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A267140 Least m>0 for which m + n^2 is a square and m + triangular(n) is a triangular number (A000217). 1
1, 35, 12, 72, 180, 336, 45, 792, 1092, 208, 1836, 2280, 112, 315, 3900, 4536, 644, 5952, 6732, 7560, 225, 715, 10332, 627, 12420, 13536, 924, 1575, 17172, 840, 396, 21240, 22692, 3267, 2565, 27336, 28980, 3392, 32412, 34200, 1881, 3795, 637, 1400, 1785, 45936, 2240 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

For n>1, a(n) <= b(n), where b(n) = 24*n^2 - 60*n + 36 = A143698(n-1), because b(n) + n^2 = (5*n-6)^2, and b(n) + n*(n+1)/2 = (7*n-9)*(7*n-8)/2 = triangular(7*n-9).

LINKS

Chai Wah Wu, Table of n, a(n) for n = 0..10000

EXAMPLE

12 + 2^2 = 16 is a square, and 12 + 2*3/2 = 15 is a triangular number, and 12 is the least such integer, so a(2)=12.

PROG

(PARI) a(n) = {m = 1; while (! (issquare(m+n^2) && ispolygonal(m+n*(n+1)/2, 3)), m++); m; } \\ Michel Marcus, Jan 11 2016

(Python)

from math import sqrt

def A267140(n):

    u, r, k, m = 2*n+1, 4*n*(n+1)+1, 0, 2*n+1

    while True:

        if int(sqrt(8*m+r))**2 == 8*m+r:

            return m

        k += 2

        m += u + k # Chai Wah Wu, Jan 13 2016

CROSSREFS

Cf. A000217, A000290, A143698, A267077.

Sequence in context: A248477 A051050 A267198 * A088830 A033355 A267402

Adjacent sequences:  A267137 A267138 A267139 * A267141 A267142 A267143

KEYWORD

nonn

AUTHOR

Alex Ratushnyak, Jan 10 2016

STATUS

approved

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Last modified October 23 22:04 EDT 2018. Contains 316541 sequences. (Running on oeis4.)