A056105 -id:A056105 - cp's OEIS frontend

A244817 The hexagonal spiral of Champernowne, read along the 150-degree ray.

1, 7, 4, 3, 5, 0, 8, 9, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 9, 1, 9, 1, 1, 2, 6, 3, 3, 7, 7, 5, 6, 6, 2, 8, 8, 9, 1, 1, 1, 6, 4, 4, 5, 7, 1, 7, 8, 2, 2, 1, 2, 1, 7, 4, 5, 4, 6, 9, 0, 1, 9, 3, 4, 2, 6, 7, 8, 7, 7, 2, 3, 6, 2, 7, 8, 9, 1, 2, 4, 6, 4, 8, 9, 7, 1, 4, 5, 2, 2, 0, 1, 1, 7

Offset: 1

Robert G. Wilson v, Jul 06 2014

			see A244807 example section for its diagram.

Cf. A007376, A003215, A244807, A244808, A244809, A244810, A244811, A244812, A244813, A244814, A244815, A244816, A244818.

almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; f[n_] := 3n^2 - 3n + 1 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(3n^2 - 3n + 1)th almost natural number (A033307), Also see formula section of A056105.

A244818 The hexagonal spiral of Champernowne, read along the 120-degree ray.

Original entry on oeis.org

1, 1, 6, 7, 1, 8, 1, 4, 1, 3, 6, 6, 6, 4, 3, 9, 3, 7, 2, 1, 2, 3, 7, 1, 7, 9, 3, 2, 8, 5, 0, 3, 5, 0, 8, 4, 8, 4, 6, 4, 7, 9, 6, 5, 2, 3, 7, 6, 3, 6, 8, 8, 0, 9, 1, 9, 3, 9, 1, 0, 0, 1, 3, 1, 2, 2, 1, 8, 1, 3, 4, 1, 6, 1, 5, 5, 1, 5, 1, 7, 7, 1, 7, 1, 9, 9, 2, 0, 2, 1, 1, 2, 6, 2, 3, 4, 2, 3, 2, 5, 6, 2, 3, 2, 8

Offset: 1

Robert G. Wilson v, Jul 06 2014

			See A244807 example section for its diagram.

Cf. A007376, A033577, A244807, A244808, A244809, A244810, A244811, A244812, A244813, A244814, A244815, A244816, A244817.

almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; f[n_] := 12n^2 - 17n + 6 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(12n^2-17n+6)th almost natural number (A033307), also see formula section of A056105.

A244805 The 240-degree spoke (or ray) of a hexagonal spiral of Ulam.

Original entry on oeis.org

1, 16, 55, 118, 205, 316, 451, 610, 793, 1000, 1231, 1486, 1765, 2068, 2395, 2746, 3121, 3520, 3943, 4390, 4861, 5356, 5875, 6418, 6985, 7576, 8191, 8830, 9493, 10180, 10891, 11626, 12385, 13168, 13975, 14806, 15661, 16540, 17443, 18370, 19321, 20296, 21295, 22318, 23365, 24436, 25531

Offset: 1

Robert G. Wilson v, Jul 06 2014

Numbers of the form 1 + k/2 + k^2/3 (associated k are in A008588). - Bruno Berselli, Jan 20 2017

			See A056105 example section for its diagram.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A056105, A244802, A056106, A244803, A056107, A244804, A056108, A056109, A244806, A003215, A033577.

Cf. A281333 (1 + floor(n/2) + floor(n^2/3)).

[12*n^2-21*n+10: n in [1..50]]; // Wesley Ivan Hurt, Jul 06 2014

A244805:=n->12*n^2 - 21*n + 10: seq(A244805(n), n=1..50); # Wesley Ivan Hurt, Jul 06 2014

f[n_] := 12 n^2 - 21 n + 10; Array[f, 47]

vector(50, n, 12*n^2 - 21*n + 10) \\ Michel Marcus, Jul 06 2014

Vec(x*(1 + 13*x + 10*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Dec 12 2016

a(n) = 12*n^2 - 21*n + 10 (see A056105).
From Colin Barker, Dec 12 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
G.f.: x*(1 + 13*x + 10*x^2) / (1 - x)^3.
(End)

A244806 The 180-degree spoke (or ray) of a hexagonal spiral of Ulam.

Original entry on oeis.org

1, 18, 59, 124, 213, 326, 463, 624, 809, 1018, 1251, 1508, 1789, 2094, 2423, 2776, 3153, 3554, 3979, 4428, 4901, 5398, 5919, 6464, 7033, 7626, 8243, 8884, 9549, 10238, 10951, 11688, 12449, 13234, 14043, 14876, 15733, 16614, 17519, 18448, 19401, 20378, 21379, 22404, 23453, 24526, 25623

Offset: 1

Robert G. Wilson v, Jul 06 2014

			See A056105 example section for its diagram.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A056105, A244802, A056106, A244803, A056107, A244804, A056108, A244805, A056109, A003215, A033577.

