A034706 Numbers which are sums of consecutive triangular numbers.

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, 66, 74, 78, 80, 81, 83, 84, 85, 91, 100, 105, 109, 110, 116, 119, 120, 121, 130, 136, 144, 145, 153, 155, 161, 164, 165, 166, 169, 171, 185, 190, 196, 199, 200, 202, 210

Offset: 1

Author

Erich Friedman

Keywords

• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• D. Subramaniam, E. Trevino, and P. Pollack, On sums of consecutive triangular numbers, INTEGERS 20A (2020) A15.

Crossrefs

Complement gives A050941.

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

Programs

Haskell

-- import Data.Set (deleteFindMin, union, fromList); import Data.List (inits)
a034706 n = a034706_list !! (n-1)
a034706_list = f 0 (tail $ inits $ a000217_list) (fromList [0]) where
   f x vss'@(vs:vss) s
     | y < x = y : f x vss' s'
     | otherwise = f w vss (union s $ fromList $ scanl1 (+) ws)
     where ws@(w:_) = reverse vs
           (y, s') = deleteFindMin s
-- Reinhard Zumkeller, May 12 2015

Maple

isA034706 := proc(n)
    local a,b;
    for a from 0 do
        if a*(a+1)/2 > n then
            return false;
        end if;
        for b from a do
            tab := (1+b-a)*(a^2+b*a+a+b^2+2*b)/6 ;
            if tab = n then
                return true;
            elif tab > n then
                break;
            end if;
        end do:
    end do:
end proc:
for n from 0 to 100 do
    if isA034706(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 14 2015

Mathematica

M = 1000; (* to get all terms <= M *)
nmax = (Sqrt[8 M + 1] - 1)/2 // Ceiling;
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 *)