cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A014633 Even pentagonal numbers.

Original entry on oeis.org

0, 12, 22, 70, 92, 176, 210, 330, 376, 532, 590, 782, 852, 1080, 1162, 1426, 1520, 1820, 1926, 2262, 2380, 2752, 2882, 3290, 3432, 3876, 4030, 4510, 4676, 5192, 5370, 5922, 6112, 6700, 6902, 7526, 7740, 8400, 8626, 9322
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Links

Crossrefs

Cf. A000326, A014632.

Programs

  • Magma
    [1/8*(1-3*(-1)^(n+1)+12*(n+1))*(1-(-1)^(n+1)+4*(n+1)): n in [0..40]]; // Vincenzo Librandi, Aug 17 2011
  • Mathematica
    LinearRecurrence[{1,2,-2,-1,1},{0,12,22,70,92},40] (* Harvey P. Dale, Aug 26 2014 *)
Select[PolygonalNumber[5,Range[0,100]],EvenQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 15 2017 *)
  • PARI
    lista(nn) = {forstep (n=0, nn, 2, if (ispolygonal(n, 5), print1(n, ", ")););} \\ Michel Marcus, Jun 20 2015

Formula

G.f.: 2*(6+5*x+12*x^2+x^3)/((1+x)^2*(1-x)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009, corrected by R. J. Mathar, Sep 16 2009
From Ant King, Aug 16 2011: (Start)
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).
a(n) = 1/8*(1-3*(-1)^(n+1)+12*(n+1))*(1-(-1)^(n+1)+4*(n+1)).(End)
Sum_{n>=1} 1/a(n) = 3*log(3)/2 - (1/sqrt(3)+1/4)*Pi - sqrt(3)*log(2-sqrt(3))/2. - Amiram Eldar, Jan 13 2024

Extensions

More terms from Patrick De Geest

A067589 Numbers k such that A067588(k) is an odd number.

Original entry on oeis.org

1, 5, 7, 15, 35, 51, 57, 77, 117, 145, 155, 187, 247, 287, 301, 345, 425, 477, 495, 551, 651, 715, 737, 805, 925, 1001, 1027, 1107, 1247, 1335, 1365, 1457, 1617, 1717, 1751, 1855, 2035, 2147, 2185, 2301, 2501, 2625, 2667, 2795, 3015, 3151, 3197, 3337
Offset: 1

Views

Author

Naohiro Nomoto, Jan 31 2002

Keywords

Comments

The terms are exactly the odd pentagonal numbers; that is, they are all the odd numbers of the form k*(3*k-1)/2 where k is an integer. - James Sellers, Jun 09 2007
Apparently groups of two odd pentagonal numbers (A000326, A014632) followed by two odd 2nd pentagonal numbers (A005449), which leads to the conjectured generating function x*(x^2+4*x+1)*(x^4-2*x^3+4*x^2-2*x+1)/((x^2+1)^2*(1-x)^3). - R. J. Mathar, Jul 26 2009
Odd generalized pentagonal numbers. - Omar E. Pol, Aug 19 2011
From Peter Bala, Jan 10 2025: (Start)
The sequence terms are the exponents in the expansion of Sum_{n >= 0} x^(2*n+1)/(Product_{k = 1..2*n+1} 1 + x^(2*k+1)) = x + x^5 - x^7 - x^15 + x^35 + x^51 - x^57 - x^77 + + - - ... (follows from Berndt et al., Theorem 3.3). Cf. A193828.
For positive integer m, define b_m(n) = Sum_{k = 1..n} k^(2*m+1)*A000009(k)*A000009(n-k). We conjecture that
i) for odd n, b(n)/ n is an integer
ii) b(2*n)/n is an integer, which is odd iff n is a member of this sequence.
Cf. A067567. (End)

Links

Crossrefs

Cf. A067567, A067588, A000009.
Cf. A001318, A193828.
Cf. A014493, A128880, A154293.

Programs

  • Mathematica
    With[{nn=50},Sort[Select[Table[(n(3n-1))/2,{n,-nn,nn}],OddQ]]] (* Harvey P. Dale, Feb 16 2014 *)

Formula

Sum_{n>=1} 1/a(n) = Pi/2. - Amiram Eldar, Aug 18 2022

Extensions

Corrected by T. D. Noe, Oct 25 2006

A014769 Squares of odd pentagonal numbers.

Original entry on oeis.org

1, 25, 1225, 2601, 13689, 21025, 61009, 82369, 180625, 227529, 423801, 511225, 855625, 1002001, 1555009, 1782225, 2614689, 2948089, 4141225, 4609609, 6255001, 6890625, 9090225, 9928801, 12794929, 13875625, 17530969, 18896409, 23474025, 25170289, 30813601
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Links

Crossrefs

Cf. A014632.
Odd terms of A100255.

Programs

  • Mathematica
    Select[Table[PolygonalNumber[5,n]^2, {n,0, 61}],OddQ] (* James C. McMahon, Dec 24 2023 *)
Select[PolygonalNumber[5,Range[70]],OddQ]^2 (* or *) LinearRecurrence[{1,4,-4,-6,6,4,-4,-1,1},{1,25,1225,2601,13689,21025,61009,82369,180625},40] (* Harvey P. Dale, Apr 07 2025 *)
  • PARI
    Vec((1 + 24*x + 1196*x^2 + 1280*x^3 + 6294*x^4 + 1976*x^5 + 2828*x^6 + 176*x^7 + 49*x^8) / ((1 - x)^5*(1 + x)^4) + O(x^40)) \\ Colin Barker, Nov 20 2018

Formula

a(n) = A014632(n)^2. - Sean A. Irvine, Nov 20 2018
From Colin Barker, Nov 20 2018: (Start)
G.f.: (1 + 24*x + 1196*x^2 + 1280*x^3 + 6294*x^4 + 1976*x^5 + 2828*x^6 + 176*x^7 + 49*x^8) / ((1 - x)^5*(1 + x)^4).
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.
a(n) = 36*n^4 + 60*n^3 + 37*n^2 + 10*n + 1 for n even.
a(n) = 36*n^4 - 12*n^3 + n^2 for n odd.
(End)

Extensions

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

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

Views

Author

Jonathan Vos Post, Feb 11 2006

Keywords

Comments

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, ...

Examples

			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.

Crossrefs

Cf. A000040, A000326, A014633.

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

Corrected and extended by Giovanni Resta, Jun 14 2016
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.