A056105 -id:A056105 - cp's OEIS frontend

A113653 Isolated semiprimes in the hexagonal spiral.

6, 51, 69, 82, 91, 183, 194, 221, 249, 265, 287, 289, 309, 314, 319, 323, 355, 371, 403, 417, 437, 469, 478, 511, 517, 519, 533, 579, 589, 649, 681, 689, 731, 749, 758, 807, 838, 849, 926, 943, 951, 961, 965, 979, 1011, 1018, 1037, 1055, 1057, 1067, 1077, 1099, 1126, 1145, 1149, 1154, 1159

Offset: 1

Jonathan Vos Post, Jan 16 2006

nonn

Isolated semiprimes in the hexagonal spiral of A003215 and A001399, which is centered on 0. Of course such a spiral can be constructed beginning with any integer. Centering on 0 gives the interesting partition and multigraph equalities of A001399.
Integers in A001358 which are not adjacent in any of six directions to any other integer in A001358 when arranged in the hexagonal spiral.
An analog of  A113688 "Isolated semiprimes in the [square] spiral," and of the hexagonal prime spiral of [Abbott 2005; Weisstein, "Prime Spiral", MathWorld].
Unfortunately the original submission (which has been preserved as the "dead" sequence A335704) omitted the number 44 from the spiral, which has caused an enormous amount of trouble. - N. J. A. Sloane, Jun 27 2020

			The spiral begins:
                120-119-118-117-116-115-114
                 /                         \
              121  85--84--83-*82*-81--80 113
               /   /                     \   \
            122  86  56--55--54--53--52  79 112
             /   /   /                 \   \   \
          123  87  57  33--32--31--30 *51* 78 111
           /   /   /   /             \   \   \   \
        124  88  58  34  16--15--14  29  50  77 110
         /   /   /   /   /         \   \   \   \   \
      125  89  59  35  17   5---4  13  28  49  76 109
       /   /   /   /   /   /     \   \   \   \   \   \
    126  90  60  36  18  *6*  0   3  12  27  48  75 108
     /   /   /   /   /   /   /   /   /   /   /   /   /
  127 *91* 61  37  19   7   1---2  11  26  47  74 107 146
     \   \   \   \   \   \         /   /   /   /   /   /
    128  92  62  38  20   8---9--10  25  46  73 106 145
       \   \   \   \   \             /   /   /   /   /
      129  93  63  39  21--22--23--24  45  72 105 144
         \   \   \   \                 /   /   /   /
        130  94  64  40--41--42--43--44  71 104 143
           \   \   \                     /   /   /
          131  95  65--66--67--68-*69*-70 103 142
             \   \                         /   /
            132  96--97--98--99-100-101-102 141
               \                             /
              133-134-135-136-137-138-139-140

Abbott, P. (Ed.). "Mathematica One-Liners: Spiral on an Integer Lattice." Mathematica J. 1, 39, 1990.

P. Abbott, Re: Hexagonal Spiral, (alt link), May 11, 2005
H. Bottomley, Spokes of a Hexagonal Spiral.
R. J. Mathar, Maple program for A113653
Eric Weisstein's World of Mathematics, Prime Spiral.

Cf. A001358, A001399, A113688.
For the sequence of isolated primes see A335916.
Related sequences:
A113519 Semiprimes in 1st spoke of a hexagonal spiral starting at 1 (A056105).
A113524 Semiprimes in 2nd spoke of a hexagonal spiral (A056106).
A113525 Semiprimes in 3rd spoke of a hexagonal spiral (A056107).
A113527 Semiprimes in 4th spoke of a hexagonal spiral (A056108).
A113528 Semiprimes in 5th spoke of a hexagonal spiral (A056109).
A113530 Semiprimes in 6th spoke of a hexagonal spiral (A003215).

Corrected and edited by N. J. A. Sloane, Jun 27 2020. Thanks to Jeffrey K. Aronson for pointing out the error in the original submission.

Terms a(4) onwards corrected by R. J. Mathar, Jun 29 2020

A062189 a(n) = 2 * 3^(n-2)n(1+2*n).

