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.
my(b=6); forprime(p=2, , if( (([0, 1; 1, b]*Mod(1, p^2))^(p-kronecker(b^2+4, p)))[1, 2]==0, print1(p, ", "); ); );
The array, A(n, k), starts in row n = 1 with columns k >= 0 as 0 1 1 2 3 5 8 0 1 2 5 12 29 70 0 1 3 10 33 109 360 0 1 4 17 72 305 1292 0 1 5 26 135 701 3640 0 1 6 37 228 1405 8658 0 1 7 50 357 2549 18200 0 1 8 65 528 4289 34840 0 1 9 82 747 6805 61992 0 1 10 101 1020 10301 104030 0 1 11 122 1353 15005 166408 Antidiagonal triangle, T(n, k), begins as: 0; 0, 1; 0, 1, 1; 0, 1, 2, 2; 0, 1, 3, 5, 3; 0, 1, 4, 10, 12, 5; 0, 1, 5, 17, 33, 29, 8; 0, 1, 6, 26, 72, 109, 70, 13; 0, 1, 7, 37, 135, 305, 360, 169, 21; 0, 1, 8, 50, 228, 701, 1292, 1189, 408, 34;
A172236:= func< n,k | k le 1 select k else Evaluate(DicksonSecond(k-1,-1), n-k) >; [A172236(n,k): k in [0..n-1], n in [1..13]]; // G. C. Greubel, Sep 29 2024
A172236[n_,k_]:=Fibonacci[k, n-k]; Table[A172236[n, k], {n,15}, {k,0,n-1}]//Flatten
A(n, k) = if (k==0, 0, if (k==1, 1, n*A(n, k-1) + A(n, k-2))); tabl(nn) = for(n=1, nn, for (k=0, nn, print1(A(n, k), ", ")); print); \\ Jianing Song, Jul 14 2018 (program from Michel Marcus; see also A316269)
A(n, k) = ([n, 1; 1, 0]^k)[2, 1] \\ Jianing Song, Nov 10 2018
def A172236(n,k): return sum(binomial(k-j-1,j)*(n-k)^(k-2*j-1) for j in range(1+(k-1)//2)) flatten([[A172236(n,k) for k in range(n)] for n in range(1,14)]) # G. C. Greubel, Sep 29 2024
G.f. = x + 6*x^2 + 37*x^3 + 228*x^4 + 1405*x^5 + 8658*x^6 + 53353*x^7 + ...
[n le 2 select n-1 else 6*Self(n-1)+Self(n-2): n in [1..25]]; // Vincenzo Librandi, Feb 23 2013
evalf(sqrt(10),200); convert(%,confrac,fractionlist); fractionlist; A005668:=-z/(-1+6*z+z**2); - Simon Plouffe in his 1992 dissertation. a := n -> `if`(n<2,n,6^(n-1)*hypergeom([1-n/2,(1-n)/2], [1-n], -1/9)): seq(simplify(a(n)), n=0..23); # Peter Luschny, Jun 28 2017
LinearRecurrence[{6,1}, {0,1}, 30] (* Vincenzo Librandi, Feb 23 2013 *)
a[ n_] := (-I)^(n - 1) ChebyshevU[ n - 1, 3 I]; (* Michael Somos, May 28 2014 *)
a[ n_] := MatrixPower[ {{0, 1}, {1, 6}}, n + 1][[1, 1]]; (* Michael Somos, May 28 2014 *)
Fibonacci[Range[0,30],6] (* G. C. Greubel, Jun 06 2019 *)
Join[{0},Convergents[Sqrt[10],30]//Denominator] (* Harvey P. Dale, Dec 28 2022 *)
{a(n) = ([0, 1; 1, 6]^(n+1)) [1, 1]}; /* Michael Somos, May 28 2014 */
{a(n) = (-I)^(n-1) * polchebyshev( n-1, 2, 3*I)}; /* Michael Somos, May 28 2014 */
from sage.combinat.sloane_functions import recur_gen3; it = recur_gen3(0,1,6,6,1,0); [next(it) for i in range(1,22)] # Zerinvary Lajos, Jul 09 2008
[lucas_number1(n,6,-1) for n in range(0, 21)]# Zerinvary Lajos, Apr 24 2009
F := proc(k,n) option remember; if n <= 1 then n; else k*procname(k,n-1)+procname(k,n-2) ; end if; end proc: Pper := proc(k,m) local cha, zer,n,fmodm ; cha := [] ; zer := [] ; for n from 0 do fmodm := F(k,n) mod m ; cha := [op(cha),fmodm] ; if fmodm = 0 then zer := [op(zer),n] ; end if; if nops(zer) = 5 then break; end if; end do ; if [op(1..zer[2],cha) ] = [ op(zer[2]+1..zer[3],cha) ] and [op(1..zer[2],cha)] = [ op(zer[3]+1..zer[4],cha) ] and [op(1..zer[2],cha)] = [ op(zer[4]+1..zer[5],cha) ] then return zer[2] ; elif [op(1..zer[3],cha) ] = [ op(zer[3]+1..zer[5],cha) ] then return zer[3] ; else return zer[5] ; end if; end proc: k := 2 ; seq( Pper(k,m),m=1..80) ;
Table[s = t = Mod[{0, 1}, n]; cnt = 1; While[tmp = Mod[2*t[[2]] + t[[1]], n]; t[[1]] = t[[2]]; t[[2]] = tmp; s != t, cnt++]; cnt, {n, 100}] (* T. D. Noe, Jul 09 2012 *)
