A056109 -id:A056109 - cp's OEIS frontend

A056106 Second spoke of a hexagonal spiral.

1, 3, 11, 25, 45, 71, 103, 141, 185, 235, 291, 353, 421, 495, 575, 661, 753, 851, 955, 1065, 1181, 1303, 1431, 1565, 1705, 1851, 2003, 2161, 2325, 2495, 2671, 2853, 3041, 3235, 3435, 3641, 3853, 4071, 4295, 4525, 4761, 5003, 5251, 5505, 5765, 6031, 6303

Offset: 0

Henry Bottomley, Jun 09 2000

First differences of A027444. - J. M. Bergot, Jun 04 2012
Numbers of the form ((h^2+h+1)^2+(-h^2+h+1)^2+(h^2+h-1)^2)/(h^2-h+1) for h=n-1. - Bruno Berselli, Mar 13 2013
For n > 0: 2*a(n) = A058331(n) + A001105(n) + A001844(n-1) = A251599(3*n-2) + A251599(3*n-1) + A251599(3*n). - Reinhard Zumkeller, Dec 13 2014
For all n >= 6, a(n+1) expressed in base n is "353". - Mathew Englander, Jan 06 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Henry Bottomley, Spokes of a hexagonal spiral (illustration of initial terms).
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

First differences of A053698, A027444, and A188947.
Cf. A113524 (semiprime terms), A002061.
Cf. A001105, A001844, A058331, A251599.
Other spokes: A003215, A056105, A056107, A056108, A056109.
Other spirals: A054552.

a056106 n = n * (3 * n - 1) + 1  -- Reinhard Zumkeller, Dec 13 2014

I:=[1,3]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2)+6: n in [1..50]]; // Vincenzo Librandi, Nov 14 2011

Table[3*n^2 - n + 1, {n,0,50}] (* G. C. Greubel, Jul 19 2017 *)

PARI
```
a(n) = 3*n^2-n+1;
```

a(n) = 3*n^2 - n + 1.
a(n) = a(n-1) + 6*n - 4 = 2*a(n-1) - a(n-2) + 6.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (1+2*x+3*x^2)*exp(x). - Paul Barry, Mar 13 2003
a(n) = A096777(3*n) for n>0. - Reinhard Zumkeller, Dec 29 2007
G.f.: (1+5*x^2)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 04 2012
a(n) = n*A002061(n+1) - (n-1)*A002061(n). - Bruno Berselli, Jan 15 2013
a(-n) = A056108(n). - Bruno Berselli, Mar 13 2013

A008810 a(n) = ceiling(n^2/3).

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 12, 17, 22, 27, 34, 41, 48, 57, 66, 75, 86, 97, 108, 121, 134, 147, 162, 177, 192, 209, 226, 243, 262, 281, 300, 321, 342, 363, 386, 409, 432, 457, 482, 507, 534, 561, 588, 617, 646, 675, 706, 737, 768, 801, 834, 867, 902, 937, 972, 1009, 1046

Offset: 0

N. J. A. Sloane

a(n+1) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and 3*w = 2*x + y. - Clark Kimberling, Jun 04 2012

a(n) is also the number of L-shapes (3-cell polyominoes) packing into an n X n square. See illustration in links. - Kival Ngaokrajang, Nov 10 2013

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, number of red blocks in Fig 2.5.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta and K.M. Tamizhmani, Blending two discrete integrability criteria: ..., arXiv:nlin/0104020 [nlin.SI], 2001.
Kival Ngaokrajang, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A000290, A056105, A056109.

Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), this sequence (m=3), A008811 (m=4), A008812 (m=5), A008813 (m=6), A008814 (m=7), A008815 (m=8), A008816 (m=9), A008817 (m=10).

a008810 = ceiling . (/ 3) . fromInteger . a000290
a008810_list = [0,1,2,3,6] ++ zipWith5
               (\u v w x y -> 2 * u - v + w - 2 * x + y)
   (drop 4 a008810_list) (drop 3 a008810_list) (drop 2 a008810_list)
   (tail a008810_list) a008810_list
-- Reinhard Zumkeller, Dec 20 2012