[12*n^2 - 19*n + 8 : n in [1..50]]; // Wesley Ivan Hurt, Jul 06 2014

A244806:=n->12*n^2 - 19*n + 8: seq(A244806(n), n=1..50); # Wesley Ivan Hurt, Jul 06 2014

Mathematica
```
f[n_] := 12n^2 - 19n + 8; Array[f, 47]
```

vector(50, n, 12*n^2 - 19*n + 8) \\ Michel Marcus, Jul 06 2014

Vec(x*(1 + 15*x + 8*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Dec 12 2016

a(n) = 12*n^2 - 19*n + 8.
See A056105 example section for its formula.
From Colin Barker, Dec 12 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
G.f.: x*(1 + 15*x + 8*x^2) / (1 - x)^3.
(End)

A244802 The 60-degree spoke (or ray) of a hexagonal spiral of Ulam.

Original entry on oeis.org

1, 10, 43, 100, 181, 286, 415, 568, 745, 946, 1171, 1420, 1693, 1990, 2311, 2656, 3025, 3418, 3835, 4276, 4741, 5230, 5743, 6280, 6841, 7426, 8035, 8668, 9325, 10006, 10711, 11440, 12193, 12970, 13771, 14596, 15445, 16318, 17215, 18136, 19081, 20050, 21043, 22060, 23101, 24166, 25255

Offset: 1

Robert G. Wilson v, Jul 06 2014

			See A056105 example section for a diagram.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A056105, A056106, A244803, A056107, A244804, A056108, A244805, A056109, A244806, A003215.

[12*n^2-27*n+16 : n in [1..50]]; // Wesley Ivan Hurt, Jul 06 2014

A244802:=n->12*n^2-27*n+16: seq(A244802(n), n=1..50); # Wesley Ivan Hurt, Jul 06 2014

Mathematica
```
f[n_] := 12n^2 - 27n + 16; Array[f, 47]
```

vector(50, n, 12*n^2 - 27*n + 16) \\ Michel Marcus, Jul 06 2014

Vec(x*(1 + 7*x + 16*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Dec 12 2016

See A056105 example section for a formula.
From Colin Barker, Dec 12 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
G.f.: x*(1 + 7*x + 16*x^2) / (1 - x)^3.
(End)

A244803 The 360 degree spoke (or ray) of a hexagonal spiral of Ulam.

Original entry on oeis.org

1, 12, 47, 106, 189, 296, 427, 582, 761, 964, 1191, 1442, 1717, 2016, 2339, 2686, 3057, 3452, 3871, 4314, 4781, 5272, 5787, 6326, 6889, 7476, 8087, 8722, 9381, 10064, 10771, 11502, 12257, 13036, 13839, 14666, 15517, 16392, 17291, 18214, 19161, 20132, 21127, 22146, 23189, 24256, 25347

Offset: 1

Robert G. Wilson v, Jul 06 2014

			See A056105 example section for a diagram.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A056105, A244802, A056106, A056107, A244804, A056108, A244805, A056109, A244806, A003215, A033577.

[12*n^2-25*n+14 : n in [1..50]]; // Wesley Ivan Hurt, Jul 06 2014

A244803:=n->12*n^2-25*n+14: seq(A244803(n), n=1..50); # Wesley Ivan Hurt, Jul 06 2014

Mathematica
```
f[n_] := 12n^2 - 25n + 14; Array[f, 47]
```

vector(50, n, 12*n^2 - 25*n + 14) \\ Michel Marcus, Jul 06 2014

Vec(x*(1 + 2*x)*((1 + 7*x) / (1 - x)^3) + O(x^50)) \\ Colin Barker, Dec 12 2016

See A056105 example section for a formula.
From Colin Barker, Dec 12 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
G.f.: x*(1 + 2*x)*((1 + 7*x) / (1 - x)^3).
(End)

A244804 The 300-degree spoke (or ray) of a hexagonal spiral of Ulam.

Original entry on oeis.org

1, 14, 51, 112, 197, 306, 439, 596, 777, 982, 1211, 1464, 1741, 2042, 2367, 2716, 3089, 3486, 3907, 4352, 4821, 5314, 5831, 6372, 6937, 7526, 8139, 8776, 9437, 10122, 10831, 11564, 12321, 13102, 13907, 14736, 15589, 16466, 17367, 18292, 19241, 20214, 21211, 22232, 23277, 24346, 25439

Offset: 1

Robert G. Wilson v, Jul 06 2014

			See A056105 example section for its diagram.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A056105, A244802, A056106, A244803, A056107, A056108, A244805, A056109, A244806, A003215, A033577.

