A034706 - OEIS

%I #25 Oct 28 2023 11:43:42

%S 0,1,3,4,6,9,10,15,16,19,20,21,25,28,31,34,35,36,45,46,49,52,55,56,64,

%T 66,74,78,80,81,83,84,85,91,100,105,109,110,116,119,120,121,130,136,

%U 144,145,153,155,161,164,165,166,169,171,185,190,196,199,200,202,210

%N Numbers which are sums of consecutive triangular numbers.

%H Reinhard Zumkeller, <a href="/A034706/b034706.txt">Table of n, a(n) for n = 1..10000</a>

%H D. Subramaniam, E. Trevino, and P. Pollack, <a href="http://math.colgate.edu/~integers/uproc15/uproc15.pdf">On sums of consecutive triangular numbers</a>, INTEGERS 20A (2020) A15.

%p isA034706 := proc(n)

%p     local a,b;

%p     for a from 0 do

%p         if a*(a+1)/2 > n then

%p             return false;

%p         end if;

%p         for b from a do

%p             tab := (1+b-a)*(a^2+b*a+a+b^2+2*b)/6 ;

%p             if tab = n then

%p                 return true;

%p             elif tab > n then

%p                 break;

%p             end if;

%p         end do:

%p     end do:

%p end proc:

%p for n from 0 to 100 do

%p     if isA034706(n) then

%p         printf("%d,",n) ;

%p     end if;

%p end do: # _R. J. Mathar_, Dec 14 2015

%t M = 1000; (* to get all terms <= M *)

%t nmax = (Sqrt[8 M + 1] - 1)/2 // Ceiling;

%t Table[Sum[n(n+1)/2, {n, j, k}], {j, 0, nmax}, {k, j, nmax}] // Flatten // Union // Select[#, # <= M&]& (* _Jean-François Alcover_, Mar 10 2019 *)

%o (Haskell)

%o -- import Data.Set (deleteFindMin, union, fromList); import Data.List (inits)

%o a034706 n = a034706_list !! (n-1)

%o a034706_list = f 0 (tail $ inits $ a000217_list) (fromList [0]) where

%o    f x vss'@(vs:vss) s

%o      | y < x = y : f x vss' s'

%o      | otherwise = f w vss (union s $ fromList $ scanl1 (+) ws)

%o      where ws@(w:_) = reverse vs

%o            (y, s') = deleteFindMin s

%o -- _Reinhard Zumkeller_, May 12 2015

%Y Complement gives A050941.

%Y Cf. A000217 (1 consec), A001110 (2 consec), A129803 (3 consec), A131557 (5 consec),  A257711 (7 consec), A034705, A269414 (subsequence of primes).

%K nonn

%O 1,3

%A _Erich Friedman_