A003215 -id:A003215 - cp's OEIS frontend

A254136 Indices of pentagonal numbers (A000326) which are also centered hexagonal numbers (A003215).

1, 73, 889, 84049, 1025713, 96992281, 1183671721, 111929008033, 1365956140129, 129165978277609, 1576312202036953, 149057427003352561, 1819062915194503441, 172012141595890577593, 2099197027822254933769, 198501862344230723189569, 2422471551043966999065793

Offset: 1

Colin Barker, Jan 26 2015

Also positive integers x in the solutions to 3*x^2 - 6*y^2 - x + 6*y - 2 = 0, the corresponding values of y being A254137.

			73 is in the sequence because the 73rd pentagonal number is 7957, which is also the 52nd centered hexagonal number.

Colin Barker, Table of n, a(n) for n = 1..653
Index entries for linear recurrences with constant coefficients, signature (1,1154,-1154,-1,1).

Cf. A000326, A003215, A254137, A254138.

LinearRecurrence[{1,1154,-1154,-1,1},{1,73,889,84049,1025713},20] (* Harvey P. Dale, Mar 24 2024 *)

Vec(-x*(x^4+72*x^3-338*x^2+72*x+1)/((x-1)*(x^2-34*x+1)*(x^2+34*x+1)) + O(x^100))

a(n) = a(n-1)+1154*a(n-2)-1154*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+72*x^3-338*x^2+72*x+1) / ((x-1)*(x^2-34*x+1)*(x^2+34*x+1)).

A254137 Indices of centered hexagonal numbers (A003215) which are also pentagonal numbers (A000326).

Original entry on oeis.org

1, 52, 629, 59432, 725289, 68583900, 836982301, 79145760592, 965876849489, 91334139138692, 1114621047327429, 105399517420289400, 1286271722739003001, 121630951768874828332, 1484356453419762135149, 140362012941764131605152, 1712946060974682764958369

Offset: 1

Colin Barker, Jan 26 2015

Also positive integers y in the solutions to 3*x^2 - 6*y^2 - x + 6*y - 2 = 0, the corresponding values of x being A254136.

			52 is in the sequence because the 52nd centered hexagonal number is 7957, which is also the 73rd pentagonal number.

Colin Barker, Table of n, a(n) for n = 1..653
Index entries for linear recurrences with constant coefficients, signature (1,1154,-1154,-1,1).

Cf. A000326, A003215, A254136, A254138.

Vec(x*(51*x^3+577*x^2-51*x-1)/((x-1)*(x^2-34*x+1)*(x^2+34*x+1)) + O(x^100))

a(n) = a(n-1)+1154*a(n-2)-1154*a(n-3)-a(n-4)+a(n-5).

G.f.: x*(51*x^3+577*x^2-51*x-1) / ((x-1)*(x^2-34*x+1)*(x^2+34*x+1)).

A254138 Pentagonal numbers (A000326) which are also centered hexagonal numbers (A003215).

Original entry on oeis.org

1, 7957, 1185037, 10596309577, 1578130224697, 14111253811878301, 2101618114050816901, 18792154258821103289617, 2798754265133491448134897, 25025774916617575492416996517, 3727140237435880812247465267837, 33327174817289665775049786996211801

Offset: 1

Colin Barker, Jan 26 2015

			7957 is in the sequence because it is the 73rd pentagonal number and the 52nd centered hexagonal number.

Colin Barker, Table of n, a(n) for n = 1..327
Index entries for linear recurrences with constant coefficients, signature (1,1331714,-1331714,-1,1).

Cf. A000326, A003215, A254136, A254137.

LinearRecurrence[{1,1331714,-1331714,-1,1},{1,7957,1185037,10596309577,1578130224697},20] (* Harvey P. Dale, Sep 26 2023 *)

Vec(-x*(x^4+7956*x^3-154634*x^2+7956*x+1)/((x-1)*(x^2-1154*x+1)*(x^2+1154*x+1)) + O(x^100))

a(n) = a(n-1)+1331714*a(n-2)-1331714*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+7956*x^3-154634*x^2+7956*x+1) / ((x-1)*(x^2-1154*x+1)*(x^2+1154*x+1)).

A254782 Indices of centered hexagonal numbers (A003215) which are also centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 11, 231, 5061, 111101, 2439151, 53550211, 1175665481, 25811090361, 566668322451, 12440892003551, 273132955755661, 5996484134620981, 131649518005905911, 2890292911995309051, 63454794545890893201, 1393115187097604341361, 30585079321601404616731

Offset: 1

Colin Barker, Feb 07 2015

Also positive integers y in the solutions to 5*x^2 - 6*y^2 - 5*x + 6*y = 0, the corresponding values of x being A133285.

The numbers (as opposed to the indices) are A133141.

			11 is in the sequence because the 11th centered hexagonal number is 331, which is also the 12th centered pentagonal number.

Colin Barker, Table of n, a(n) for n = 1..746
Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.
Index entries for linear recurrences with constant coefficients, signature (23,-23,1).

