A016933 -id:A016933 - cp's OEIS frontend

A304506 a(n) = 2(3n+1)(9n+8).

16, 136, 364, 700, 1144, 1696, 2356, 3124, 4000, 4984, 6076, 7276, 8584, 10000, 11524, 13156, 14896, 16744, 18700, 20764, 22936, 25216, 27604, 30100, 32704, 35416, 38236, 41164, 44200, 47344, 50596, 53956, 57424, 61000, 64684, 68476, 72376, 76384, 80500, 84724, 89056

Offset: 0

Emeric Deutsch, May 14 2018

a(n) is the second Zagreb index of the single-defect 4-gonal nanocone CNC(4,n) (see definition in the Doslic et al. reference, p. 27).
The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph.
The M-polynomial of CNC(4,n) is M(CNC(4,n);x,y) = 4*x^2*y^2 + 8*n*x^2*y^3 + 2*n*(3*n+1)*x^3*y^3.
More generally, the M-polynomial of CNC(k,n) is M(CNC(k,n); x,y) = k*x^2*y^2 + 2*k*n*x^2*y^3 + k*n*(3*n + 1)*x^3*y^3/2.
6*a(n) + 25 is a square. - Bruno Berselli, May 14 2018

Colin Barker, Table of n, a(n) for n = 0..1000
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian Journal of Mathematical Chemistry, Vol. 6, No. 2, 2015, pp. 93-102.
T. Doslic and M. Saheli, Augmented eccentric connectivity index of single-defect nanocones, Journal of Mathematical Nanoscience, Vol. 1, No. 1, 2011, pp. 25-31.
A. Khaksar, M. Ghorbani, and H. R. Maimani, On atom bond connectivity and GA indices of nanocones, Optoelectronics and Advanced Materials - Rapid Communications, Vol. 4, No. 11, 2010, pp. 1868-1870.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016933, A017257, A304505.

List([0..50],n->2*(3*n+1)*(9*n+8)); # Muniru A Asiru, May 14 2018

Maple
```
seq((2*(9*n+8))*(3*n+1), n = 0 .. 40);
```

Table[2(3n+1)(9n+8),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{16,136,364},50] (* Harvey P. Dale, Aug 15 2022 *)

a(n) = 2*(3*n+1)*(9*n+8); \\ Altug Alkan, May 14 2018

Vec(4*(4 + 22*x + x^2) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 14 2018

From Colin Barker, May 14 2018: (Start)
G.f.: 4*(4 + 22*x + x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: 2*exp(x)*(8 + 60*x + 27*x^2).
a(n) = A016933(n)*A017257(n). (End)

A326728 A(n, k) = n(k - 1)k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.

Original entry on oeis.org

0, 0, -1, 0, -1, -2, 0, -1, -1, -3, 0, -1, 0, 0, -4, 0, -1, 1, 3, 2, -5, 0, -1, 2, 6, 8, 5, -6, 0, -1, 3, 9, 14, 15, 9, -7, 0, -1, 4, 12, 20, 25, 24, 14, -8, 0, -1, 5, 15, 26, 35, 39, 35, 20, -9, 0, -1, 6, 18, 32, 45, 54, 56, 48, 27, -10

Offset: 0

Peter Luschny, Aug 04 2019

A formal extension of the figurative numbers A139600 to negative n.

			[0] 0, -1, -2, -3, -4, -5, -6,  -7,  -8,  -9, -10, ... A001489
[1] 0, -1, -1,  0,  2,  5,  9,  14,  20,  27,  35, ... A080956
[2] 0, -1,  0,  3,  8, 15, 24,  35,  48,  63,  80, ... A067998
[3] 0, -1,  1,  6, 14, 25, 39,  56,  76,  99, 125, ... A095794
[4] 0, -1,  2,  9, 20, 35, 54,  77, 104, 135, 170, ... A014107
[5] 0, -1,  3, 12, 26, 45, 69,  98, 132, 171, 215, ... A326725
[6] 0, -1,  4, 15, 32, 55, 84, 119, 160, 207, 260, ... A270710
[7] 0, -1,  5, 18, 38, 65, 99, 140, 188, 243, 305, ...

Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.

Cf. A001489 (n=0), A080956 (n=1), A067998 (n=2), A095794 (n=3), A014107 (n=4), A326725 (n=5), A270710 (n=6).
Columns include A008585, A016933, A017329.
Cf. A139600.

A := (n, k) -> n*(k - 1)*k/2 - k:
seq(seq(A(n - k, k), k=0..n), n=0..11);

def A326728Row(n):
    x, y = 1, 1
    yield 0
    while True:
        yield -x
        x, y = x + y - n, y - n
for n in range(8):
    R = A326728Row(n)
print([next(R) for _ in range(11)])

A016944 a(n) = (6*n + 2)^12.

