A062728 -id:A062728 - cp's OEIS frontend

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

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

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.

Original entry on oeis.org

2, 4, 12, 18, 31, 41, 59, 73, 96, 114, 142, 164, 197, 223, 261, 291, 334, 368, 416, 454, 507, 549, 607, 653, 716, 766, 834, 888, 961, 1019, 1097, 1159, 1242, 1308, 1396, 1466, 1559, 1633, 1731, 1809, 1912, 1994, 2102, 2188, 2301, 2391, 2509, 2603, 2726, 2824, 2952

Offset: 0

Bruno Berselli, Oct 25 2011

For an origin of this sequence, see the triangular spiral illustrated in the Links section.

First bisection gives A117625 (without the initial term).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A152832 (by Superseeker).

Cf. sequences related to the triangular spiral: A022266, A022267, A027468, A038764, A045946, A051682, A062708, A062725, A062728, A062741, A064225, A064226, A081266-A081268, A081270-A081272, A081275 [incomplete list].

[(6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1: n in [0..50]];

LinearRecurrence[{1,2,-2,-1,1},{2,4,12,18,31},60] (* Harvey P. Dale, Jun 15 2022 *)

for(n=0, 50, print1((6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1", "));

G.f.: (2+2*x+4*x^2+2*x^3-x^4)/((1+x)^2*(1-x)^3).
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
a(n)-a(-n-1) = A168329(n+1).
a(n)+a(n-1) = A102214(n).
a(2n)-a(2n-1) = A016885(n).
a(2n+1)-a(2n) = A016825(n).

A350522 a(n) = 18*n + 16.

Original entry on oeis.org

16, 34, 52, 70, 88, 106, 124, 142, 160, 178, 196, 214, 232, 250, 268, 286, 304, 322, 340, 358, 376, 394, 412, 430, 448, 466, 484, 502, 520, 538, 556, 574, 592, 610, 628, 646, 664, 682, 700, 718, 736, 754, 772, 790, 808, 826, 844, 862, 880, 898, 916, 934, 952, 970

Offset: 0

Omar E. Pol, Jan 03 2022

Sixth column of A006370 (the Collatz or 3x+1 map) when it is interpreted as a rectangular array with six columns read by rows.

Leo Tavares, Illustration: Triple Hexagonal Rings
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Bisection of A017245.

Cf. A006370, A008588, A008600, A016969, A017257, A062728, A239129.

GAP
```
List([0..53], n-> 18*n+16)
```
Magma
```
[18*n+16: n in [0..53]];
```
Maple
```
seq(18*n+16, n=0..53);
```
Mathematica
```
Table[18n+16, {n, 0, 53}]
```
Maxima
```
makelist(18*n+16, n, 0, 53);
```
PARI
```
a(n)=18*n+16
```
Python
```
[18*n+16 for n in range(53)]
```

a(n) = A239129(n+1) - 1.
From Stefano Spezia, Jan 04 2022: (Start)
O.g.f.: 2*(8 + x)/(1 - x)^2.
E.g.f.: 2*exp(x)*(8 + 9*x).
a(n) = 2*a(n-1) - a(n-2) for n > 1. (End)
a(n) = 3*A008588(n+1) - 2. - Leo Tavares, Sep 14 2022
From Elmo R. Oliveira, Apr 12 2024: (Start)
a(n) = 2*A017257(n) = A006370(A016969(n)).
a(n) = 2*(A062728(n+1) - A062728(n)). (End)

A257488 Triangle, read by rows, T(n,k) = kSum_{i=0..n-k} C(2i+2k,i)C(n-i-1,k-1)/(i+k) for 1 <= k <= n.

Original entry on oeis.org

1, 3, 1, 8, 6, 1, 22, 25, 9, 1, 64, 92, 51, 12, 1, 196, 324, 237, 86, 15, 1, 625, 1128, 996, 484, 130, 18, 1, 2055, 3934, 3966, 2377, 860, 183, 21, 1, 6917, 13812, 15335, 10744, 4845, 1392, 245, 24, 1, 23713, 48884, 58359, 46068, 24603, 8859, 2107, 316, 27, 1

Offset: 1

Vladimir Kruchinin, Apr 26 2015

			Triangle starts:
1;
3,   1;
8,   6,  1;
22, 25,  9,  1;
64, 92, 51, 12, 1;

Cf. A014138.

Flatten@ Table[k Sum[Binomial[2 i + 2 k, i] Binomial[n - i - 1, k - 1]/(i + k), {i, 0, n - k}], {n, 10}, {k, n}] (* Michael De Vlieger, Apr 27 2015 *)

T(n,k):=k*sum((binomial(2*i+2*k,i)*binomial(n-i-1,k-1))/(i+k),i,0,n-k);

T(n,k)=k*sum(i=0,n-k,(binomial(2*i+2*k,i)*binomial(n-i-1,k-1))/(i+k))
for(n=1,10,for(k=1,n,print1(T(n,k),", "))) \\ Derek Orr, Apr 27 2015

G.f.: 1/(1-(C(x)-1)/(1-x)*y)-1, where C(x) is g.f. of Catalan numbers (A000108).
T(n,n-1) = 3*(n-1) for n > 1. - Derek Orr, Apr 27 2015
T(n,n-2) = A062728(n-2) for n > 2. - Derek Orr, Apr 27 2015
T(n,1) = A014138(n). - Derek Orr, Apr 27 2015

A341768 a(n) = n * (binomial(n,2) - 2).

Original entry on oeis.org

0, -2, -2, 3, 16, 40, 78, 133, 208, 306, 430, 583, 768, 988, 1246, 1545, 1888, 2278, 2718, 3211, 3760, 4368, 5038, 5773, 6576, 7450, 8398, 9423, 10528, 11716, 12990, 14353, 15808, 17358, 19006, 20755, 22608, 24568, 26638, 28821, 31120, 33538, 36078, 38743, 41536, 44460

Offset: 0

Ilya Gutkovskiy, Feb 19 2021

The n-th second n-gonal number.

			a(7) = A147875(7) = A000566(-7) = 133.

Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005449, A005564, A006002, A014105, A033954, A034856, A045944, A060354, A062728, A135705, A147875, A179986, A292551.

Table[n (Binomial[n, 2] - 2), {n, 0, 45}]
LinearRecurrence[{4, -6, 4, -1}, {0, -2, -2, 3}, 46]
CoefficientList[Series[-x (2 - 6 x + x^2)/(1 - x)^4, {x, 0, 45}], x]

G.f.: -x*(2 - 6*x + x^2)/(1 - x)^4.
E.g.f.: -exp(x)*x*(4 - 2*x - x^2)/2.
a(n) = n^2*(n - 1)/2 - 2*n.

cp's OEIS Frontend

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

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Formula

A198392 a(n) = (6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16 + 1.

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Magma

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A350522 a(n) = 18*n + 16.

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GAP

Magma

Maple

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Python

Formula

A257488 Triangle, read by rows, T(n,k) = k*Sum_{i=0..n-k} C(2*i+2*k,i)*C(n-i-1,k-1)/(i+k) for 1 <= k <= n.

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Author

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Examples

Crossrefs

Programs

Mathematica

Maxima

PARI

Formula

A341768 a(n) = n * (binomial(n,2) - 2).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.

A257488 Triangle, read by rows, T(n,k) = kSum_{i=0..n-k} C(2i+2k,i)C(n-i-1,k-1)/(i+k) for 1 <= k <= n.