Cf. A003215, A005891, A133141, A133285.

LinearRecurrence[{23,-23,1},{1,11,231},20] (* Harvey P. Dale, Mar 01 2022 *)

Vec(-x*(x^2-12*x+1)/((x-1)*(x^2-22*x+1)) + O(x^100))

a(n) = 23*a(n-1)-23*a(n-2)+a(n-3).
G.f.: -x*(x^2-12*x+1) / ((x-1)*(x^2-22*x+1)).
a(n) = 1/2+1/24*(11+2*sqrt(30))^(-n)*(6+sqrt(30)-(-6+sqrt(30))*(11+2*sqrt(30))^(2*n)). - Colin Barker, Mar 03 2016

A254964 Indices of heptagonal numbers (A000566) that are also centered hexagonal numbers (A003215).

Original entry on oeis.org

1, 2, 14, 37, 301, 806, 6602, 17689, 144937, 388346, 3182006, 8525917, 69859189, 187181822, 1533720146, 4109474161, 33671984017, 90221249714, 739249928222, 1980758019541, 16229826436861, 43486455180182, 356316931682714, 954721255944457, 7822742670582841

Offset: 1

Colin Barker, Feb 11 2015

Also positive integers x in the solutions to 5*x^2 - 6*y^2 - 3*x + 6*y - 2 = 0, the corresponding values of y being A254965.

			14 is in the sequence because the 14th heptagonal number is 469, which is also the 13th centered hexagonal number.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,22,-22,-1,1).

Cf. A000566, A003215, A254965, A254966.

LinearRecurrence[{1,22,-22,-1,1},{1,2,14,37,301},30] (* Harvey P. Dale, Apr 13 2018 *)

Vec(-x*(x^2-3*x+1)*(x^2+4*x+1)/((x-1)*(x^4-22*x^2+1)) + O(x^100))

a(n) = a(n-1)+22*a(n-2)-22*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^2-3*x+1)*(x^2+4*x+1) / ((x-1)*(x^4-22*x^2+1)).

A254965 Indices of centered hexagonal numbers (A003215) that are also heptagonal numbers (A000566).

Original entry on oeis.org

1, 2, 13, 34, 275, 736, 6027, 16148, 132309, 354510, 2904761, 7783062, 63772423, 170872844, 1400088535, 3751419496, 30738175337, 82360356058, 674839768869, 1808176413770, 14815736739771, 39697520746872, 325271368506083, 871537280017404, 7141154370394045

Offset: 1

Colin Barker, Feb 11 2015

Also positive integers y in the solutions to 5*x^2 - 6*y^2 - 3*x + 6*y - 2 = 0, the corresponding values of x being A254964.

			13 is in the sequence because the 13th centered hexagonal number is 469, which is also the 14th heptagonal number.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,22,-22,-1,1).

Cf. A000566, A003215, A254964, A254966.

Vec(x*(x^3+11*x^2-x-1)/((x-1)*(x^4-22*x^2+1)) + O(x^100))

a(n) = a(n-1)+22*a(n-2)-22*a(n-3)-a(n-4)+a(n-5).

G.f.: x*(x^3+11*x^2-x-1) / ((x-1)*(x^4-22*x^2+1)).

A254966 Heptagonal numbers (A000566) that are also centered hexagonal numbers (A003215).

Original entry on oeis.org

1, 7, 469, 3367, 226051, 1622881, 108956107, 782225269, 52516617517, 377030956771, 25312900687081, 181728138938347, 12200765614555519, 87592585937326477, 5880743713315073071, 42219444693652423561, 2834506269052250664697, 20349684749754530829919

Offset: 1

Colin Barker, Feb 11 2015

			469 is in the sequence because it is the 14th heptagonal number and the 13th centered hexagonal number.

Colin Barker, Table of n, a(n) for n = 1..746
Index entries for linear recurrences with constant coefficients, signature (1,482,-482,-1,1).

Cf. A000566, A003215, A254964, A254965.

LinearRecurrence[{1,482,-482,-1,1},{1,7,469,3367,226051},20] (* Harvey P. Dale, May 17 2019 *)

Vec(-x*(x^4+6*x^3-20*x^2+6*x+1)/((x-1)*(x^2-22*x+1)*(x^2+22*x+1)) + O(x^100))

a(n) = a(n-1)+482*a(n-2)-482*a(n-3)-a(n-4)+a(n-5).

G.f.: -x*(x^4+6*x^3-20*x^2+6*x+1) / ((x-1)*(x^2-22*x+1)*(x^2+22*x+1)).