[Ceiling(n^2/3): n in [0..60]]; // G. C. Greubel, Sep 12 2019

seq(ceil(n^2/3), n=0..60); # G. C. Greubel, Sep 12 2019

Ceiling[Range[0,60]^2/3] (* Vladimir Joseph Stephan Orlovsky, Mar 15 2011 *)
LinearRecurrence[{2,-1,1,-2,1},{0,1,2,3,6},60] (* Harvey P. Dale, Jun 20 2011 *)

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

[ceil(n^2/3) for n in (0..60)] # G. C. Greubel, Sep 12 2019

a(-n) = a(n) = ceiling(n^2/3).
G.f.: x*(1 + x^3)/((1 - x)^2*(1 - x^3)) = x*(1 - x^6)/((1 - x)*(1 - x^3))^2.
From Michael Somos, Aug 03 2006: (Start)
Euler transform of length 6 sequence [ 2, 0, 2, 0, 0, -1].
a(3n-1) = A056105(n).
a(3n+1) = A056109(n). (End)
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n > 4. - Harvey P. Dale, Jun 20 2011
a(A008585(n)) = A033428(n). - Reinhard Zumkeller, Dec 20 2012
9*a(n) = 4 + 3*n^2 - 2*A099837(n+3). - R. J. Mathar, May 02 2013
a(n) = n^2 - 2*A000212(n). - Wesley Ivan Hurt, Jul 07 2013
Sum_{n>=1} 1/a(n) = Pi^2/18 + sqrt(2)*Pi*sinh(2*sqrt(2)*Pi/3)/(1+2*cosh(2*sqrt(2)*Pi/3)). - Amiram Eldar, Aug 13 2022
E.g.f.: (exp(x)*(4 + 3*x*(1 + x)) - 4*exp(-x/2)*cos(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 28 2022

A081436 Fifth subdiagonal in array of n-gonal numbers A081422.

Original entry on oeis.org

1, 7, 24, 58, 115, 201, 322, 484, 693, 955, 1276, 1662, 2119, 2653, 3270, 3976, 4777, 5679, 6688, 7810, 9051, 10417, 11914, 13548, 15325, 17251, 19332, 21574, 23983, 26565, 29326, 32272, 35409, 38743, 42280, 46026, 49987, 54169, 58578, 63220

Offset: 0

Paul Barry, Mar 21 2003

One of a family of sequences with palindromic generators.
Also as A(n) = (1/6)*(6*n^3 - 3*n^2 + 3*n), n>0: structured pentagonal diamond numbers (vertex structure 5). (Cf. A004068 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers.) - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF4 denominators of A156933. See A157705 for background information. - Johannes W. Meijer, Mar 07 2009
Row 1 of the convolution arrays A213831 and A213833. - Clark Kimberling, Jul 04 2012
Partial sums of A056109. - J. M. Bergot, Jun 22 2013
Number of ordered pairs of intersecting multisets of size 2, each chosen with repetition from {1,...,n}. - Robin Whitty, Feb 12 2014
Row sums of A244418. - L. Edson Jeffery, Jan 10 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Christian Barrientos, The number of spanning trees of cyclic snakes, Indones. J. Comb. (2025) Vol. 9, No. 1, 21-30. See p. 29.
J. A. Dias da Silva and Pedro J. Freitas, Counting Spectral Radii of Matrices with Positive Entries, arXiv:1305.1139 [math.CO], 2013.
Theorem of the Day, Lovász Local Lemma example involving intersecting pairs of multisets
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A081434, A081435, A081437, A156933, A157705, A244418.

List([0..40], n-> (n+1)*(2*(n+1)^2-n)/2); # G. C. Greubel, Aug 14 2019

[(2*n^3+5*n^2+5*n+2)/2: n in [0..40]]; // Vincenzo Librandi, Jul 19 2011

A081436 := proc(n)
    (n+1)*(2*n^2+3*n+2)/2 ;
end proc:
seq(A081436(n),n=0..60) ; # R. J. Mathar, Jun 26 2013

