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A034706
Numbers which are sums of consecutive triangular numbers.
14
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
1,3
LINKS
D. Subramaniam, E. Trevino, and P. Pollack, On sums of consecutive triangular numbers, INTEGERS 20A (2020) A15.
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 *)
PROG
(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
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).
Sequence in context: A166161 A368136 A130904 * A245810 A054686 A005214
KEYWORD
nonn
STATUS
approved