[ 12*n^2 - 23*n + 12 : n in [1..50] ]; // Wesley Ivan Hurt, Jul 06 2014

A244804:=n->12*n^2 - 23*n + 12: seq(A244804(n), n=1..50); # Wesley Ivan Hurt, Jul 06 2014

Mathematica
```
f[n_] := 12n^2 - 23n + 12; Array[f, 47]
```

vector(50, n, 12*n^2 - 23*n + 12) \\ Michel Marcus, Jul 06 2014

Vec(x*(1 + 11*x + 12*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Dec 12 2016

See A056105 example section for its formula.
From Colin Barker, Dec 12 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>3.
G.f.: x*(1 + 11*x + 12*x^2) / (1 - x)^3.
(End)

A270710 a(n) = 3n^2 + 2n - 1.

Original entry on oeis.org

-1, 4, 15, 32, 55, 84, 119, 160, 207, 260, 319, 384, 455, 532, 615, 704, 799, 900, 1007, 1120, 1239, 1364, 1495, 1632, 1775, 1924, 2079, 2240, 2407, 2580, 2759, 2944, 3135, 3332, 3535, 3744, 3959, 4180, 4407, 4640, 4879, 5124, 5375, 5632, 5895, 6164, 6439, 6720, 7007, 7300, 7599

Offset: 0

Ilya Gutkovskiy, Mar 22 2016

In general, the ordinary generating function for the values of quadratic polynomial p*n^2 + q*n + k, is (k + (p + q - 2*k)*x + (p - q + k)*x^2)/(1 - x)^3.
From Bruno Berselli, Mar 25 2016: (Start)
This sequence and A140676 provide all integer m such that 3*m + 4 is a square.
Numbers related to A135713 by A135713(n) = n*a(n) - Sum_{k=0..n-1} a(k).
After -1, second bisection of A184005. (End)

			a(0) = 3*0^2 + 2*0 - 1 = -1;
a(1) = 3*1^2 + 2*1 - 1 =  4;
a(2) = 3*2^2 + 2*2 - 1 = 15;
a(3) = 3*3^2 + 2*3 - 1 = 32, etc.

Ilya Gutkovskiy, Examples of the ordinary generating function for the values of quadratic polynomial.
Leo Tavares, Triple Diamond Illustration.
Eric Weisstein's World of Mathematics, Quadratic Polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033428, A045944, A056105, A056109, A060747, A135713, A140676, A184005.

Cf. A000290.

List([0..50], n -> 3*n^2+2*n-1); # Bruno Berselli, Feb 16 2018

[3*n^2+2*n-1: n in [0..50]]; // Bruno Berselli, Mar 25 2016

Table[3 n^2 + 2 n - 1, {n, 0, 50}]
LinearRecurrence[{3, -3, 1}, {-1, 4, 15}, 51]

makelist(3*n^2+2*n-1, n, 0, 50); /* Bruno Berselli, Mar 25 2016 */

Vec((-1 + 7*x)/(1 - x)^3 + O(x^60)) \\ Michel Marcus, Mar 22 2016

lista(nn) = {for(n=0, nn, print1(3*n^2 + 2*n - 1, ", ")); } \\ Altug Alkan, Mar 25 2016

vector(50, n, n--; 3*n^2+2*n-1) \\ Bruno Berselli, Mar 25 2016

[3*n^2+2*n-1 for n in (0..50)] # Bruno Berselli, Mar 25 2016

G.f.: (-1 + 7*x)/(1 - x)^3.
E.g.f.: exp(x)*(-1 + 5*x + 3*x^2).
a(n)  = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n)  = A033428(n) + A060747(n).
a(n)  = A045944(n) - 1 = A056109(n) - 2.
a(-n) = A140676(n-1), with A140676(-1) = -1.
Sum_{n>=0} 1/a(n) = 3*(log(3) - 2)/8 - Pi/(8*sqrt(3)) = -0.564745312278736...
a(n) = Sum_{i = n-1..2*n-1} (2*i + 1). - Bruno Berselli, Feb 16 2018
a(n) = A000290(n+1) + 2*A000290(n) - 2. - Leo Tavares, May 28 2023
Sum_{n>=0} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)) + 3/4. - Amiram Eldar, Jul 20 2023

A285792 Primes equal to an octagonal number plus 1.

