A133151 -id:A133151 - cp's OEIS frontend

A184751 a(n) = A184750(n)/A133151(n) unless A133151(n) = 0 in which case a(n) = 0.

0, 0, 0, 0, 1, 1, 2, 1, 1, 3, 2, 1, 3, 4, 4, 3, 1, 2, 6, 1, 1, 9, 8, 1, 3, 2, 2, 3, 1, 4, 12, 1, 1, 6, 2, 1, 3, 16, 16, 9, 1, 2, 19, 1, 1, 12, 4, 19, 9, 2, 2, 3, 1, 8, 24, 1, 1, 18, 23, 1, 3, 19, 4, 3, 1, 23, 19, 1, 1, 32, 16, 1, 3, 2, 2, 27, 1, 4, 12, 1, 19, 23

Offset: 1

Rémi Eismann, Jan 21 2011

nonn

a(n) is the "level" of pentagonal numbers (A000326).
The decomposition of pentagonal numbers into weight * level + gap is A000326(n) = A133151(n) * a(n) + A016777(n) if a(n) > 0.
A184750(n) = A000326(n) - A016777(n) if A000326(n) - A016777(n) > A016777(n), 0 otherwise.

			For n = 3 we have A133151(3) = 0, hence a(3) = 0.
For n = 5 we have A184750(5)/A133151(5) = 19 / 19 = 1, hence a(5) = 1.
For n = 25 we have A184750(25)/A133151(25) = 849 / 283 = 5, hence a(25) = 3.

Rémi Eismann, Table of n, a(n) for n = 1..1000

Cf. A000326, A016777, A133151, A184750, A117078, A117563, A001223, A118534.

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

Original entry on oeis.org

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

Rémi Eismann, Jan 21 2011

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.

			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.

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

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

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

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

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

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

Edited - Wolfdieter Lang, Apr 19 2014

cp's OEIS Frontend

A184751 a(n) = A184750(n)/A133151(n) unless A133151(n) = 0 in which case a(n) = 0.

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

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