A184750 a(n) = largest k such that A000326(n+1) = A000326(n) + (A000326(n) mod k), or 0 if no such k exists.

0, 0, 0, 0, 19, 32, 48, 67, 89, 114, 142, 173, 207, 244, 284, 327, 373, 422, 474, 529, 587, 648, 712, 779, 849, 922, 998, 1077, 1159, 1244, 1332, 1423, 1517, 1614, 1714, 1817, 1923, 2032, 2144, 2259, 2377, 2498, 2622, 2749

Offset: 1

Author

Rémi Eismann, Jan 21 2011

Keywords

• nonn
• easy

Comments

From the definition, a(n) = A000326(n) - A016777(n) if A000326(n) - A016777(n) > A016777(n), 0 otherwise, where A000326 are the pentagonal numbers and A016777 are the gaps between pentagonal numbers: 3n + 1.

Examples

			For n = 3 we have A000326(3) = 12, A000326(4) = 22; there is no k such that 22 - 12 = 10 = (12 mod k), hence a(3) = 0.
For n = 5 we have A000326(5) = 35, A000326(6) = 51; 19 is the largest k such that 51 - 35 = 16 = (35 mod k), hence a(5) = 19; a(5) = (75-35-2)/2 = 19.
For n = 25 we have A000326(25) = 925, A000326(26) = 1001; 849 is the largest k such that 1001 - 925 = 76 = (925 mod k), hence a(25) = 849; a(25) = (1875-175-2)/2 = 849.
		

Links

• Rémi Eismann, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000326, A016777, A133151, A184751, A117078, A117563, A001223, A118534.

Programs

Maple

A184750:=n->(3*n^2 - 7*n - 2)*signum(floor(n/5))/2; seq(A184750(n), n=1..50); # Wesley Ivan Hurt, Apr 05 2014

Mathematica

Table[(3 n^2 - 7 n - 2) Sign[Floor[n/5]]/2, {n, 50}] (* Wesley Ivan Hurt, Apr 05 2014 *)

PARI

concat([0,0,0,0], Vec(-x^5*(9*x^2-25*x+19)/(x-1)^3 + O(x^100))) \\ Colin Barker, Apr 05 2014

Formula

a(n) = (3n^2-7n-2)/2 for n >= 5 and a(n) = 0 for n <= 4.

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>7. G.f.: x^5*(9*x^2-25*x+19) / (1-x)^3. - Colin Barker, Apr 05 2014

a(n) = A000326(n) - A016777(n), n>=5, (see a comment above). - Wolfdieter Lang, Apr 19 2014

Extensions

Edited - Wolfdieter Lang, Apr 19 2014