LinearRecurrence[{4, -6, 4, -1}, {1, 7, 24, 58}, 40] (* Jean-François Alcover, Sep 21 2017 *)

a(n)=n^3+5/2*n*(n+1)+1 \\ Charles R Greathouse IV, Jun 20 2013

[(n+1)*(2*(n+1)^2-n)/2 for n in (0..40)] # G. C. Greubel, Aug 14 2019

a(n) = (n+1)*(2*n^2 + 3*n + 2)/2.
G.f.: (1+x)*(1+2*x)/(1-x)^4. (Convolution of A005408 and A016777.)
a(n) = A110449(n, n-1), for n>1.
a(n) = (n+1)*T(n+1) + n*T(n), where T( ) are triangular numbers. Binomial transform of [1, 6, 11, 6, 0, 0, 0, ...]. - Gary W. Adamson, Dec 28 2007
E.g.f.: exp(x)*(2 + 12*x + 11*x^2 + 2*x^3)/2. - Stefano Spezia, Apr 13 2021
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Apr 14 2021

G.f. simplified and crossrefs added by Johannes W. Meijer, Mar 07 2009

A056578 a(n) = 1 + 2n + 3n^2 + 4*n^3.

Original entry on oeis.org

1, 10, 49, 142, 313, 586, 985, 1534, 2257, 3178, 4321, 5710, 7369, 9322, 11593, 14206, 17185, 20554, 24337, 28558, 33241, 38410, 44089, 50302, 57073, 64426, 72385, 80974, 90217, 100138, 110761, 122110, 134209, 147082, 160753, 175246, 190585, 206794, 223897, 241918

Offset: 0

Henry Bottomley, Jun 29 2000

			For n>4 this is 4321 translated from base n to base 10.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Note: 1 + 2*x + 3*x^2 + 4*x^3 is the first derivative of 1 + x + x^2 + x^3 + x^4, i.e., A053699.

Cf. A000012, A005408, A056109, A056579.

f[n_]:=1+2*n+3*n^2+4*n^3; lst={}; Do[AppendTo[lst,f[n]],{n,0,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 12 2010 *)

a(n) = (A053699(n+1) - A053699(n-1))/2 - 4*n - 1.
G.f.: (1 + 6*x + 15*x^2 + 2*x^3)/(1-x)^4. - Colin Barker, Jan 10 2012
From Elmo R. Oliveira, Apr 20 2025: (Start)
E.g.f.: exp(x)*(1 + 9*x + 15*x^2 + 4*x^3).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

More terms from Elmo R. Oliveira, Apr 20 2025

A244807 The hexagonal spiral of Champernowne, read along the East (or 90-degree) ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

Inspired by Stanislaw M. Ulam's hexagonal spiral, circa 1963. See example section of A056105.

When A056105, A056106, A056107, A056108, A056109 & A003215 were submitted, the offsets were 0. Here the offset is 1.

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

Cf. A007376, A054552, A056105, A244808, A244809, A244810, A244811, A244812, A244813, A244814, A244815, A244816, A244817, 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- 8n +6 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

For each 30 degrees of the compass, the corresponding spoke (or ray) has a generating formula as follows:
 3n^2- 8n +6
12n^2-27n+16
 3n^2- 7n+ 5
12n^2-25n+14
 3n^2 -6n +4
12n^2-23n+12
 3n^2 -5n +3
12n^2-21n+10
 3n^2 -4n +2
12n^2-19n +8
 3n^2 -3n +1
12n^2-17n+ 6
Also see formula section of A056105.

A033577 a(n) = (3n+1) (4*n+1).

Original entry on oeis.org

1, 20, 63, 130, 221, 336, 475, 638, 825, 1036, 1271, 1530, 1813, 2120, 2451, 2806, 3185, 3588, 4015, 4466, 4941, 5440, 5963, 6510, 7081, 7676, 8295, 8938, 9605, 10296, 11011, 11750, 12513, 13300, 14111, 14946, 15805, 16688, 17595, 18526, 19481, 20460, 21463

Offset: 0

N. J. A. Sloane

Also the 120º spoke (or ray) of a hexagonal spiral of Ulam. - Robert G. Wilson v, Jul 06 2014

If two independent real random variables x and y are distributed according to the same exponential distribution with pdf(x) = lambda * exp(-lambda * x) for some lambda > 0, then the probability that 3 <= x/(n*y) < 4 is given by n/a(n) for n>1. - Andres Cicuttin, Dec 11 2016

			See A056105 example section for hexagonal spiral of Ulam diagram. - _Robert G. Wilson v_, Jul 06 2014

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Subsequence of A281333.

Cf. A003215, A056105, A056106, A056107, A056108, A056109, A244802, A244803, A244804, A244805, A244806.

