A298397 Pentagonal numbers divisible by 4.

0, 12, 92, 176, 376, 532, 852, 1080, 1520, 1820, 2380, 2752, 3432, 3876, 4676, 5192, 6112, 6700, 7740, 8400, 9560, 10292, 11572, 12376, 13776, 14652, 16172, 17120, 18760, 19780, 21540, 22632, 24512, 25676, 27676, 28912, 31032, 32340, 34580, 35960, 38320, 39772, 42252

Offset

1,2

Comments

If b(n) is the n-th octagonal number multiple of 32 then a(n) = b(n)/8.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

O.g.f.: 4*x^2*(3 + 20*x + 15*x^2 + 10*x^3)/((1 + x)^2*(1 - x)^3).

E.g.f.: (33 - 32*x + 24*x^2)*exp(x) + (7 + 6*x)*exp(-x) - 40.

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

a(n) = 8*n*(3*n - 7) - (6*n - 7)*(-1)^n + 33.

From Colin Barker, Jan 20 2018: (Start)

a(n) = 24*n^2 - 62*n + 40 for n even.

a(n) = 24*n^2 - 50*n + 26 for n odd. (End)

Example

A000326(8) = 92 is in the sequence because 92 = 4*23.

Maple

P:=proc(n) local x; x:=n*(3*n-1)/2; if x mod 4=0 then x; fi; end:
seq(P(i), i=0..2*10^2); # Paolo P. Lava, Jan 19 2018

Mathematica

Table[8 n (3 n - 7) - (6 n - 7) (-1)^n + 33, {n, 1, 50}]
(* Second program (using definition): *)
Select[Table[k*(3*k - 1)/2, {k, 0, 200}], Divisible[#, 4]&] (* Jean-François Alcover, Jan 19 2018 *)

Prog

(PARI) vector(50, n, nn; 8*n*(3*n-7)-(6*n-7)*(-1)^n+33)
(Sage) [8*n*(3*n-7)-(6*n-7)*(-1)^n+33 for n in (1..50)]
(Maxima) makelist(8*n*(3*n-7)-(6*n-7)*(-1)^n+33, n, 1, 50);
(Magma) [8*n*(3*n-7)-(6*n-7)*(-1)^n+33: n in [1..50]];
(GAP) List([1..50], n -> 8*n*(3*n-7)-(6*n-7)*(-1)^n+33);
(PARI) concat(0, Vec(4*x^2*(3 + 20*x + 15*x^2 + 10*x^3) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ Colin Barker, Jan 20 2018

Crossrefs

Cf. A000326, A000567.

Subsequence of A047217, A047388.

Cf. pentagonal numbers divisible by k: A014633 (k=2), A268351 (k=3), this sequence (k=4), A117793 (k=5).

Sequence in context: A160869 A026074 A339715 * A246585 A120990 A220330

Adjacent sequences: A298394 A298395 A298396 * A298398 A298399 A298400

Keyword

nonn,easy

Author

Bruno Berselli, Jan 18 2018

Status

approved