A133312 a(n) is the first pentagonal number that is nontrivially the sum of two pentagonal numbers of the type P(p) + P(p+n) (we always have P(k) = P(0) + P(k)).

70, 1926, 6305, 92, 22632, 34580, 49051, 66045, 85562, 1426, 925, 159251, 188860, 220992, 255647, 292825, 852, 2625, 7107, 466767, 516560, 568876, 623715, 681077, 5192, 803370, 7957, 935755, 1005732, 1078232, 22265, 8626, 1310870, 3577

Offset

1,1

Comments

The sequence is globally increasing with, at first sight, "unpredictible holes"; the regular values satisfy a(n) = (2523/2)*n^2 + (1189/2)*n + 70, which is approximately (35.5176013*n + 8.3690808)^2. In fact, we have to solve (6*k-1)^2 = 2*(6*p + 3*n - 1)^2 + 18*n^2 - 1 (Eq. 1) with given n that is X^2 = 2Y^2 + 18*n^2 - 1 where X = 6*k - 1 and Y = 6*p + 3*n - 1. p = 20*n + 5 and k = 29*n + 7 always give a solution of Eq. 1 but it is not certain that it is the best, i.e., the first.

Links

Table of n, a(n) for n=1..34.

Example

a(0) = 70 because the first interesting relation is P(70) = P(35) + P(35), i.e., 70 = 2*35.
a(1) = 1926 because the first nonobvious relation is P(36) = 1926 = P(25) + P(26).

Maple

for n from 1 to n ; a:=proc(k) if type (sqrt(18*k^2-6*k+1-9*n^2)/6-(3*n-1)/6, integer)=true then k*(3*k-1)/2 else fi end : seq (a(k), k=n..100000) od;

Crossrefs

Sequence in context: A076430 A006296 A047835 * A333967 A093757 A163022

Adjacent sequences: A133309 A133310 A133311 * A133313 A133314 A133315

Keyword

nonn

Author

Richard Choulet, Dec 20 2007

Status

approved