A257175 The smallest m such that the m-th triangular number is greater than or equal to half the n-th triangular number.

0, 1, 2, 2, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 16, 17, 17, 18, 19, 19, 20, 21, 22, 22, 23, 24, 24, 25, 26, 27, 27, 28, 29, 29, 30, 31, 31, 32, 33, 34, 34, 35, 36, 36, 37, 38, 39, 39, 40, 41, 41, 42, 43, 43, 44, 45, 46, 46, 47, 48, 48, 49, 50, 51, 51, 52, 53, 53, 54, 55, 56, 56, 57

Offset

0,3

Comments

The a(n)-th triangular number is the smallest triangular number that is greater than or equal to half of the n-th triangular number.

Links

Matthew Scroggs, Table of n, a(n) for n = 0..10000

Formula

a(n) = ceiling((sqrt(2n^2 + 2n + 1) - 1)/2). - Charles R Greathouse IV, Apr 17 2015

Example

For n=4, the 4th triangular number is 10. a(4)=3 as the 3rd triangular number is the first which is 5 or more.

Mathematica

f[n_] := Block[{t = Accumulate[Range@ n], k, m}, {1}~Join~Rest@ Flatten@ Reap@ For[k = 1, k < n, m = 1; While[t[[m]] < t[[k]]/2, m++]; Sow[m], k++]]; f@ 80 (* Michael De Vlieger, Apr 17 2015 *)

Prog

(Python)
def tri(n):
... return .5*n*(n+1)
.
for n in range(1, 10001):
... k = 1
... while 2*tri(k)<tri(n):
....... k+=1
... print k
(PARI) a(n) = my(t = n*(n+1)/4, k = 0); while(k*(k+1)/2 < t, k++); k; \\ Michel Marcus, Apr 17 2015
(Magma) [Ceiling((Sqrt(2*n^2 + 2*n + 1) - 1)/2): n in [1..80]]; // Vincenzo Librandi, Apr 18 2015
(MIT/GNU Scheme) (define (A257175 n) (ceiling->exact (/ (+ -1 (sqrt (+ (* 2 n n) n n 1))) 2))) ;; After Greathouse's formula - Antti Karttunen, Apr 18 2015

Crossrefs

Cf. A000217.

Sequence in context: A057561 A064726 A274616 * A210357 A057359 A076538

Adjacent sequences: A257172 A257173 A257174 * A257176 A257177 A257178

Keyword

nonn,easy

Author

Matthew Scroggs, Apr 17 2015

Status

approved