A045944 -id:A045944 - cp's OEIS frontend

A140678 a(n) = n(3n + 10).

0, 13, 32, 57, 88, 125, 168, 217, 272, 333, 400, 473, 552, 637, 728, 825, 928, 1037, 1152, 1273, 1400, 1533, 1672, 1817, 1968, 2125, 2288, 2457, 2632, 2813, 3000, 3193, 3392, 3597, 3808, 4025, 4248, 4477, 4712, 4953, 5200, 5453, 5712

Offset: 0

Omar E. Pol, May 22 2008

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

Cf. A033428, A045944, A140676, A067725, A140677, A067707, A140679, A140680, A140681, A140689.

Table[n (3 n + 10), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 13, 32}, 50] (* Harvey P. Dale, Jun 05 2012 *)

a(n)=n*(3*n+10) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*n^2 + 10*n.
a(n) = 6*n + a(n-1) + 7, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: x*(13 - 7*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=0, a(1)=13, a(2)=32. - Harvey P. Dale, Jun 05 2012
E.g.f.: (3*x^2 + 13*x)*exp(x). - G. C. Greubel, Jul 20 2017

A211013 Second 13-gonal numbers: a(n) = n(11n+9)/2.

Original entry on oeis.org

0, 10, 31, 63, 106, 160, 225, 301, 388, 486, 595, 715, 846, 988, 1141, 1305, 1480, 1666, 1863, 2071, 2290, 2520, 2761, 3013, 3276, 3550, 3835, 4131, 4438, 4756, 5085, 5425, 5776, 6138, 6511, 6895, 7290, 7696, 8113, 8541, 8980, 9430, 9891, 10363

Offset: 0

Omar E. Pol, Aug 04 2012

Sequence found by reading the line from 0, in the direction 0, 31... and the line from 10, in the direction 10, 63,..., in the square spiral whose vertices are the generalized 13-gonal numbers A195313.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195313.
Second k-gonal numbers (k=5..14): A005449, A014105, A147875, A045944, A179986, A033954, A062728, A135705, this sequence, A211014.
Cf. A051865.

List([0..50], n-> n*(11*n+9)/2); # G. C. Greubel, Jul 04 2019

[n*(11*n+9)/2: n in [0..50]]; // G. C. Greubel, Jul 04 2019

Table[n*(11*n+9)/2, {n,0,50}] (* G. C. Greubel, Jul 04 2019 *)

a(n)=n*(11*n+9)/2 \\ Charles R Greathouse IV, Jun 17 2017

[n*(11*n+9)/2 for n in (0..50)] # G. C. Greubel, Jul 04 2019

G.f.: x*(10+x)/(1-x)^3. - Philippe Deléham, Mar 27 2013
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 0, a(1) = 10, a(2) = 31. - Philippe Deléham, Mar 27 2013
a(n) = A051865(n) + 9n = A180223(n) + 8n = A022268(n) + 5n = A022269(n) + 4n = A152740(n) - n. - Philippe Deléham, Mar 27 2013
a(n) = A218530(11n+9). - Philippe Deléham, Mar 27 2013
E.g.f.: x*(20 + 11*x)*exp(x)/2. - G. C. Greubel, Jul 04 2019

A307011 First coordinate in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. Second and third coordinates are given in A307012 and A345978.

Original entry on oeis.org

0, 1, 0, -1, -1, 0, 1, 2, 2, 1, 0, -1, -2, -2, -2, -1, 0, 1, 2, 3, 3, 3, 2, 1, 0, -1, -2, -3, -3, -3, -3, -2, -1, 0, 1, 2, 3, 4, 4, 4, 4, 3, 2, 1, 0, -1, -2, -3, -4, -4, -4, -4, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5

