A285792 -id:A285792 - cp's OEIS frontend

A056105 First spoke of a hexagonal spiral.

1, 2, 9, 22, 41, 66, 97, 134, 177, 226, 281, 342, 409, 482, 561, 646, 737, 834, 937, 1046, 1161, 1282, 1409, 1542, 1681, 1826, 1977, 2134, 2297, 2466, 2641, 2822, 3009, 3202, 3401, 3606, 3817, 4034, 4257, 4486, 4721, 4962, 5209, 5462, 5721, 5986, 6257

Offset: 0

Henry Bottomley, Jun 09 2000

Also the number of (not necessarily maximal) cliques in the n X n grid graph. - Eric W. Weisstein, Nov 29 2017

			The spiral begins:
                   49--48--47--46--45
                   /                 \
                 50  28--27--26--25  44
                 /   /             \   \
               51  29  13--12--11  24  43
               /   /   /         \   \   \
             52  30  14   4---3  10  23  42  67
             /   /   /   /     \   \   \   \   \
           53  31  15   5   1===2===9==22==41==66==>
             \   \   \   \         /   /   /   /
             54  32  16   6---7---8  21  40  65
               \   \   \             /   /   /
               55  33   17--18--19--20  39  64
                 \   \                 /   /
                 56  34--35--36--37--38  63
                   \                     /
                   57--58--59--60--61--62

G. C. Greubel, Table of n, a(n) for n = 0..5000
Henry Bottomley, Illustration of initial terms
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Grid Graph
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A285792 (prime terms), A113519 (semiprime terms).
Other spokes: A003215, A056106, A056107, A056108, A056109.
Other spirals: A054552.
Cf. A000290, A000567, A008810, A033577.

List([0..50], n -> 3*n^2-2*n+1); # G. C. Greubel, Dec 02 2018

[3*n^2-2*n+1: n in [0..50]]; // Wesley Ivan Hurt, Jul 06 2014

A056105:=n->3*n^2 - 2*n + 1: seq(A056105(n), n=0..50); # Wesley Ivan Hurt, Jul 06 2014

LinearRecurrence[{3, -3, 1}, {1, 2, 9}, 50] (* Harvey P. Dale, Nov 02 2011 *)
Table[3 n^2 - 2 n + 1, {n, 0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
CoefficientList[Series[(-1 + x - 6 x^2)/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 29 2017 *)

a(n)=3*n^2-2*n+1 /* Michael Somos, Aug 03 2006 */

[3*n^2-2*n+1 for n in range(50)] # G. C. Greubel, Dec 02 2018

a(n) = 3*n^2 - 2*n + 1.
a(n) = a(n-1) + 6*n - 5.
a(n) = 2*a(n-1) - a(n-2) + 6.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A056106(n) - n = A056107(n) - 2*n.
a(n) = A056108(n) - 3*n = A056109(n) - 4*n = A003215(n) - 5*n.
A008810(3*n-1) = A056109(-n) = a(n). - Michael Somos, Aug 03 2006
G.f.: (1-x+6*x^2)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 04 2012
From Robert G. Wilson v, Jul 05 2014: (Start)
Each of the 6 primary spokes or rays has a generating formula as stated here:
1st:  90 degrees A056105 3n^2 - 2n + 1
2nd:  30 degrees A056106 3n^2 -  n + 1
3rd: 330 degrees A056107 3n^2      + 1
4th: 270 degrees A056108 3n^2 +  n + 1
5th: 210 degrees A056109 3n^2 + 2n + 1
6th: 150 degrees A003215 3n^2 + 3n + 1
Each of the 6 secondary spokes or rays has a generating formula as stated here:
1st:  60 degrees 12n^2 - 27n + 16
2nd: 360 degrees 12n^2 - 25n + 14
3rd: 300 degrees 12n^2 - 23n + 12
4th: 240 degrees 12n^2 - 21n + 10
5th: 180 degrees 12n^2 - 19n +  8
6th: 120 degrees 12n^2 - 17n +  6 = A033577(n+1)
(End)
a(n) = 1 + A000567(n). - Omar E. Pol, Apr 26 2017
a(n) = A000290(n-1) + 2*A000290(n), n >= 1. - J. M. Bergot, Mar 03 2018
E.g.f.: (1 + x + 3*x^2)*exp(x). - G. C. Greubel, Dec 02 2018

A285790 Primes equal to a hexagonal number plus 1.

