A136117 Pentagonal numbers (A000326) which are the sum of 2 other positive pentagonal numbers.

70, 92, 852, 925, 1247, 1426, 1926, 2625, 3577, 5192, 6305, 6501, 7107, 7740, 7957, 8177, 8626, 9560, 10292, 12927, 13207, 14652, 15555, 16172, 18095, 20475, 20827, 21901, 22265, 22632, 23002, 23751, 24130, 28497, 29330, 31032, 33227, 33675

Offset: 1

Author

M. F. Hasler, Dec 15 2007; corrected Dec 25 2007

Keywords

• nonn

Comments

It is conjectured that every integer and hence every pentagonal number, greater than 33066, hence greater than A000326(149) = 33227, can be represented as the sum of three pentagonal numbers. - Jonathan Vos Post, Dec 18 2007

Examples

			a(1)=70=P(7) is the least pentagonal number which can be written as sum of two other pentagonal numbers, P(7)=P(5)+P(5).
		

Crossrefs

Cf. A000326, A136112-A136118, A007527.

Programs

PARI

P(n)=n*(3*n-1)>>1 /* a.k.a. A000326 */
isPent(t)=P(sqrtint(t<<1\3)+1)==t
for(i=1,299,for(j=1,(i+1)\sqrt(2),isPent(P(i)-P(j)) && print1(P(i)",") || next(2)))
/* The following is much faster, at the cost of implementing sum2sqr(), cf. A133388*/
A136117next(i)=i=sqrtint(i\3*2)*6+5; until(0, for(j=2,#t=sum2sqr((i+=6)^2+1), t[j]%6==[5,5] && break(2)));i^2\24
A136117vect(n,i)=vector(n,j,i=A136117next(i)) /* 2nd arg =0 by default but allows one to start elsewhere */
A136117(n,i)=until(!n--,i=A136117next(i));i \\ M. F. Hasler, Dec 25 2007

Formula

a(n) = A000326(A136116(n)) = A000326(m)+A136114(m) where m is the index of the n-th nonzero term in A136114 or A136115.