A056108 -id:A056108 - cp's OEIS frontend

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

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

A244813 The hexagonal spiral of Champernowne, read along the West (or 270-degree) ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

			see A244807 example section for its diagram.

Cf. A007376, A056108, A244807, A244808, A244809, A244810, A244811, A244812, 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 - 5n + 3 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(3n^2 - 5n + 3)th almost natural number (A033307), Also see formula section of A056105.

A096777 a(n) = a(n-1) + Sum_{k=1..n-1}(a(k) mod 2), a(1) = 1.

Original entry on oeis.org

1, 2, 3, 5, 8, 11, 15, 20, 25, 31, 38, 45, 53, 62, 71, 81, 92, 103, 115, 128, 141, 155, 170, 185, 201, 218, 235, 253, 272, 291, 311, 332, 353, 375, 398, 421, 445, 470, 495, 521, 548, 575, 603, 632, 661, 691, 722, 753, 785, 818, 851, 885, 920, 955, 991, 1028

Offset: 1

Reinhard Zumkeller, Jul 09 2004

a(n) = a(n-1) + (number of odd terms so far in the sequence). Example: 15 is 11 + 4 odd terms so far in the sequence (they are 1,3,5,11). See A007980 for the same construction with even integers. - Eric Angelini, Aug 05 2007

A016789 and A032766 give positions where even and odd terms occur; a(3*n)=A056106(n); a(3*n-1)=A077588(n); a(3*n-2)=A056108(n). - Reinhard Zumkeller, Dec 29 2007

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 11*x^6 + 15*x^7 + 20*x^8 + ... - _Michael Somos_, Apr 18 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J.-L. Baril, T. Mansour, A. Petrossian, Equivalence classes of permutations modulo excedances, 2014.
Eric Weisstein's World of Mathematics, Odd Number
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A004396, A007980, A016789, A032766, A056106, A056108, A061347, A077588, A097602, A131093.

a096777 n = a096777_list !! (n-1)
a096777_list = 1 : zipWith (+) a096777_list
                               (scanl1 (+) (map (`mod` 2) a096777_list))
-- Reinhard Zumkeller, Mar 11 2014

[Floor((n-2)^2/3)+n: n in [1..60]]; // Vincenzo Librandi, Dec 27 2015

A096777:=n->n + floor((n-2)^2/3); seq(A096777(n), n=1..100); # Wesley Ivan Hurt, Mar 06 2014

Table[n + Floor[(n-2)^2/3], {n, 100}] (* Wesley Ivan Hurt, Mar 06 2014 *)

a(n)=(n-2)^2\3+n \\ Charles R Greathouse IV, Mar 06 2014

a(n+1) - a(n) = A004396(n).
a(n) = floor(n/3) * (3*floor(n/3) + 2*(n mod 3) - 1) + n mod 3 + 0^(n mod 3). - Reinhard Zumkeller, Dec 29 2007
a(n) = floor((n-2)^2/3) + n. - Christopher Hunt Gribble, Mar 06 2014
G.f.: -x*(x^4+1) / ((x-1)^3*(x^2+x+1)). - Colin Barker, Mar 07 2014
Euler transform of finite sequence [2, 0, 1, 1, 0, 0, 0, -1]. - Michael Somos, Apr 18 2020
a(n) = (10 + 3*n*(n - 1) - A061347(n+1))/9. - Stefano Spezia, Sep 22 2022

A247608 a(n) = Sum_{k=0..3} binomial(6,k)*binomial(n,k).

Original entry on oeis.org

1, 7, 28, 84, 195, 381, 662, 1058, 1589, 2275, 3136, 4192, 5463, 6969, 8730, 10766, 13097, 15743, 18724, 22060, 25771, 29877, 34398, 39354, 44765, 50651, 57032, 63928, 71359, 79345, 87906, 97062, 106833, 117239, 128300, 140036, 152467, 165613, 179494

Offset: 0

Vincenzo Librandi, Sep 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Krattenthaler, Advanced determinant calculus Séminaire Lotharingien de Combinatoire, B42q (1999), 67 pp, (see p. 54).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005408, A056108.

[(6+31*n-15*n^2+20*n^3)/6: n in [0..40]];

[1+6*Binomial(n,1)+15*Binomial(n,2)+20*Binomial(n,3): n in [0..40]];

I:=[1, 7, 28, 84]; [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]];

Table[(6 + 31 n - 15 n^2 + 20 n^3)/6, {n, 0, 50}] (* or *) CoefficientList[Series[(1 + 3 x + 6 x^2 + 10 x^3)/(1-x)^4,{x, 0, 50}], x]

Vec((1+3*x+6*x^2+10*x^3)/(1-x)^4 + O (x^50)) \\ Michel Marcus, Sep 22 2014

m=3; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # Bruno Berselli, Sep 22 2014

G.f.: (1+3*x+6*x^2+10*x^3)/(1-x)^4.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4).
a(n) = (6+31*n-15*n^2+20*n^3)/6.
a(n) = 1+6*Binomial(n,1)+15*Binomial(n,2)+20*Binomial(n,3).

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)

A156140 Accumulation of Stern's diatomic series: a(0)=-1, a(1)=0, and a(n+1) = (2e(n)+1)*a(n) - a(n-1) for n > 1, where e(n) is the highest power of 2 dividing n.