Original entry on oeis.org

0, 2, 20, 126, 648, 2970, 12636, 51030, 198288, 747954, 2755620, 9959598, 35429400, 124357194, 431530092, 1482720390, 5050815264, 17075199330, 57338232372, 191385721566, 635369601960, 2099044209402, 6903833113980

Offset: 0

Henry Bottomley, Jun 13 2001

Define a triangle with left (first) column T(n,0)=n^2 for n=0,1,2,3.. and the remaining terms T(r,c) = T(r-1,c-1) + 2*T(r,c-1). Then T(n,n) = a(n) on the diagonal. T(n,1) = A056105(n). - J. M. Bergot, Jan 26 2013

Harry J. Smith, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

List([0..30], n-> 2*3^(n-2)*n*(1+2*n)); # G. C. Greubel, Jun 06 2019

[2*3^(n-2)*n*(1+2*n): n in [0..30]]; // G. C. Greubel, Jun 06 2019

Table[2*3^(n-2)*n*(1+2*n), {n,0,30}] (* G. C. Greubel, Jun 06 2019 *)
LinearRecurrence[{9,-27,27},{0,2,20},30] (* Harvey P. Dale, Jun 08 2022 *)

{ for (n=0, 200, write("b062189.txt", n, " ", n*(4*n + 2)*3^(n - 2)) ) } \\ Harry J. Smith, Aug 02 2009

[2*3^(n-2)*n*(1+2*n) for n in (0..30)] # G. C. Greubel, Jun 06 2019

a(n) = A002943(n)*A000244(n-2). Binomial transform of A007758.
G.f.: 2*x*(1+x)/(1-3*x)^3. - Ralf Stephan, Mar 13 2003
a(n) = 2*A077616(n). - R. J. Mathar, Jan 29 2013
E.g.f.: 2*x*(1+2*x)*exp(3*x). - G. C. Greubel, Jun 06 2019

A131464 a(n) = 4n^3 - 3n^2 + 2*n - 1.

Original entry on oeis.org

2, 23, 86, 215, 434, 767, 1238, 1871, 2690, 3719, 4982, 6503, 8306, 10415, 12854, 15647, 18818, 22391, 26390, 30839, 35762, 41183, 47126, 53615, 60674, 68327, 76598, 85511, 95090, 105359, 116342, 128063, 140546, 153815, 167894, 182807, 198578, 215231, 232790

Offset: 1

Mohammad K. Azarian, Jul 26 2007

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

Cf. A084849, A056109, A056105, A056578.

[4*n^3-3*n^2+2*n-1: n in [1..40]]; // Vincenzo Librandi, Feb 12 2013

I:=[2, 23, 86, 215]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 12 2013

CoefficientList[Series[(2 + 15 x + 6 x^2 + x^3)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2013 *)
Table[4n^3-3n^2+2n-1,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{2,23,86,215},40] (* Harvey P. Dale, May 05 2018 *)

From Vincenzo Librandi, Feb 12 2013: (Start)
G.f.: x*(2 + 15*x + 6*x^2 + x^3)/(1 - x)^4.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). (End)
E.g.f.: 1 - exp(x)*(1 - 3*x - 9*x^2 - 4*x^3). - Stefano Spezia, Dec 06 2024

A363765 Lexicographically earliest sequence of numbers in a hexagonal spiral such that their neighbors have no common digit.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 44, 11, 20, 13, 22, 14, 23, 15, 24, 16, 25, 17, 26, 33, 27, 30, 28, 34, 55, 29, 40, 35, 60, 57, 46, 37, 18, 39, 47, 19, 36, 45, 70, 12, 48, 51, 49, 31, 50, 21, 38, 41, 53, 61, 42, 58, 32, 80, 52, 64, 59, 62, 81, 56, 43, 72

Offset: 1

Carole Dubois, Jun 20 2023

This sequence ends with 672 terms.

			a(26) = 27 because the neighbors of the 26th hexagon are 10, 30, 33, 44, 48, 51 which have no common digit with 27.
a(673) can't be calculated because its neighbors would be 216, 397, and 548, and they use all digits.

Carole Dubois, Table of n, a(n) for n = 1..672
Carole Dubois, Hexagonal spiral for A363765

Cf. A000384, A003458, A014105, A056105.

A130884 3n^3 + 2n^2 + n + 1.

Original entry on oeis.org

1, 7, 35, 103, 229, 431, 727, 1135, 1673, 2359, 3211, 4247, 5485, 6943, 8639, 10591, 12817, 15335, 18163, 21319, 24821, 28687, 32935, 37583, 42649, 48151, 54107, 60535, 67453, 74879, 82831, 91327, 100385, 110023, 120259, 131111

Offset: 0

Mohammad K. Azarian, Jul 26 2007

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

Cf. A084849, A056109, A056105.

[3*n^3+2*n^2+n+1: n in [0..35]]; // Vincenzo Librandi, Feb 12 2013

Table[3n^3+2n^2+n+1,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,7,35,103},40] (* Harvey P. Dale, Jan 17 2012 *)

G.f.: (1+13*x^2+x^3+3*x)/(-1+x)^4. - R. J. Mathar, Nov 14 2007

a(0)=1, a(1)=7, a(2)=35, a(3)=103, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)- a(n-4) [Harvey P. Dale, Jan 17 2012]

A130885 3n^3 - 2n^2 + n - 1.