F := proc(k,n) option remember; if n <= 1 then n; else k*procname(k,n-1)+procname(k,n-2) ; end if; end proc: Pper := proc(k,m) local cha, zer,n,fmodm ; cha := [] ; zer := [] ; for n from 0 do fmodm := F(k,n) mod m ; cha := [op(cha),fmodm] ; if fmodm = 0 then zer := [op(zer),n] ; end if; if nops(zer) = 5 then break; end if; end do ; if [op(1..zer[2],cha) ] = [ op(zer[2]+1..zer[3],cha) ] and [op(1..zer[2],cha)] = [ op(zer[3]+1..zer[4],cha) ] and [op(1..zer[2],cha)] = [ op(zer[4]+1..zer[5],cha) ] then return zer[2] ; elif [op(1..zer[3],cha) ] = [ op(zer[3]+1..zer[5],cha) ] then return zer[3] ; else return zer[5] ; end if; end proc: k := 3 ; seq( Pper(k,m),m=1..80) ;
Table[s = t = Mod[{0, 1}, n]; cnt = 1; While[tmp = Mod[3*t[[2]] + t[[1]], n]; t[[1]] = t[[2]]; t[[2]] = tmp; s!= t, cnt++]; cnt, {n, 100}] (* Vincenzo Librandi, Dec 20 2012, T. D. Noe *)
F := proc(k,n) option remember; if n <= 1 then n; else k*procname(k,n-1)+procname(k,n-2) ; end if; end proc: Pper := proc(k,m) local cha, zer,n,fmodm ; cha := [] ; zer := [] ; for n from 0 do fmodm := F(k,n) mod m ; cha := [op(cha),fmodm] ; if fmodm = 0 then zer := [op(zer),n] ; end if; if nops(zer) = 5 then break; end if; end do ; if [op(1..zer[2],cha) ] = [ op(zer[2]+1..zer[3],cha) ] and [op(1..zer[2],cha)] = [ op(zer[3]+1..zer[4],cha) ] and [op(1..zer[2],cha)] = [ op(zer[4]+1..zer[5],cha) ] then return zer[2] ; elif [op(1..zer[3],cha) ] = [ op(zer[3]+1..zer[5],cha) ] then return zer[3] ; else return zer[5] ; end if; end proc: k := 4 ; seq( Pper(k,m),m=1..80) ;
Table[s = t = Mod[{0, 1}, n]; cnt=1; While[tmp = Mod[4*t[[2]] + t[[1]], n]; t[[1]] = t[[2]]; t[[2]] = tmp; s!= t, cnt++]; cnt, {n, 100}] (* Vincenzo Librandi, Dec 20 2012, after T. D. Noe *)
F := proc(k,n) option remember; if n <= 1 then n; else k*procname(k,n-1)+procname(k,n-2) ; end if; end proc: Pper := proc(k,m) local cha, zer,n,fmodm ; cha := [] ; zer := [] ; for n from 0 do fmodm := F(k,n) mod m ; cha := [op(cha),fmodm] ; if fmodm = 0 then zer := [op(zer),n] ; end if; if nops(zer) = 5 then break; end if; end do ; if [op(1..zer[2],cha) ] = [ op(zer[2]+1..zer[3],cha) ] and [op(1..zer[2],cha)] = [ op(zer[3]+1..zer[4],cha) ] and [op(1..zer[2],cha)] = [ op(zer[4]+1..zer[5],cha) ] then return zer[2] ; elif [op(1..zer[3],cha) ] = [ op(zer[3]+1..zer[5],cha) ] then return zer[3] ; else return zer[5] ; end if; end proc: k := 5 ; seq( Pper(k,m),m=1..80) ;
Table[s = t = Mod[{0, 1}, n]; cnt = 1; While[tmp = Mod[5*t[[2]] + t[[1]], n]; t[[1]] = t[[2]]; t[[2]] = tmp; s!= t, cnt++]; cnt, {n, 100}] (* Vincenzo Librandi, Dec 20 2012, after T. D. Noe *)
A271782(k) = forprime(p=2,10^8, if( (([0,1;1,k]*Mod(1,p^2))^(p-kronecker(k^2+4,p)))[1,2]==0, return(p);); ); \\ Max Alekseyev, Apr 22 2016, corrected by Richard N. Smith, Jul 16 2019 to include p=2 and p divides k^2+4
Reading 0, 1, 2, 6, 16, 44, 120, 328, 896, 2448,.. modulo 12 gives 0, 1, 2, 6, 4, 8, 0, 4, 8, 0, 4, 8 ,.. with period length a(n=12)= 3.
a={1};For[n=2,n<=80,n++,{x={{0,1}}; t={1,1}; While[ !MemberQ[x,t], {xl = x[[ -1]]; AppendTo[x,t]; t={Mod[2*(t[[1]]+xl[[1]]),n], Mod[2*(t[[2]] + xl[[2]]),n]};}]; p = Flatten[Position[x,t]][[1]]; AppendTo[a, Length[x] - p+1];}]; Print[a]; (* John W. Layman, Aug 10 2010 *)
Reading 0, 1, 3, 12, 45, 171, 648, 2457, 9315, 35316, 133893 .. modulo n=3 gives 0, 1, 0, 0, 0, 0, 0, .. with period length a(n=3)= 1. Reading modulo n=6 gives 0, 1, 3, 0, 3, 3, 0, 3, 3 with period length a(n=6)=3.
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