Offset: 0

Hugo Pfoertner, Mar 19 2019

From Peter Munn, Jul 22 2021: (Start)
The points of the spiral are equally the points of a hexagonal lattice, the points of an isometric (triangular) grid and the center points of the cells of a honeycomb (regular hexagonal tiling or grid). The coordinate system can be described using 3 axes that pass through spiral point 0 and one of points 1, 2 or 3. Along each axis, one of the coordinates is 0.
a(n) is the signed distance from spiral point n to the axis that passes through point 2. The distance is measured along either of the lines through point n that are parallel to one of the other 2 axes and the sign is such that point 1 has positive distance.
This coordinate can be paired with either of the other coordinates to form oblique coordinates as described in A307012. Alternatively, all 3 coordinates can be used together, symmetrically, as described in A345978.
There is a negated variant of the 3rd coordinate, which is the conventional sense of this coordinate for specifying (with the 2nd coordinate) the Eisenstein integers that can be the points of the spiral when it is embedded in the complex plane. See A307013.
(End)

Hugo Pfoertner, Table of n, a(n) for n = 0..10034
Margherita Barile, Oblique Coordinates, entry in Eric Weisstein's World of Mathematics.
HandWiki, Hexagonal Lattice.
Peter Munn, Illustration of signed distance of spiral points.
Hugo Pfoertner, Illustration of A307012 vs A307011, spiral.
Hugo Pfoertner, Illustration of A345978 vs A307011, spiral.
Wikipedia, Signed distance function.

Cf. A307012, A307013, A307014, A307016, A307017, A328818, A345978.
Numbers on the spokes of the spiral: A000567, A028896, A033428, A045944, A049450, A049451.
Positions on the spiral that correspond to Eisenstein primes: A345435.

r=-1;d=-1;print1(m=0,", ");for(k=0,8,for(j=1,r,print1(s,", "));if(k%2,,m++;r++);for(j=-m,m+1,if(d*j>=-m,print1(s=d*j,", ")));d=-d)

Name revised by Peter Munn, Jul 08 2021

A140679 a(n) = n(3n+14).

Original entry on oeis.org

0, 17, 40, 69, 104, 145, 192, 245, 304, 369, 440, 517, 600, 689, 784, 885, 992, 1105, 1224, 1349, 1480, 1617, 1760, 1909, 2064, 2225, 2392, 2565, 2744, 2929, 3120, 3317, 3520, 3729, 3944, 4165, 4392, 4625, 4864, 5109, 5360, 5617, 5880, 6149, 6424, 6705, 6992

Offset: 0

Omar E. Pol, May 22 2008

			a(1)=6*1+0+11=17; a(2)=6*2+17+11=40; a(3)=6*3+40+11=69. See 2nd formula.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033428, A045944, A140676, A067725, A140677, A140678, A067707, A140680, A140681, A140689.

Table[n(3n+14),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,17,40},50] (* Harvey P. Dale, Apr 29 2011 *)

a(n)=n*(3*n+14) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*n^2 + 14*n.
a(n) = a(n-1) + 6*n + 11, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(1)=0, a(2)=17, a(3)=40. - Harvey P. Dale, Apr 29 2011
E.g.f.: (3*x^2 + 17*x)*exp(x). - G. C. Greubel, Jul 20 2017

A140680 a(n) = n(3n+16).

Original entry on oeis.org

0, 19, 44, 75, 112, 155, 204, 259, 320, 387, 460, 539, 624, 715, 812, 915, 1024, 1139, 1260, 1387, 1520, 1659, 1804, 1955, 2112, 2275, 2444, 2619, 2800, 2987, 3180, 3379, 3584, 3795, 4012, 4235, 4464, 4699, 4940, 5187, 5440, 5699

Offset: 0

Omar E. Pol, May 22 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033428, A045944, A140676, A067725, A140677, A140678, A067707, A140679, A140681, A140689.

a[n_]:=Sum[6*i+13, {i, 1, n}]; (* Vladimir Joseph Stephan Orlovsky, Dec 04 2008 *)
Table[n(3n+16),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,19,44},50] (* Harvey P. Dale, Aug 23 2020 *)

a(n)=n*(3*n+16) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*n^2 + 16*n.
a(n) = 6*n + a(n-1) + 13 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
E.g.f.: (3*x^2 + 19*x)*exp(x). - G. C. Greubel, Jul 20 2017

A140689 a(n) = n(3n + 20).

Original entry on oeis.org

0, 23, 52, 87, 128, 175, 228, 287, 352, 423, 500, 583, 672, 767, 868, 975, 1088, 1207, 1332, 1463, 1600, 1743, 1892, 2047, 2208, 2375, 2548, 2727, 2912, 3103, 3300, 3503, 3712, 3927, 4148, 4375, 4608, 4847, 5092, 5343, 5600, 5863

Offset: 0

Omar E. Pol, May 22 2008

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

Cf. A033428, A045944, A140676, A067725, A140677, A140678, A067707, A140679, A140680, A140681.

a[n_]:=Sum[6*i+17, {i, 1, n}]; (* Vladimir Joseph Stephan Orlovsky, Dec 04 2008 *)
Table[n(3n+20),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,23,52},50] (* Harvey P. Dale, Apr 29 2016 *)

a(n)=n*(3*n+20) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*n^2 + 20*n.
a(n) = a(n-1) + 6*n + 17 (with a(0)=0). - Vincenzo Librandi, Dec 15 2010
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3), with a(0)=0, a(1)=23, a(2)=52. - Harvey P. Dale, Apr 29 2016
From G. C. Greubel, Jul 21 2017: (Start)
G.f.: x*(23 - 17*x)/(1 - x)^3.
E.g.f.: x*(3*x + 23)*exp(x). (End)

A165367 Trisection a(n) = A026741(3n + 2).

Original entry on oeis.org

1, 5, 4, 11, 7, 17, 10, 23, 13, 29, 16, 35, 19, 41, 22, 47, 25, 53, 28, 59, 31, 65, 34, 71, 37, 77, 40, 83, 43, 89, 46, 95, 49, 101, 52, 107, 55, 113, 58, 119, 61, 125, 64, 131, 67, 137, 70, 143, 73, 149, 76, 155, 79, 161, 82, 167, 85, 173, 88, 179, 91, 185, 94, 191, 97, 197

Offset: 0

Paul Curtz, Sep 17 2009

The other trisections are A165351 and A165355.

John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1)

Cf. A000326, A016777, A016969, A022998, A026741, A045944, A165351, A165355.

A026741 := proc(n) if type(n,'odd') then n; else n/2 ; fi; end:
A165367 := proc(n) A026741(3*n+2) ; end: seq(A165367(n),n=0..100) ; # R. J. Mathar, Nov 22 2009

LinearRecurrence[{0, 2, 0, -1}, {1, 5, 4, 11}, 66] (* Jean-François Alcover, Nov 15 2017 *)

a(n) = (3*n+2)>>!(n%2); \\ Ruud H.G. van Tol, Oct 09 2023

a(n)*A022998(n) = A045944(n).
a(n)*A026741(n+1) = A000326(n+1).
a(2n) = A016777(n); a(2n+1) = A016969(n).
From R. J. Mathar Nov 22 2009: (Start)
a(n) = 2*a(n-2) - a(n-4).
G.f.: (1 + 5*x + 2*x^2 + x^3)/((1-x)^2*(1+x)^2). (End)

All comments rewritten as formulas by R. J. Mathar, Nov 22 2009

A190816 a(n) = 5n^2 - 4n + 1.

Original entry on oeis.org

1, 2, 13, 34, 65, 106, 157, 218, 289, 370, 461, 562, 673, 794, 925, 1066, 1217, 1378, 1549, 1730, 1921, 2122, 2333, 2554, 2785, 3026, 3277, 3538, 3809, 4090, 4381, 4682, 4993, 5314, 5645, 5986, 6337, 6698, 7069, 7450, 7841, 8242, 8653, 9074

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 20 2011

For n >= 2, hypotenuses of primitive Pythagorean triangles with m = 2*n-1, where the sides of the triangle are a = m^2 - n^2, b = 2*n*m, c = m^2 + n^2; this sequence is the c values, short sides (a) are A045944(n-1), and long sides (b) are A002939(n).

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Short sides (a) A045944(n-1), long sides (b) A002939(n).
Cf. A017281 (first differences), A051624 (a(n)-1), A202141.
Sequences of the form m*n^2 - 4*n + 1: -A131098 (m=0), A028872 (m=1), A056220 (m=2), A045944 (m=3), A016754 (m=4), this sequence (m=5), A126587 (m=6), A339623 (m=7), A080856 (m=8).

[5*n^2 - 4*n + 1: n in [0..50]]; // Vincenzo Librandi, Jun 19 2011

Table[5*n^2 - 4*n + 1, {n, 0, 100}]
LinearRecurrence[{3,-3,1},{1,2,13},100] (* or *) CoefficientList[ Series[ (-10 x^2+x-1)/(x-1)^3,{x,0,100}],x] (* Harvey P. Dale, May 24 2011 *)

a(n)=5*n^2-4*n+1 \\ Charles R Greathouse IV, Oct 16 2015

[5*n^2-4*n+1 for n in range(41)] # G. C. Greubel, Dec 03 2023

From Harvey P. Dale, May 24 2011: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=2, a(2)=13.
G.f.: (1 - x + 10*x^2)/(1-x)^3. (End)
E.g.f.: (1 + x + 5*x^2)*exp(x). - G. C. Greubel, Dec 03 2023

Edited by Franklin T. Adams-Watters, May 20 2011

A211014 Second 14-gonal numbers: n(6n+5).

Original entry on oeis.org

0, 11, 34, 69, 116, 175, 246, 329, 424, 531, 650, 781, 924, 1079, 1246, 1425, 1616, 1819, 2034, 2261, 2500, 2751, 3014, 3289, 3576, 3875, 4186, 4509, 4844, 5191, 5550, 5921, 6304, 6699, 7106, 7525, 7956, 8399, 8854, 9321, 9800, 10291, 10794, 11309, 11836, 12375

Offset: 0

Omar E. Pol, Aug 04 2012

Sequence found by reading the line from 0, in the direction 0, 34, ... and the line from 11 in the direction 11, 69, ..., in the square spiral whose vertices are the generalized 14-gonal numbers A195818.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Mark W. Coffey, Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers, arXiv:1601.01673 [math.NT], 2016. See 3rd formula in Proposition 3 p. 36 giving a(n+1).
Leo Tavares, Star illustration
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195818.
Second k-gonal numbers (k=5..14): A005449, A014105, A147875, A045944, A179986, A033954, A062728, A135705, A211013, this sequence.
Cf. A051866.
Cf. A003154.

List([0..50], n-> n*(6*n+5)); # G. C. Greubel, Jul 04 2019

[n*(6*n+5): n in [0..50]]; // G. C. Greubel, Jul 04 2019

Table[n*(6*n+5), {n,0,50}] (* G. C. Greubel, Jul 04 2019 *)
LinearRecurrence[{3,-3,1},{0,11,34},50] (* Harvey P. Dale, Dec 12 2022 *)

a(n)=n*(6*n+5) \\ Charles R Greathouse IV, Jun 17 2017

[n*(6*n+5) for n in (0..50)] # G. C. Greubel, Jul 04 2019

a(n) = -2*Sum_{k=0..n-1} binomial(6*n+5, 6*k+8)*Bernoulli(6*k+8). - Michel Marcus, Jan 11 2016
From G. C. Greubel, Jul 04 2019: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(11+x)/(1-x)^3.
E.g.f.: x*(11+6*x)*exp(x). (End)
From Amiram Eldar, Feb 28 2022: (Start)
Sum_{n>=1} 1/a(n) = sqrt(3)*Pi/10 + 6/25 - 3*log(3)/10 - 2*log(2)/5.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/5 + log(2)/5 - 6/25 - sqrt(3)*log(sqrt(3)+2)/5. (End)
a(n) = A003154(n+1) - n - 1. - Leo Tavares, Jan 29 2023

A267137 Numbers of the form x^2 + x + x*y + y + y^2 where x and y are integers.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 14, 16, 17, 20, 21, 22, 24, 25, 26, 30, 32, 33, 34, 36, 37, 40, 41, 42, 44, 46, 49, 50, 52, 54, 56, 57, 58, 60, 64, 65, 66, 69, 70, 72, 74, 76, 80, 81, 82, 85, 86, 89, 90, 92, 94, 96, 97, 100, 101, 102, 104, 105, 108, 110, 112, 114, 116

Offset: 1

Altug Alkan, Jan 10 2016

nonn

Inspired by relation between A003136 and A202822. See comment section of A202822.
Prime terms of this sequence are 2, 5, 17, 37, 41, 89, 97, 101, 137, 149, ...
Perfect power terms of this sequence are 1, 4, 8, 9, 16, 25, 32, 36, 49, 64, 81, 100, 121, 144, 169, ...
Obviously, A000290, A002378 and A045944 are subsequences.
The complement of this sequence is A322430. - Kemoneilwe Thabo Moseki, Dec 12 2019

			1 is a term because (-1)^2 + (-1) + (-1)*(-1) + (-1) + (-1)^2 = 1.
4 is a term because 2^2 + 2 + 2*(-2) + (-2) + (-2)^2 = 4.
24 is a term because 2^2 + 2 + 2*3 + 3 + 3^2 = 24.

Michael De Vlieger, Table of n, a(n) for n = 1..10583
Alexandre Chaduteau, Nyan Raess, Henry Davenport, and Frank Schindler, Hilbert Space Fragmentation in the Chiral Luttinger Liquid, arXiv:2409.10359 [cond-mat.str-el], 2024. See pp. 5, 8.
Alexandre Chaduteau, Nyan Raess, Henry Davenport, and Frank Schindler, Momentum-space modulated symmetries in the Luttinger liquid, Phys. Rev. B (2025) Vol. 111, Art. No. 165105. See p. 4.

Cf. A000290, A002378, A003136, A045944, A202822.

f[{i_, j_}] := (i^2 + i*j + j^2 + i + j); Union@ Map[f, Tuples[Range[-10, 10], 2] ] (* Michael De Vlieger, Sep 23 2024, after Harvey P. Dale at A202822 *)

x='x+O('x^500); p=eta(x)^3/eta(x^3); for(n=0, 499, if(polcoeff(p, n) != 0 && n%3==1, print1((n-1)/3, ", ")));

is(n) = sumdiv( n, d, kronecker( -3, d));
for(n=0, 1e3, if(is(3*n+1), print1(n, ", ")));

is(n) = #bnfisintnorm(bnfinit(z^2+z+1), n);
for(n=0, 1e3, if(is(3*n+1), print1(n, ", ")));

a(n) = (A202822(n) - 1) / 3.

cp's OEIS Frontend

A140678 a(n) = n*(3*n + 10).

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A211013 Second 13-gonal numbers: a(n) = n*(11*n+9)/2.

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A307011 First coordinate in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. Second and third coordinates are given in A307012 and A345978.

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A140679 a(n) = n*(3*n+14).

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A140680 a(n) = n*(3*n+16).

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A140689 a(n) = n*(3*n + 20).

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A165367 Trisection a(n) = A026741(3n + 2).

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A190816 a(n) = 5*n^2 - 4*n + 1.

A140678 a(n) = n(3n + 10).

A211013 Second 13-gonal numbers: a(n) = n(11n+9)/2.

A140679 a(n) = n(3n+14).

A140680 a(n) = n(3n+16).

A140689 a(n) = n(3n + 20).

A190816 a(n) = 5n^2 - 4n + 1.

A211014 Second 14-gonal numbers: n(6n+5).