List([0..50], n-> (3*n+1)*(4*n+1)); # G. C. Greubel, Oct 12 2019

[(3*n+1)*(4*n+1) : n in [0..50]]; // Wesley Ivan Hurt, Jul 06 2014

A033577:=n->(3*n+1)*(4*n+1): seq(A033577(n), n=0..50); # Wesley Ivan Hurt, Jul 06 2014

f[n_] := (3n + 1)(4n + 1); Array[f, 50, 0] (* Robert G. Wilson v, Jul 06 2014 *)
LinearRecurrence[{3,-3,1},{1,20,63},50] (* Harvey P. Dale, Jul 16 2020 *)

vector(50, m, 12*m^2 - 17*m + 6) \\ Michel Marcus, Jul 06 2014

Vec((1 + 17*x + 6*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Dec 12 2016

[(3*n+1)*(4*n+1) for n in range(50)] # G. C. Greubel, Oct 12 2019

From Colin Barker, Dec 12 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.
G.f.: (1 + 17*x + 6*x^2)/(1-x)^3. (End)
E.g.f.: (1 + 19*x + 12*x^2)*exp(x). - G. C. Greubel, Oct 12 2019

More terms from Wesley Ivan Hurt, Jul 06 2014

A244815 The hexagonal spiral of Champernowne, read along the 210-degree ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

			see A244807 example section for its diagram.

Cf. A007376, A056109, A244807, A244808, A244809, A244810, A244811, A244812, A244813, A244814, A244816, A244817, 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 - 4n + 2 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(3n^2 - 4n + 2)th almost natural number (A033307), Also see formula section of A056105.

A056579 1+2n+3n^2+4n^3+5n^4.

Original entry on oeis.org

1, 15, 129, 547, 1593, 3711, 7465, 13539, 22737, 35983, 54321, 78915, 111049, 152127, 203673, 267331, 344865, 438159, 549217, 680163, 833241, 1010815, 1215369, 1449507, 1715953, 2017551, 2357265, 2738179, 3163497, 3636543

Offset: 0

Henry Bottomley, Jun 29 2000

			For n>5 this is 54321 translated from base n to base 10

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Note: 1+2x+3x^2+4x^3+5x^4 is derivative of 1+x+x^2+x^3+x^4 +x^5, i.e. A053700. Cf. A000012, A005408, A056109, A056578.

Join[{1},Table[Total[Table[i n^(i-1),{i,5}]],{n,30}]] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,15,129,547,1593},30] (* Harvey P. Dale, Sep 20 2017 *)

a(n)=1+2*n+3*n^2+4*n^3+5*n^4 \\ Charles R Greathouse IV, Oct 07 2015

a(n) =(A053700(n+1)-A053700(n-1))/2-10n^2-4n-2.

G.f.: -(3*x^4+42*x^3+64*x^2+10*x+1) / (x-1)^5. - Colin Barker, May 04 2013

A086285 Numbers k such that 1 + 2k + 3k^2 is prime.

Original entry on oeis.org

2, 12, 14, 18, 24, 26, 32, 38, 44, 56, 62, 68, 90, 92, 98, 114, 120, 126, 128, 144, 150, 158, 164, 168, 170, 176, 180, 186, 192, 200, 210, 212, 216, 230, 246, 254, 260, 266, 276, 278, 282, 290, 300, 318, 332, 344, 354, 362, 366, 378, 396, 398, 408, 420, 432

Offset: 1

Zak Seidov, Aug 28 2003

Excluding a(1), these are also the bases in which 321 represents a prime number.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A056109 (3n^2+2n+1).

Select[Range[500],PrimeQ[1+2#+3#^2]&] (* Harvey P. Dale, Jun 23 2012 *)

is(n)=isprime(1+2*n+3*n^2) \\ Charles R Greathouse IV, Feb 17 2017

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)

cp's OEIS Frontend

A056106 Second spoke of a hexagonal spiral.

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A008810 a(n) = ceiling(n^2/3).

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A081436 Fifth subdiagonal in array of n-gonal numbers A081422.

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A056578 a(n) = 1 + 2*n + 3*n^2 + 4*n^3.

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A244807 The hexagonal spiral of Champernowne, read along the East (or 90-degree) ray.

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A033577 a(n) = (3*n+1) * (4*n+1).

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A056578 a(n) = 1 + 2n + 3n^2 + 4*n^3.

A033577 a(n) = (3n+1) (4*n+1).