A047388 -id:A047388 - cp's OEIS frontend

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

Bruno Berselli, Jan 18 2018

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

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

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).

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).

List([1..50], n -> 8*n*(3*n-7)-(6*n-7)*(-1)^n+33);

[8*n*(3*n-7)-(6*n-7)*(-1)^n+33: n in [1..50]];

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

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 *)

makelist(8*n*(3*n-7)-(6*n-7)*(-1)^n+33, n, 1, 50);

vector(50, n, nn; 8*n*(3*n-7)-(6*n-7)*(-1)^n+33)

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

[8*n*(3*n-7)-(6*n-7)*(-1)^n+33 for n in (1..50)]

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)

A299646 a(n) = Sum_{k = n..2*n+1} k^2.

Original entry on oeis.org

1, 14, 54, 135, 271, 476, 764, 1149, 1645, 2266, 3026, 3939, 5019, 6280, 7736, 9401, 11289, 13414, 15790, 18431, 21351, 24564, 28084, 31925, 36101, 40626, 45514, 50779, 56435, 62496, 68976, 75889, 83249, 91070, 99366, 108151, 117439, 127244, 137580, 148461, 159901

Offset: 0

Bruno Berselli, Feb 20 2018

Inverse binomial transform is 1, 13, 27, 14, 0, 0, 0, ... (0 continued).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Subsequence of A008854, A047388, A174070 (after 1).

Cf. A050409: Sum_{k = n..2*n} k^2; A050410: Sum_{k = n..2*n-1} k^2.

List([0..50], n -> (n+2)*(14*n^2+11*n+3)/6);

[(n+2)*(14*n^2+11*n+3)/6: n in [0..50]];

seq((n + 2)*(14*n^2 + 11*n + 3)/6, n=0..50); # Peter Luschny, Feb 21 2018

Table[(n + 2) (14 n^2 + 11 n + 3)/6, {n, 0, 50}]
(* Second program: *)
LinearRecurrence[{4, -6, 4, -1}, {1, 14, 54, 135}, 41] (* Jean-François Alcover, Feb 21 2018 *)

makelist((n+2)*(14*n^2+11*n+3)/6, n, 0, 50);

a(n)=(n+2)*(14*n^2+11*n+3)/6 \\ Charles R Greathouse IV, Feb 21 2018

Vec((1 + 10*x + 4*x^2 - x^3)/(1 - x)^4 + O(x^60)) \\ Colin Barker, Feb 22 2018

[(n+2)*(14*n^2+11*n+3)/6 for n in (0..50)]

O.g.f.: (1 + 10*x + 4*x^2 - x^3)/(1 - x)^4.
E.g.f.: (6 + 78*x + 81*x^2 + 14*x^3)*exp(x)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = (n + 2)*(14*n^2 + 11*n + 3)/6. Therefore:
a(6*k + r) = 504*k^3 + 18*(14*r + 13)*k^2 + (42*r^2 + 78*r + 25)*k + a(r), with 0 <= r <= 5. Example: for r=5, a(6*k + 5) = (6*k + 7)*(84*k^2 + 151*k + 68).

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A298397 Pentagonal numbers divisible by 4.

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A299646 a(n) = Sum_{k = n..2*n+1} k^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

Maxima

PARI

PARI

Sage

Formula