Original entry on oeis.org

1, 17, 65, 163, 329, 581, 937, 1415, 2033, 2809, 3761, 4907, 6265, 7853, 9689, 11791, 14177, 16865, 19873, 23219, 26921, 30997, 35465, 40343, 45649, 51401, 57617, 64315, 71513, 79229, 87481, 96287, 105665, 115633, 126209, 137411

Offset: 1

Mohammad K. Azarian, Jul 26 2007

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

Cf. A084849; A056109; A056105.

[3*n^3-2*n^2+n-1: n in [1..40]]; // Vincenzo Librandi, Feb 12 2013

I:=[1, 17, 65, 163]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 12 2013

Table[3*n^3 - 2*n^2 + n - 1, {n, 1, 40}] (* Vincenzo Librandi, Feb 12 2013 *)
LinearRecurrence[{4,-6,4,-1},{1,17,65,163},40] (* Harvey P. Dale, Nov 21 2019 *)

G.f.: x*(1+13*x+3*x^2+x^3)/(-1+x)^4. - R. J. Mathar, Nov 14 2007

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Vincenzo Librandi, Feb 12 2013

A130886 4n^4 + 3n^3 + 2n^2 + n + 1.

Original entry on oeis.org

1, 11, 99, 427, 1253, 2931, 5911, 10739, 18057, 28603, 43211, 62811, 88429, 121187, 162303, 213091, 274961, 349419, 438067, 542603, 664821, 806611, 969959, 1156947, 1369753, 1610651, 1882011, 2186299, 2526077, 2904003

Offset: 0

Mohammad K. Azarian, Jul 26 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A084849, A056109, A056105.

[4*n^4+3*n^3+2*n^2+n+1: n in [0..40]]; // Vincenzo Librandi, Feb 12 2013

CoefficientList[Series[(1 + 6 x + 54 x^2 + 32 x^3 + 3 x^4)/(1 - x)^5, {x, 0, 35}], x] (* Vincenzo Librandi, Feb 12 2013 *)

G.f.: (1 + 6*x + 54*x^2 + 32*x^3 + 3*x^4)/(1 - x)^5. - Vincenzo Librandi, Feb 12 2013

A131466 a(n) = 5n^4 - 4n^3 + 3n^2 - 2n + 1.

Original entry on oeis.org

1, 3, 57, 319, 1065, 2691, 5713, 10767, 18609, 30115, 46281, 68223, 97177, 134499, 181665, 240271, 312033, 398787, 502489, 625215, 769161, 936643, 1130097, 1352079, 1605265, 1892451, 2216553, 2580607, 2987769, 3441315

Offset: 0

Mohammad K. Azarian, Jul 26 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A084849, A056109, A056105, A056578, A056579.

Table[5n^4-4n^3+3n^2-2n+1,{n,0,30}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,3,57,319,1065},30] (* Harvey P. Dale, Nov 01 2020 *)

a(n)=5*n^4-4*n^3+3*n^2-2*n+1 \\ Charles R Greathouse IV, Oct 21 2022

From Chai Wah Wu, Nov 13 2018: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
G.f.: (-15*x^4 - 54*x^3 - 52*x^2 + 2*x - 1)/(x - 1)^5. (End)

A275673 List of numbers that are in a spoke of a hexagonal spiral.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 22, 25, 28, 31, 34, 37, 41, 45, 49, 53, 57, 61, 66, 71, 76, 81, 86, 91, 97, 103, 109, 115, 121, 127, 134, 141, 148, 155, 162, 169, 177, 185, 193, 201, 209, 217, 226, 235, 244, 253, 262, 271, 281, 291, 301, 311, 321

Offset: 1

Peter Kagey, Aug 04 2016

nonn

This sequence contains k if and only if k is in one of the following sequences: A056105, A056106, A056107, A056108, A056109, A003215.

Alternatively, this sequence consists of the numbers of the form 3k^2 + bk + 1 for nonnegative k and -2 <= b <= 3.

Peter Kagey, Table of n, a(n) for n = 1..10003

Cf. A056105, A056106, A056107, A056108, A056109, A003215.

a275673 n = a275673_list !! (n - 1)
a275673_list = scanl (+) 1 $ concatMap (replicate 6) [1..]

Conjectures from Colin Barker, Aug 05 2016: (Start)
a(n) = 2*a(n-1)-a(n-2)+a(n-6)-2*a(n-7)+a(n-8) for n>8.
G.f.: x*(1-x^6+x^7) / ((1-x)^3*(1+x)*(1-x+x^2)*(1+x+x^2)).
(End)
Conjecture: a(n) = (n+4-3*floor((n+4)/6)-2)*floor((n+4)/6)+1. - Luce ETIENNE, May 25 2017

A335704 Erroneous version of A113653.

