35513,c-1) ;
        if W1 < 0 and abs(W2) > abs(W1) then
            r :=  A035513inv(abs(W1),abs(W2)) ;
            if r >= 0 then
                return r;
            end if;
        end if;
    end do:
end proc:
seq(A269729(n),n=0..120) ; # R. J. Mathar, May 08 2019
  • Mathematica
    W[n_, k_] := W[n, k] = Fibonacci[k+1] Floor[n*GoldenRatio] + (n-1)* Fibonacci[k];
Winv[w1_, w2_] := Winv[w1, w2] = Module[{r, c, W1, W2}, For[r = 1, True, r++, If[W[r, 1] > w2, Return[-1]]; For[c = 1, True, c++, W1 = W[r, c]; W2 = W[r, c+1]; If[W1 == w1 && W2 == w2, Return[r-1], If[W2 > w2, Break[]]]]]];
a[n_] := a[n] = Module[{c, W1, W2, r, nw}, nw = n+1; For[c = 1, True, c--, W1 = W[nw, c]; W2 = W[nw, c-1]; If[W1 < 0 && Abs[W2] > Abs[W1], r = Winv[Abs[W1], Abs[W2]]; If[r >= 0, Return[r]]]]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 120}] (* Jean-François Alcover, Aug 09 2023, after R. J. Mathar *)

Extensions

Terms from a(18) on by R. J. Mathar, May 08 2019
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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