Original entry on oeis.org

-1, 0, 1, 3, 2, 7, 5, 8, 3, 13, 10, 17, 7, 18, 11, 15, 4, 21, 17, 30, 13, 35, 22, 31, 9, 32, 23, 37, 14, 33, 19, 24, 5, 31, 26, 47, 21, 58, 37, 53, 16, 59, 43, 70, 27, 65, 38, 49, 11, 50, 39, 67, 28, 73, 45, 62, 17, 57, 40, 63, 23, 52, 29, 35, 6, 43, 37, 68, 31, 87, 56, 81, 25, 94, 69

Offset: 0

Arie Werksma (Werksma(AT)Tiscali.nl), Feb 04 2009

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A002487, A007814.
From Yosu Yurramendi, Mar 09 2018: (Start)
a(2^m +  0) = A000027(m),   m >= 0.
a(2^m +  1) = A002061(m+2), m >= 1.
a(2^m +  2) = A002522(m),   m >= 2.
a(2^m +  3) = A033816(m-1), m >= 2.
a(2^m +  4) = A002061(m),   m >= 2.
a(2^m +  5) = A141631(m),   m >= 3.
a(2^m +  6) = A084849(m-1), m >= 3.
a(2^m +  7) = A056108(m-1), m >= 3.
a(2^m +  8) = A000290(m-1), m >= 3.
a(2^m +  9) = A185950(m-1), m >= 4.
a(2^m + 10) = A144390(m-1), m >= 4.
a(2^m + 12) = A014106(m-2), m >= 4.
a(2^m + 16) = A028387(m-3), m >= 4.
a(2^m + 18) = A250657(m-4), m >= 5.
a(2^m + 20) = A140677(m-3), m >= 5.
a(2^m + 32) = A028872(m-2), m >= 5.
a(2^m -  1) = A005563(m-1), m >= 0.
a(2^m -  2) = A028387(m-2), m >= 2.
a(2^m -  3) = A033537(m-2), m >= 2.
a(2^m -  4) = A008865(m-1), m >= 3.
a(2^m -  7) = A140678(m-3), m >= 3.
a(2^m -  8) = A014209(m-3), m >= 4.
a(2^m - 16) = A028875(m-2), m >= 5.
a(2^m - 32) = A108195(m-5), m >= 6.
(End)

A156140 := proc(n)
    option remember ;
    if n <= 1 then
        n-1 ;
    else
        (2*A007814(n-1)+1)*procname(n-1)-procname(n-2) ;
    end if;
end proc:
seq(A156140(n),n=0..80) ; # R. J. Mathar, Mar 14 2009

Fold[Append[#1, (2 IntegerExponent[#2, 2] + 1) #1[[-1]] - #1[[-2]] ] &, {-1, 0}, Range[73]] (* Michael De Vlieger, Mar 09 2018 *)

first(n)=my(v=vector(n+1)); v[1]=-1; v[2]=0; for(k=1,n-1,v[k+2]=(2*valuation(k,2)+1)*v[k+1] - v[k]); v \\ Charles R Greathouse IV, Apr 05 2016

fusc(n)=my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); b
a(n)=my(m=1,s,t); if(n==0, return(-1)); while(n%2==0, s+=fusc(n>>=1)); while(n>1, t=logint(n,2); n-=2^t; s+=m*fusc(n)*(t^2+t+1); m*=-t); m*(n-1) + s \\ Charles R Greathouse IV, Dec 13 2016

a <- c(0,1)
maxlevel <- 6 # by choice
for(m in 1:maxlevel) {
  a[2^(m+1)] <- m + 1
  for(k in 1:(2^m-1)) {
    r <- m - floor(log2(k)) - 1
    a[2^r*(2*k+1)] <- a[2^r*(2*k)] + a[2^r*(2*k+2)]
}}
a
# Yosu Yurramendi, May 08 2018

Let b(n) = A002487(n), Stern's diatomic series.
a(n+1)*b(n) - a(n)*b(n+1) = 1 for n >= 0.
a(2n+1) = a(n) + a(n+1) + b(n) + b(n+1) for n >= 0.
a(2n) = a(n) + b(n) for n >= 0.
a(2^n + k) = -n*a(k) + (n^2 + n + 1)*b(k) for 0 <= k <= 2^n.
b(2^n + k) = -a(k) + (n + 1)*b(k) for 0 <= k <= 2^n.
a(2^m + k) = b(2^m+k)*m + b(k), m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Mar 09 2018
a(2^(m+1)+2^m+1) = 2*m+1, m >= 0. - Yosu Yurramendi, Mar 09 2018
From Yosu Yurramendi, May 08 2018: (Start)
a(2^m) = m, m >= 0.
a(2^r*(2*k+1)) = a(2^r*(2*k)) + a(2^r*(2*k+2)), r = m - floor(log_2(k)) - 1, m > 0, 1 <= k < 2^m.
(End)

cp's OEIS Frontend

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

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A244813 The hexagonal spiral of Champernowne, read along the West (or 270-degree) ray.

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A096777 a(n) = a(n-1) + Sum_{k=1..n-1}(a(k) mod 2), a(1) = 1.

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A247608 a(n) = Sum_{k=0..3} binomial(6,k)*binomial(n,k).

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

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

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