Original entry on oeis.org

6, 51, 55, 69, 82, 183, 194, 249, 259, 287, 309, 314, 319

Offset: 1

dead

This is the erroneous version of A113653 that was submitted to the OEIS by Jonathan Vos Post on Jan 16 2006. Because 44 was omitted from the spiral, not only are the terms here incorrect, but a large number of other sequences will also need to be corrected. For this reason the whole of the original submission has been preserved here with a different A-number. - N. J. A. Sloane, Jun 27 2020

Isolated semiprimes in the hexagonal spiral, embedded in the triangular lattice, are the analogy to A113688 "Isolated semiprimes in the [square] spiral," as well as analogous in another way to the hexagonal prime spiral of [Abbott 2005; Weisstein, "Prime Spiral", MathWorld]. A113519 Semiprimes in first spoke of a hexagonal spiral (A056105). A113524 Semiprimes in second spoke of a hexagonal spiral (A056106). A113525 Semiprimes in third spoke of a hexagonal spiral (A056107). A113527 Semiprimes in fourth spoke of a hexagonal spiral (A056108). A113528 Semiprimes in fifth spoke of a hexagonal spiral (A056109). A113530 Semiprimes in sixth spoke of a hexagonal spiral (A003215). This is embedded in the hexagonal spiral of A003215 and A001399, which is centered on zero; of course such a spiral can be constructed beginning with any integer. Centering on zero gives the interesting partition and multigraph equalities of A001399.

			Copy this as proportionally spaced text, make semiprimes bold, draw boundaries around clumps of adjacent semiprimes. For example, there is a triangular clump of three semiprimes: {4, 14, 15}; a linear clump of three semiprimes {49, 77, 111}; a linear clump of two semiprimes {247, 305}; an irregular clump of seven {115, 155, 201, 202, 203, 253, 254}; a clump of eighteen whose least element is 33 and greatest is 206; and a long branching clump of sixteen whose least element is 9 and greatest is 129.
.................209.208.207.206.205.204.203.202.201
................210.162.161.160.159.158.157.156.155.200
..............211.163.121.120.119.118.117.116.115.154.199
............212.164.122.86..85..84..83..82..81.114.153.198
..........213.165.123.87..57..56..55..54..53..80.113.152.197
........214.166.124.88..58..33..32..31..30..52..79.112.151.196
......215.167.125.89..59..34..16..15..14..29..51..78.111.150.195
....216.168.126.90..60..35..17..5...4...13..28..50..77.110.149.194
..217.169.127.91..61..36..18..6...0...3...12..27..49..76.109.148.193
218.170.128.92..62..37..19..7...1...2...11..26..48..75.108.147.192.243
..219.171.129.93..63..38..20..8...9...10..25..47..74.107.146.191.242
....220.172.130.94..64..39..21..22..23..24..46..73.106.145.190.241
......221.173.131.95..65..40..41..42..43..45..72.105.144.189.240
........222.174.132.96..66..67..68..69..70..71.104.143.188.239
..........223.175.133.97..98..99.100.101.102.103.142.187.238
............224.176.134.135.136.137.138.139.140.141.186.237
..............225.177.178.179.180.181.182.183.184.185.236
................226.227.228.229.230.231.232.233.234.235

Abbott, P. (Ed.). "Mathematica One-Liners: Spiral on an Integer Lattice." Mathematica J. 1, 39, 1990.

P. Abbott, Re: Hexagonal Spiral, (alt link), May 11, 2005
H. Bottomley, Spokes of a Hexagonal Spiral.
Eric Weisstein's World of Mathematics, Prime Spiral.

Cf. A001358, A001399, A003215, A056105-A056109, A113688, A113519, A113524, A113525, A113528, A113527, A113530, A113688.

{a(n)} = {integers in A001358 which are not adjacent in any of six directions to any other integers in A001358 when arranged as the hexagonal spiral of A003215}.

cp's OEIS Frontend

A113653 Isolated semiprimes in the hexagonal spiral.

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A062189 a(n) = 2 * 3^(n-2)*n*(1+2*n).

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A131464 a(n) = 4*n^3 - 3*n^2 + 2*n - 1.

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A363765 Lexicographically earliest sequence of numbers in a hexagonal spiral such that their neighbors have no common digit.

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A130884 3n^3 + 2n^2 + n + 1.

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Magma

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A130885 3n^3 - 2n^2 + n - 1.

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Magma

Magma

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A130886 4n^4 + 3n^3 + 2n^2 + n + 1.

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Magma

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A131466 a(n) = 5n^4 - 4n^3 + 3n^2 - 2n + 1.

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A062189 a(n) = 2 * 3^(n-2)n(1+2*n).

A131464 a(n) = 4n^3 - 3n^2 + 2*n - 1.