A014633 -id:A014633 - cp's OEIS frontend

A014632 Odd pentagonal numbers.

1, 5, 35, 51, 117, 145, 247, 287, 425, 477, 651, 715, 925, 1001, 1247, 1335, 1617, 1717, 2035, 2147, 2501, 2625, 3015, 3151, 3577, 3725, 4187, 4347, 4845, 5017, 5551, 5735, 6305, 6501, 7107, 7315, 7957, 8177, 8855, 9087, 9801, 10045, 10795, 11051, 11837

Offset: 0

Mohammad K. Azarian

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1, 2, -2, -1, 1).

Cf. A000326, A014633.

[1/8*(11+3*(-1)^(n+1)-12*(n+1))*(3+(-1)^(n+1)-4*(n+1)): n in [0..40]]; // Vincenzo Librandi, Aug 17 2011

A014632:=n->(3*(-1)^n+12*n+1)*((-1)^n+4*n+1)/8: seq(A014632(n), n=0..100); # Wesley Ivan Hurt, Apr 28 2017

Select[Table[n (3 n - 1)/2, {n,100}], OddQ]  (* Harvey P. Dale, Feb 03 2011 *)

a(n) = a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).
a(n) = 48+2*a(n-2)-a(n-4). - Ant King, Aug 16 2011
G.f.: (1+4*x+28*x^2+8*x^3+7*x^4)/((1+x)^2*(1-x)^3). - R. J. Mathar, Jul 25 2009
a(n) = (3*(-1)^n+12*n+1)*((-1)^n+4*n+1)/8. - Ant King, Aug 16 2011
Sum_{n>=0} 1/a(n) = Pi/4 + 3*log(3)/2 + sqrt(3)*log(2-sqrt(3))/2. - Amiram Eldar, Jan 13 2024

More terms from Patrick De Geest

A193866 Even pentagonal numbers divided by 2.

Original entry on oeis.org

0, 6, 11, 35, 46, 88, 105, 165, 188, 266, 295, 391, 426, 540, 581, 713, 760, 910, 963, 1131, 1190, 1376, 1441, 1645, 1716, 1938, 2015, 2255, 2338, 2596, 2685, 2961, 3056, 3350, 3451, 3763, 3870, 4200, 4313, 4661, 4780, 5146, 5271, 5655, 5786, 6188

Offset: 0

Omar E. Pol, Aug 18 2011

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000326, A001105, A033991.

[1/16*(1-3*(-1)^n+12*n)*(1-(-1)^n+4*n): n in [0..60]]; // Vincenzo Librandi, Jun 20 2015

Table[(1/16 (1 - 3 (-1)^n + 12 n) (1 - (-1)^n + 4 n)), {n, 0, 50}] (* Vincenzo Librandi, Jun 20 2015 *)
Select[PolygonalNumber[5,Range[0,100]],EvenQ]/2 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 13 2018 *)

a(n)=3*n^2+if(n%2,5*n+1,-n)/2 \\ Charles R Greathouse IV, Aug 18 2011

a(n) = 1/16*(1-3*(-1)^n+12*n)*(1-(-1)^n+4*n).
a(n) = A014633(n)/2.
a(0)=0, a(1)=6, a(2)=11, a(3)=35, a(4)=46, a(n)=a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5). G.f.: x*(6+5*x+12*x^2+x^3)/(1-x-2*x^2+2*x^3+x^4-x^5). [Colin Barker, Jan 25 2012]

A298397 Pentagonal numbers divisible by 4.

Original entry on oeis.org

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)

A014770 Squares of even pentagonal numbers.

Original entry on oeis.org

0, 144, 484, 4900, 8464, 30976, 44100, 108900, 141376, 283024, 348100, 611524, 725904, 1166400, 1350244, 2033476, 2310400, 3312400, 3709476, 5116644, 5664400, 7573504, 8305924, 10824100, 11778624, 15023376, 16240900, 20340100

Offset: 0

Mohammad K. Azarian

Cf. A014633.

Even terms of A100255.

a(n) = A014633(n)^2. - Sean A. Irvine, Nov 20 2018
From Chai Wah Wu, Jan 23 2020: (Start)
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n > 8.
G.f.: -4*x*(x^7 + 168*x^6 + 227*x^5 + 1428*x^4 + 551*x^3 + 960*x^2 + 85*x + 36)/((x - 1)^5*(x + 1)^4). (End)

More terms from Erich Friedman.

a(0)=0 inserted for consistency with A014633 by Sean A. Irvine, Nov 20 2018

A114409 Length of all-prime chain of prime[n] + successive even pentagonal numbers.

Original entry on oeis.org

1, 2, 3, 2, 1, 2, 4, 1, 2, 4, 1, 2, 1, 2, 1, 2, 4, 4, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 4, 1, 2, 2, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 4, 1, 2, 3, 1, 2, 1, 2, 2, 1, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 2, 4, 1, 4, 2, 4, 2, 1, 1, 1, 2, 1

Offset: 2

Jonathan Vos Post, Feb 11 2006

a(1) is undefined, as prime(1) is the only even prime, for which the length-5 chain is of 2 + successive odd pentagonal numbers A014632: 2 prime, 2+1 = 3 prime, 2+5 = 7 prime, 2+35 = 37 prime, 2+51 = 53 prime, but 2+117 = 119 = 7 * 17 nonprime. Pentagonal numbers A000326 = n*(3*n-1)/2 = 0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ... Even pentagonal numbers A014633 = 12, 22, 70, 92, 176, 210, 330, 376, 532, 590, 782, 852, 1080, ...

			a(2) = 1 because prime(2) = 3 is prime, but prime(2) + EvenPent(1) = 3 + 12 = 15 = 3 * 5 is nonprime, giving a chain of just 1 successive prime.
a(3) = 2 because 5 is prime, prime(3) + EvenPent(1) = 5 + 12 = 17 is prime, but prime(3) + EvenPent(2) = 5 + 22 = 27 = 3^3 is nonprime, giving a chain of 2 successive primes.
a(4) = 3 because 7 is prime, 7+12 = 19 is prime, 7+22 = 29 is prime, but 7+70 = 77 = 7*11 is nonprime, for a chain of 3 successive primes.

Cf. A000040, A000326, A014633.

evp = Select[#*(3*# - 1)/2 &@ Range[200], EvenQ]; a[n_] := Block[{s = Prime@n, c = 1}, While[PrimeQ[s + evp[[c]]], c++]; c]; a /@ Range[2, 90] (* Giovanni Resta, Jun 14 2016 *)

a(n) = k = length of chain prime[n] + A014633(1) + ... + A014633(k) such that each term in the chain is prime.

Corrected and extended by Giovanni Resta, Jun 14 2016

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A014632 Odd pentagonal numbers.

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A193866 Even pentagonal numbers divided by 2.

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A298397 Pentagonal numbers divisible by 4.

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A014770 Squares of even pentagonal numbers.

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A114409 Length of all-prime chain of prime[n] + successive even pentagonal numbers.

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