Original entry on oeis.org

2, 7, 29, 67, 191, 277, 379, 631, 947, 1129, 1327, 2017, 2557, 2851, 4561, 4951, 5779, 6217, 8647, 9181, 12721, 13367, 14029, 15401, 16111, 17579, 20707, 21529, 22367, 24091, 24977, 31627, 36857, 37951, 42487, 43661, 44851, 47279, 53629, 58997, 64621, 66067

Offset: 1

Colin Barker, Apr 26 2017

nonn

Apart from the leading 2 the same as A176616. - R. J. Mathar, Apr 27 2017

Primes in A130883. - Omar E. Pol, Apr 27 2017

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

Cf. A000040, A000384, A002496, A055469, A285789, A285791, A176616, A285792.

Select[PolygonalNumber[6,Range[200]]+1,PrimeQ] (* Harvey P. Dale, Jun 16 2022 *)

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
maxk=300; L=List(); for(k=1, maxk, if(isprime(p=pg(6, k) + 1), listput(L, p))); Vec(L)

A285789 Primes equal to a pentagonal number plus 1.

Original entry on oeis.org

2, 13, 23, 71, 211, 331, 853, 1163, 1427, 2381, 2753, 3433, 3877, 5923, 6113, 6701, 7741, 8627, 9323, 11311, 12377, 14653, 14951, 17443, 18427, 23003, 27271, 31033, 32341, 32783, 34127, 38321, 43777, 52361, 55201, 56941, 57527, 62323, 64171, 64793, 69877

Offset: 1

Colin Barker, Apr 26 2017

nonn

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

Cf. A000040, A000326, A002496, A055469, A285790, A285791, A285792.

Select[PolygonalNumber[5,Range[300]]+1,PrimeQ] (* Harvey P. Dale, Dec 04 2024 *)

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
maxk=300; L=List(); for(k=1, maxk, if(isprime(p=pg(5, k) + 1), listput(L, p))); Vec(L)

A285791 Primes equal to a heptagonal number plus 1.

Original entry on oeis.org

2, 19, 113, 149, 541, 617, 971, 1289, 1783, 2357, 3011, 3187, 5689, 6427, 7481, 7757, 9829, 12497, 12853, 14327, 15881, 17099, 18793, 21023, 24851, 28463, 30637, 31193, 45361, 50909, 54539, 60607, 63761, 66179, 69473, 70309, 83449, 88079, 90917, 91873, 94771

Offset: 1

Colin Barker, Apr 26 2017

nonn

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

Cf. A000040, A000566, A002496, A055469, A285789, A285790, A285792.

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
maxk=300; L=List(); for(k=1, maxk, if(isprime(p=pg(7, k) + 1), listput(L, p))); Vec(L)

from sympy import isprime
def heptagonal(n): return n*(5*n-3)//2
def aupto(limit):
  alst, n, hn = [], 1, heptagonal(1)
  while hn < limit:
    if isprime(hn+1): alst.append(hn+1)
    n, hn = n+1, heptagonal(n+1)
  return alst
print(aupto(94771)) # Michael S. Branicky, Feb 19 2021

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A056105 First spoke of a hexagonal spiral.

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A285790 Primes equal to a hexagonal number plus 1.

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A285789 Primes equal to a pentagonal number plus 1.

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A285791 Primes equal to a heptagonal number plus 1.

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