Original entry on oeis.org

2, 41, 97, 281, 409, 937, 1409, 2297, 4721, 5209, 6257, 8009, 8641, 12161, 14561, 18097, 21001, 23057, 24121, 26321, 27457, 37409, 42961, 50441, 52009, 55217, 56857, 60209, 70841, 76481, 90481, 139537, 147409, 152777, 161009, 169457, 172321, 227977, 238009

Offset: 1

Colin Barker, Apr 26 2017

nonn

Primes in A056105. - Omar E. Pol, Apr 26 2017

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

Cf. A000040, A000567, A002496, A055469, A285789, A285790, A285791.

Select[PolygonalNumber[8,Range[300]]+1,PrimeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 22 2017 *)

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(8, k) + 1), listput(L, p))); Vec(L)

A238410 a(n) = floor((3(n-1)^2 + 1)/2).

Original entry on oeis.org

0, 2, 6, 14, 24, 38, 54, 74, 96, 122, 150, 182, 216, 254, 294, 338, 384, 434, 486, 542, 600, 662, 726, 794, 864, 938, 1014, 1094, 1176, 1262, 1350, 1442, 1536, 1634, 1734, 1838, 1944, 2054, 2166, 2282, 2400, 2522, 2646, 2774, 2904, 3038, 3174, 3314, 3456, 3602, 3750, 3902, 4056, 4214, 4374, 4538, 4704

Offset: 1

Emeric Deutsch, Feb 27 2014

a(n) = the eccentric connectivity index of the path P[n] on n vertices. The eccentric connectivity index of a simple connected graph G is defined to be the sum over all vertices i of G of the product E(i)D(i), where E(i) is the eccentricity and D(i) is the degree of vertex i. For example, a(4)=14 because the vertices of P[4] have degrees 1,2,2,1 and eccentricities 3,2,2,3; we have 1*3 + 2*2 + 2*2 + 1*3 = 14.
From Paul Curtz, Feb 23 2023: (Start)
East spoke of the hexagonal spiral using A004526 with a single 0:
.
            43  42  42  41  41  40
          43  28  28  27  27  26  40
        44  29  17  16  16  15  26  39
      44  29  17   8   8   7  15  25  39
    45  30  18   9   3   2   7  14  25  38
  45  30  18   9   3   0---2---6--14--24--38-->
    31  19  10   4   1   1   6  13  24  37
      31  19  10   4   5   5  13  23  37
        32  20  11  11  12  12  23  36
          32  20  21  21  22  22  36
            33  33  34  34  35  35
.
Other spokes are A032528, A005449, A227017, A045943, A005448, A212959, A000326, A282513, A143689, and A104249. (End)

Matthew House, Table of n, a(n) for n = 1..10000
M. J. Morgan, S. Mukwembi and H. C. Swart, On the eccentric connectivity index of a graph, Discrete Math., 311, 2011, 1229-1234.
B. Zhou and Zh. Du, On eccentric connectivity index, Comm. Math. Comp. Chem. (MATCH), 63, 2010, 181-198.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A004526, A077043.

(Cf. A056105, A056107.)

a := proc (n) options operator, arrow: floor((3/2)*(n-1)^2+1/2) end proc: seq(a(n), n = 1 .. 70);

Table[Floor[(3(n-1)^2+1)/2],{n,80}]  (* or *) LinearRecurrence[{2,0,-2,1},{0,2,6,14},80] (* Harvey P. Dale, Apr 30 2022 *)

a(n)=(3*(n-1)^2 + 1)\2 \\ Charles R Greathouse IV, Feb 15 2017

a(n) = (3*n)^2/6 for n even and a(n) = ((3*n)^2 + 3)/6 for n odd. - Miquel Cerda, Jun 17 2016
From Ilya Gutkovskiy, Jun 17 2016: (Start)
G.f.: 2*x^2*(1 + x + x^2)/((1 - x)^3*(1 + x)).
a(n) = (6*n^2 - 12*n + 7 + (-1)^n)/4.
a(n) = 2* A077043(n-1). (End)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Matthew House, Feb 15 2017
Sum_{n>=2} 1/a(n) = Pi^2/36 + tanh(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - Amiram Eldar, Mar 12 2023

cp's OEIS Frontend

A244817 The hexagonal spiral of Champernowne, read along the 150-degree ray.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A244818 The hexagonal spiral of Champernowne, read along the 120-degree ray.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A244805 The 240-degree spoke (or ray) of a hexagonal spiral of Ulam.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

A244806 The 180-degree spoke (or ray) of a hexagonal spiral of Ulam.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

A244802 The 60-degree spoke (or ray) of a hexagonal spiral of Ulam.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

A244803 The 360 degree spoke (or ray) of a hexagonal spiral of Ulam.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

A244804 The 300-degree spoke (or ray) of a hexagonal spiral of Ulam.

Views

Author

Keywords

A270710 a(n) = 3n^2 + 2n - 1.