A068149 Triangular numbers in which neighboring digits differ at most by 1. Allowed neighbors of 9 are 0, 8 and 9.

0, 1, 3, 6, 10, 21, 45, 55, 66, 78, 210, 666, 990, 2211, 3321, 5565, 6555, 8778, 10011, 90100, 112101, 222111, 232221, 443211, 887778, 5433456, 5456556, 5656566, 5676765, 22221111, 22321221, 34565455, 88877778, 211099878, 212210901

Offset: 1

Author

Amarnath Murthy, Feb 23 2002

Keywords

• base
• easy
• nonn

Comments

Includes (2 * 10^(2*k) - 10^k - 1)/9 and (8 * 10^(2*k) - 10^(k+1) + 2)/9 for k >= 1, and (2 * 10^(2*k) + 89 * 10^k + 989)/9 for k >= 2. - Robert Israel, Sep 22 2024

Links

• Robert Israel, Table of n, a(n) for n = 1..95 (n = 1 .. 65 from Andrew Howroyd)

Crossrefs

Intersection of A000217 and A376425.

Programs

Maple

f:= proc(n) local i;
   seq(10*n+i, i= sort([n-1, n, n+1] mod 10))
end proc:
istri:= proc(n) issqr(1+8*n) end proc:
S:= [$1..9]: R:= 0,1,3,6: count:= 4:
for i from 1 while count < 95 do
  for k from i to i+1 do
    for s in S do
      tmin:= ceil(sqrt(8*s*10^k+1));
      if tmin::even then tmin:= tmin+1 fi;
      for t from tmin to floor(sqrt(8*(s+1)*10^k-7)) by 2 do
        x:= (t-1)/2; y:= x*(x+1)/2;
        L:= convert(y,base,10);
        if convert(L[2..-1]-L[1..-2] mod 10, set) subset {0,1,9} then
          R:= R,y; count:= count+1;
  fi od od od;
  if count < 95 then S:= map(f, S) fi;
od:
R; # Robert Israel, Sep 23 2024

Mathematica

Do[a = IntegerDigits[n(n + 1)/2]; k = 1; l = Length[a]; While[k < l && (Abs[a[[k]]- a[[k + 1]]] < 2 || Abs[a[[k]] - a[[k + 1]]] > 8), k++ ]; If[k == l, Print[n(n + 1)/2]], {n, 0, 10^5} ]
Select[Accumulate[Range[0,30000]],Max[Select[Abs[Differences[ IntegerDigits[ #]]], #!=9&]]<2&] (* Harvey P. Dale, Oct 09 2013 *)

Extensions

Edited and extended by Robert G. Wilson v and Sascha Kurz, Mar 01 2002

Offset changed by Andrew Howroyd, Sep 22 2024