A322802 Number of compositions (ordered partitions) of n into centered hexagonal numbers (A003215).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 17, 22, 28, 36, 45, 56, 70, 88, 111, 140, 178, 226, 286, 361, 455, 573, 721, 909, 1148, 1451, 1834, 2318, 2928, 3695, 4661, 5880, 7420, 9366, 11826, 14935, 18860, 23812, 30059, 37941, 47888, 60445, 76302, 96327

Offset: 0

Ilya Gutkovskiy, Dec 26 2018

nonn

Eric Weisstein's World of Mathematics, Hex Number
Index entries for sequences related to compositions

Cf. A003215, A280863, A280953, A281084, A282502, A282504, A322798, A322801, A322803.

h:= proc(n) option remember; `if`(n<0, 0, (t->
      `if`(3*t*(t+1)+1>n, t-1, t))(1+h(n-1)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-(3*i*(i+1)+1)), i=0..h(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 28 2018

nmax = 53; CoefficientList[Series[1/(1 - Sum[x^(3 k (k + 1) + 1), {k, 0, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=0} x^(3*k*(k+1)+1)).

A107118 Numbers that are both centered triangular numbers (A005448) and centered hexagonal numbers (A003215).

Original entry on oeis.org

1, 19, 631, 21421, 727669, 24719311, 839728891, 28526062969, 969046412041, 32919051946411, 1118278719765919, 37988557420094821, 1290492673563457981, 43838762343737476519, 1489227427013510743651, 50589893756115627807601, 1718567160280917834714769

Offset: 1

Richard Choulet, Sep 18 2007

The centered hexagonal numbers are given by 3*p^2 - 3*p + 1 while the centered triangular numbers are given by (3*r^2 + 3*r + 2)/2. A natural number is both of the above numbers if and only if there exist numbers p and r such that 2*(2p-1)^2 = (2*r+1)^2+1. The Diophantine equation X^2 = 2*Y^2 - 1 has the following solutions: X is given by 1, 7, 41, 239, ..., i.e., A002315, and Y is given by A001653. The first equation gives r with 0, 3, 20, 119, 6906, i.e., A001652, and p with 1, 3, 15, 85, 493, ..., i.e., A011900.

Colin Barker, Table of n, a(n) for n = 1..654
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Cf. A003215 (Centered hexagonal numbers), A005448 (Centered triangular numbers).

Cf. A001652, A001653, A002315, A011900.

a[n_] := 17*n - 7 + Sqrt[288*n^2 - 252*n + 45]; NestList[a, 1, 20] (* Stefan Steinerberger, Sep 18 2007 *)
LinearRecurrence[{35,-35,1},{1,19,631},30] (* Harvey P. Dale, Jan 16 2016 *)

Vec(-x*(x^2-16*x+1)/((x-1)*(x^2-34*x+1)) + O(x^100)) \\ Colin Barker, Jan 02 2015

a(n+2) = 34*a(n+1) - a(n) - 14.
a(n+1) = 17*a(n) - 7 + sqrt(288*a(n)^2 - 252*a(n) + 45).
G.f.: h(z)=(z*(1-16*z+z^2))/((1-z)*(1-34*z+z^2)).
a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3). - Colin Barker, Jan 02 2015
a(n) = (14+(9+6*sqrt(2))*(17+12*sqrt(2))^(-n)+(9-6*sqrt(2))*(17+12*sqrt(2))^n)/32. - Colin Barker, Mar 02 2016

More terms from Stefan Steinerberger, Sep 18 2007

A298125 The hex numbers (A003215) together with 3.

Original entry on oeis.org

1, 3, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, 1027, 1141, 1261, 1387, 1519, 1657, 1801, 1951, 2107, 2269, 2437, 2611, 2791, 2977, 3169, 3367, 3571, 3781, 3997, 4219, 4447, 4681, 4921, 5167, 5419, 5677, 5941, 6211, 6487

Offset: 1

Brian Watson, Jan 13 2018

Apart from the term 3, these are precisely the sizes of hexagonal clusters of circles in the A_2 (or hexagonal) lattice.

These numbers include many common numbers of strands used in cables.

Colin Barker, Table of n, a(n) for n = 1..1000
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
Wikipedia, Wire: Number of strands
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A003215.

Vec(x*(1 + x^2 + 6*x^3 - 2*x^4) / (1 - x)^3 + O(x^60)) \\ Colin Barker, Feb 05 2018

From Colin Barker, Feb 05 2018: (Start)
G.f.: x*(1 + x^2 + 6*x^3 - 2*x^4) / (1 - x)^3.
a(n) = 7 - 9*n + 3*n^2 for n>2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>5.
(End)

Entry revised by N. J. A. Sloane, Jan 22 2018

cp's OEIS Frontend

A254136 Indices of pentagonal numbers (A000326) which are also centered hexagonal numbers (A003215).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A254137 Indices of centered hexagonal numbers (A003215) which are also pentagonal numbers (A000326).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A254138 Pentagonal numbers (A000326) which are also centered hexagonal numbers (A003215).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A254782 Indices of centered hexagonal numbers (A003215) which are also centered pentagonal numbers (A005891).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A254964 Indices of heptagonal numbers (A000566) that are also centered hexagonal numbers (A003215).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A254965 Indices of centered hexagonal numbers (A003215) that are also heptagonal numbers (A000566).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A254966 Heptagonal numbers (A000566) that are also centered hexagonal numbers (A003215).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI