A051682 -id:A051682 - cp's OEIS frontend

A081266 Staggered diagonal of triangular spiral in A051682.

0, 6, 21, 45, 78, 120, 171, 231, 300, 378, 465, 561, 666, 780, 903, 1035, 1176, 1326, 1485, 1653, 1830, 2016, 2211, 2415, 2628, 2850, 3081, 3321, 3570, 3828, 4095, 4371, 4656, 4950, 5253, 5565, 5886, 6216, 6555, 6903, 7260, 7626, 8001, 8385, 8778, 9180

Offset: 0

Paul Barry, Mar 15 2003

Staggered diagonal of triangular spiral in A051682, between (0,4,17) spoke and (0,7,23) spoke.
Binomial transform of (0, 6, 9, 0, 0, 0, ...).
If Y is a fixed 3-subset of a (3n+1)-set X then a(n) is the number of (3n-1)-subsets of X intersecting Y. - Milan Janjic, Oct 28 2007
Partial sums give A085788. - Leo Tavares, Nov 23 2023

			a(1)=9*1+0-3=6, a(2)=9*2+6-3=21, a(3)=9*3+21-3=45.
For n=3, a(3) = -0^2+1^2-2^2+3^2-4^2+5^2-6^2+7^2-8^2+9^2 = 45.

Muniru A Asiru, Table of n, a(n) for n = 0..10000
Tomislav Došlić and Luka Podrug, Sweet division problems: from chocolate bars to honeycomb strips and back, arXiv:2304.12121 [math.CO], 2023.
Milan Janjic, Two Enumerative Functions
Milan Janjic and B. Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019).
Leo Tavares, Star illustration
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000290, A005449, A014105, A022266, A033585, A062725, A144312, A144314, A218470.

Cf. A003154, A133694.

List([0..50],n->Binomial(3*n+1,2)); # Muniru A Asiru, Feb 28 2019

seq(binomial(3*n+1,2), n=0..45); # Zerinvary Lajos, Jan 21 2007

LinearRecurrence[{3,-3,1},{0,6,21},50] (* Harvey P. Dale, Aug 29 2015 *)

a(n)=3*n*(3*n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 6*C(n,1) + 9*C(n,2).
a(n) = 3*n*(3*n+1)/2.
G.f.: (6*x+3*x^2)/(1-x)^3.
a(n) = A000217(3*n); a(2*n) = A144314(n). - Reinhard Zumkeller, Sep 17 2008
a(n) = 3*A005449(n). - R. J. Mathar, Mar 27 2009
a(n) = 9*n+a(n-1)-3 for n>0, a(0)=0. - Vincenzo Librandi, Aug 08 2010
a(n) = A218470(9n+5). - Philippe Deléham, Mar 27 2013
a(n) = Sum_{k=0..3n} (-1)^(n+k)*k^2. - Bruno Berselli, Aug 29 2013
E.g.f.: 3*exp(x)*x*(4 + 3*x)/2. - Stefano Spezia, Jun 06 2021
From Amiram Eldar, Aug 11 2022: (Start)
Sum_{n>=1} 1/a(n) = 2 - Pi/(3*sqrt(3)) - log(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/(3*sqrt(3)) + 4*log(2)/3 - 2. (End)
From Leo Tavares, Nov 23 2023: (Start)
a(n) = 3*A000217(n) + 3*A000290(n).
a(n) = A003154(n+1) - A133694(n+1). (End)

A081267 Diagonal of triangular spiral in A051682.

Original entry on oeis.org

1, 9, 26, 52, 87, 131, 184, 246, 317, 397, 486, 584, 691, 807, 932, 1066, 1209, 1361, 1522, 1692, 1871, 2059, 2256, 2462, 2677, 2901, 3134, 3376, 3627, 3887, 4156, 4434, 4721, 5017, 5322, 5636, 5959, 6291, 6632, 6982, 7341, 7709, 8086, 8472, 8867, 9271

Offset: 0

Paul Barry, Mar 15 2003

Binomial transform of (1, 8, 9, 0, 0, 0, ...).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjić, Two Enumerative Functions
Richard J. Mathar, Bivariate generating functions enumerating non-bonding dominoes on rectangular boards, arXiv:2404.18806 (2024) Table 2.
Richard J. Mathar, Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards, arXiv:2404.18806 [math.CO], 2024. See p. 7.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A050509, A051682, A081268.

Cf. A220083 for a list of numbers of the form n*P(s,n)-(n-1)*P(s,n-1), where P(s,n) is the n-th polygonal number with s sides.

[(9*n^2 + 7*n + 2)/2: n in [0..50]]; // Vincenzo Librandi, Aug 14 2014

LinearRecurrence[{3,-3,1},{1,9,26},50] (* Harvey P. Dale, Aug 13 2014 *)
CoefficientList[Series[(1 + 6 x + 2 x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 14 2014 *)

a(n)=(9*n^2+7*n+2)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = C(n, 0) + 8*C(n, 1) + 9*C(n, 2).
a(n) = (9*n^2 + 7*n + 2)/2.
G.f.: (1 + 6*x + 2*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), for n > 2. a(n) = right term in M^n * [1 1 1], where M = the 3 X 3 matrix [1 0 0 / 3 1 0 / 5 3 1]. M^n * [1 1 1] = [1 3n+1 a(n)]. - Gary W. Adamson, Dec 22 2004
a(n) = 9*n + a(n-1) - 1 with n > 0, a(0)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = (n+1)*A000326(n+1) - (n)*A000326(n). - Bruno Berselli, Dec 10 2012
a(n) = A050509(n) - A050509(n-1). - Bill McEachen, Nov 01 2020
E.g.f.: exp(x)*(2 + 16*x + 9*x^2)/2. - Stefano Spezia, Dec 25 2022

A081271 Vertical of triangular spiral in A051682.

Original entry on oeis.org

1, 13, 34, 64, 103, 151, 208, 274, 349, 433, 526, 628, 739, 859, 988, 1126, 1273, 1429, 1594, 1768, 1951, 2143, 2344, 2554, 2773, 3001, 3238, 3484, 3739, 4003, 4276, 4558, 4849, 5149, 5458, 5776, 6103, 6439, 6784, 7138, 7501, 7873, 8254, 8644, 9043, 9451

Offset: 0

Paul Barry, Mar 15 2003

Lies to the right of the y-axis of the triangle.

Binomial transform of (1, 12, 9, 0, 0, 0, ...).

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A062741, A283394 (see Crossrefs section).

LinearRecurrence[{3,-3,1},{1,13,34},50] (* Harvey P. Dale, Aug 30 2025 *)

a(n)=(9*n^2+15*n+2)/2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (1 + 10*x - 2*x^2)/(1 - x)^3.
a(n) = binomial(n,0) + 12*binomial(n,1) + 9*binomial(n,2).
a(n) = (9*n^2 + 15*n + 2)/2.
a(0) = 1, a(n) = a(n-1) + 9*n + 3 for n > 0 - Gerald McGarvey, Aug 18 2004
From Elmo R. Oliveira, Oct 25 2024: (Start)
E.g.f.: exp(x)*(1 + 12*x + 9*x^2/2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A210981 A062725 and positive terms of A051682 interleaved.

Original entry on oeis.org

0, 1, 7, 11, 23, 30, 48, 58, 82, 95, 125, 141, 177, 196, 238, 260, 308, 333, 387, 415, 475, 506, 572, 606, 678, 715, 793, 833, 917, 960, 1050, 1096, 1192, 1241, 1343, 1395, 1503, 1558, 1672, 1730, 1850, 1911, 2037, 2101, 2233, 2300, 2438, 2508, 2652, 2725, 2875, 2951, 3107, 3186, 3348

Offset: 0

Omar E. Pol, Aug 03 2012

Vertex number of a square spiral similar to A195160.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Members of this family are A093005, A210977, A006578, A210978, A181995, this sequence, A210982.

Cf. A051682, A062725, A195160.

LinearRecurrence[{1,2,-2,-1,1},{0,1,7,11,23},70] (* Harvey P. Dale, Jun 29 2023 *)

G.f.: -x*(1+6*x+2*x^2) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Aug 07 2012

a(n) = ( 18*n^2+14*n-5+(6*n+5)*(-1)^n )/16. - Luce ETIENNE, Oct 14 2014

A244648 Decimal expansion of the sum of the reciprocals of the hendecagonal numbers (A051682).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 03 2014

			1.195434116529627974352499234698499354884682627084658062386021603017...

Wikipedia, Polygonal number

Cf. A000038, A013661, A244639, A244644, A244645, A244646, A244647, A244649.

RealDigits[ Sum[2/(9n^2 - 7n), {n, 1 , Infinity}], 10, 111][[1]]

Sum_{n=1..infinity} 2/(9n^2 - 7n).

Equals (5*log(3) + Pi*cot(2*Pi/9) - 4*cos(2*Pi/9)*log(cos(Pi/18)) + 4*cos(Pi/9)*log(sin(2*Pi/9)) - 4*log(sin(Pi/9))*sin(Pi/18))/7. - Vaclav Kotesovec, Jul 04 2014

A081268 Diagonal of triangular spiral in A051682.

Original entry on oeis.org

1, 12, 32, 61, 99, 146, 202, 267, 341, 424, 516, 617, 727, 846, 974, 1111, 1257, 1412, 1576, 1749, 1931, 2122, 2322, 2531, 2749, 2976, 3212, 3457, 3711, 3974, 4246, 4527, 4817, 5116, 5424, 5741, 6067, 6402, 6746, 7099, 7461, 7832, 8212, 8601, 8999, 9406

Offset: 0

Paul Barry, Mar 15 2003

			a(1) = 9*1 +  1 + 2 = 12.
a(2) = 9*2 + 12 + 2 = 32.
a(3) = 9*3 + 32 + 2 = 61.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A064226, A081267.

a(n)=(9*n^2+13*n+2)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = C(n,0) + 11*C(n,1) + 9*C(n,2); binomial transform of (1, 11, 9, 0, 0, 0, ...).
a(n) = (9*n^2 + 13*n + 2)/2.
G.f.: (1 + 9*x - x^2)/(1-x)^3.
a(n) = 9*n + a(n-1) + 2 (with a(0)=1). - Vincenzo Librandi, Aug 08 2010
From Elmo R. Oliveira, Nov 16 2024: (Start)
E.g.f.: exp(x)*(9*x^2 + 22*x + 2)/2.
a(n) = A064226(n) - 2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A081272 Downward vertical of triangular spiral in A051682.

Original entry on oeis.org

1, 25, 85, 181, 313, 481, 685, 925, 1201, 1513, 1861, 2245, 2665, 3121, 3613, 4141, 4705, 5305, 5941, 6613, 7321, 8065, 8845, 9661, 10513, 11401, 12325, 13285, 14281, 15313, 16381, 17485, 18625, 19801, 21013, 22261, 23545, 24865, 26221, 27613, 29041, 30505

Offset: 0

Paul Barry, Mar 15 2003

Reflection of A081271 in the horizontal A051682.
Binomial transform of (1, 24, 36, 0, 0, 0, .....).
One of the six verticals of a triangular spiral which starts with 1 (see the link). Other verticals are A060544, A081589, A080855, A157889, A038764. - Yuriy Sibirmovsky, Sep 18 2016.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kaie Kubjas, Luca Sodomaco, and Elias Tsigaridas, Exact solutions in low-rank approximation with zeros, arXiv:2010.15636 [math.AG], 2020.
Yuriy Sibirmovsky, Six verticals of the triangular spiral.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051682.

Cf. A062741, A060544, A081589, A080855, A157889, A038764.

Table[n^2 + (n + 1)^2, {n, 0, 300, 3}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 25, 85}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2012 *)
Table[n^2 + (n + 1)^2, {n, 0, 150, 3}] (* Vincenzo Librandi, Aug 07 2013 *)

x='x+O('x^99); Vec((1+22*x+13*x^2)/(1-x)^3) \\ Altug Alkan, Sep 18 2016

a(n) = C(n, 0) + 24*C(n, 1) + 36*C(n, 2).
a(n) = 18*n^2 + 6*n + 1.
G.f.: (1 + 22*x + 13*x^2)/(1 - x)^3.
E.g.f.: exp(x)*(1 + 24*x + 18*x^2). - Stefano Spezia, Mar 07 2023

A081270 Diagonal of triangular spiral in A051682.

Original entry on oeis.org

3, 16, 38, 69, 109, 158, 216, 283, 359, 444, 538, 641, 753, 874, 1004, 1143, 1291, 1448, 1614, 1789, 1973, 2166, 2368, 2579, 2799, 3028, 3266, 3513, 3769, 4034, 4308, 4591, 4883, 5184, 5494, 5813, 6141, 6478, 6824, 7179, 7543, 7916, 8298, 8689, 9089, 9498, 9916

Offset: 0

Paul Barry, Mar 15 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hacène Belbachir, Toufik Djellal, Jean-Gabriel Luque, On the self-convolution of generalized Fibonacci numbers, arXiv:1703.00323 [math.CO], 2017.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A064226, A081268.

[(9*n^2+17*n+6)/2: n in [0..50]]; // Vincenzo Librandi, Jul 08 2012

CoefficientList[Series[(3+7x-x^2)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 08 2012 *)

a(n)=(9*n^2+17*n+6)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A064226(n) + 2*n.
a(n) = 3*binomial(n,0) + 13*binomial(n,1) + 9*binomial(n,2); binomial transform of (3, 13, 9, 0, 0, 0, ...).
a(n) = (9*n^2 + 17*n + 6)/2.
G.f.: (3 + 7*x - x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 08 2012
E.g.f.: exp(x)*(6 + 26*x + 9*x^2)/2. - Elmo R. Oliveira, Nov 13 2024

A081275 Shallow diagonal of triangular spiral in A051682.

Original entry on oeis.org

1, 31, 97, 199, 337, 511, 721, 967, 1249, 1567, 1921, 2311, 2737, 3199, 3697, 4231, 4801, 5407, 6049, 6727, 7441, 8191, 8977, 9799, 10657, 11551, 12481, 13447, 14449, 15487, 16561, 17671, 18817, 19999, 21217, 22471, 23761, 25087, 26449, 27847, 29281, 30751, 32257

Offset: 0

Paul Barry, Mar 15 2003

Reflection of A060544 in the horizontal A051682.

Binomial transform of (1, 30, 36, 0, 0, 0, ...).

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

Cf. A051682, A060544, A092296.

Table[30Binomial[n,1]+36Binomial[n,2]+1,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{1,31,97},40] (* Harvey P. Dale, Jun 30 2011 *)
CoefficientList[Series[(1 + 28 x + 7 x^2) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 07 2013 *)

a(n)=18*n^2+12*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = C(n,0) + 30*C(n,1) + 36*C(n,2).
a(n) = 18*n^2 + 12*n + 1.
G.f.: (1 + 28*x + 7*x^2)/(1-x)^3.
a(0)=1, a(1)=31, a(2)=97, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jun 30 2011
E.g.f.: exp(x)*(1 + 30*x + 18*x^2). - Elmo R. Oliveira, Nov 13 2024

A280071 Indices of 11-gonal numbers (A051682) that are also centered 11-gonal numbers (A060544).

Original entry on oeis.org

1, 12, 232, 4621, 92181, 1838992, 36687652, 731914041, 14601593161, 291299949172, 5811397390272, 115936647856261, 2312921559734941, 46142494546842552, 920536969377116092, 18364596892995479281, 366371400890532469521, 7309063420917653911132

Offset: 1

Colin Barker, Dec 25 2016

Also positive integers x in the solutions to 9*x^2 - 11*y^2 - 7*x + 11*y - 2 = 0, the corresponding values of y being A280072.

			12 is in the sequence because the 12th 11-gonal number is 606, which is also the 11th centered 11-gonal number.

Colin Barker, Table of n, a(n) for n = 1..750
Index entries for linear recurrences with constant coefficients, signature (21,-21,1).

Cf. A051682, A060544, A131215, A280072

LinearRecurrence[{21,-21,1},{1,12,232},20] (* Harvey P. Dale, May 27 2025 *)

Vec(x*(1 - 9*x + x^2) / ((1 - x)*(1 - 20*x + x^2)) + O(x^30))

a(n) = (14 + (11-3*sqrt(11))*(10+3*sqrt(11))^n + (10+3*sqrt(11))^(-n)*(11+3*sqrt(11)))/36.
a(n) = 21*a(n-1) - 21*a(n-2) + a(n-3) for n>3.
G.f.: x*(1 - 9*x + x^2) / ((1 - x)*(1 - 20*x + x^2)).

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A081266 Staggered diagonal of triangular spiral in A051682.

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A081267 Diagonal of triangular spiral in A051682.

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A081271 Vertical of triangular spiral in A051682.

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A210981 A062725 and positive terms of A051682 interleaved.

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A244648 Decimal expansion of the sum of the reciprocals of the hendecagonal numbers (A051682).

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A081268 Diagonal of triangular spiral in A051682.

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A081272 Downward vertical of triangular spiral in A051682.

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A081270 Diagonal of triangular spiral in A051682.