Original entry on oeis.org

4096, 68719476736, 56693912375296, 4096000000000000, 95428956661682176, 1152921504606846976, 9065737908494995456, 52654090776777588736, 244140625000000000000, 951166013805414055936, 3226266762397899821056, 9774779120406941925376, 26963771415920784510976

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A016788, A016933, A016934, A016935, A016936, A016937, A016938, A016939, A016940, A016941, A016942, A016943.

Subsequence of A008456.

[(6*n+2)^12: n in [0..20]]; // Vincenzo Librandi, May 05 2011

(6*Range[0,20]+2)^12 (* or *) LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{4096,68719476736,56693912375296,4096000000000000,95428956661682176,1152921504606846976,9065737908494995456,52654090776777588736,244140625000000000000,951166013805414055936,3226266762397899821056,9774779120406941925376,26963771415920784510976},20] (* Harvey P. Dale, Aug 03 2021 *)

From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016933(n)^12 = A016934(n)^6 = A016935(n)^4 = A016936(n)^3 = A016938(n)^2.
a(n) = 2^12*A016788(n).
Sum_{n>=0} 1/a(n) = PolyGamma(11, 1/3)/86890185149644800. (End)

A078689 Continued fraction expansion of e^(1/3).

Original entry on oeis.org

1, 2, 1, 1, 8, 1, 1, 14, 1, 1, 20, 1, 1, 26, 1, 1, 32, 1, 1, 38, 1, 1, 44, 1, 1, 50, 1, 1, 56, 1, 1, 62, 1, 1, 68, 1, 1, 74, 1, 1, 80, 1, 1, 86, 1, 1, 92, 1, 1, 98, 1, 1, 104, 1, 1, 110, 1, 1, 116, 1, 1, 122, 1, 1, 128, 1, 1, 134, 1, 1, 140, 1, 1, 146

Offset: 0

Benoit Cloitre, Dec 17 2002

Thomas J. Osler, A proof of the continued fraction expansion of e^(1/M), Amer. Math. Monthly, 113 (No. 1, 2006), 62-66.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A016933, A058281, A092041.

ContinuedFraction[Exp[1/3], 100] (* Amiram Eldar, May 20 2022 *)

a(3k+1) = 6k+2, otherwise a(i) = 1.
G.f.: -(x^2-x+1)*(x^3-3*x^2-3*x-1) / ((x-1)^2*(x^2+x+1)^2). - Colin Barker, Jun 24 2013
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/18 + log(2)/6. - Amiram Eldar, May 04 2025

A101468 Triangle read by rows: T(n,k)=(n+1-k)(3k+1).

Original entry on oeis.org

1, 2, 4, 3, 8, 7, 4, 12, 14, 10, 5, 16, 21, 20, 13, 6, 20, 28, 30, 26, 16, 7, 24, 35, 40, 39, 32, 19, 8, 28, 42, 50, 52, 48, 38, 22, 9, 32, 49, 60, 65, 64, 57, 44, 25, 10, 36, 56, 70, 78, 80, 76, 66, 50, 28, 11, 40, 63, 80, 91, 96, 95, 88, 75, 56, 31, 12, 44, 70, 90, 104, 112, 114

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 21 2005

The triangle is generated from the product A*B
of the infinite lower triangular matrices A =
1 0 0 0...
1 1 0 0...
1 1 1 0...
1 1 1 1...
... and B =
1 0 0 0...
1 4 0 0...
1 4 7 0...
1 4 7 10...
...
Row sums give pentagonal pyramidal numbers A002411 T(n+0,0)= 1*n=A000027(n) T(n+0,1)= 4*n=A008586(n) T(n+1,2)= 7*n=A008589(n) T(n+2,3)=10*n=A008592(n) ...
so for example T(n+1,n-0)=6*n+2=A016933(n) T(n+1,n-1)=9*n+3=A017197(n) T(n+2,n-1)=12*n+4=A017569(n)
T(n,0)*T(n,1) = A033996(n) (8 times triangular numbers)
T(n,n)*T(n,0) = A000567(n+1) (Octagonal numbers) etc.

			Triangle begins:
1,
2,  4,
3,  8,  7,
4,  12, 14, 10,
5,  16, 21, 20, 13,
6,  20, 28, 30, 26, 16,
7,  24, 35, 40, 39, 32, 19,
8,  28, 42, 50, 52, 48, 38, 22,
9,  32, 49, 60, 65, 64, 57, 44, 25,
10, 36, 56, 70, 78, 80, 76, 66, 50, 28,
11, 40, 63, 80, 91, 96, 95, 88, 75, 56, 31, etc.
[_Bruno Berselli_, Feb 10 2014]

Cf. A095871 (product B*A), A002411.

t[n_, k_] := If[n < k, 0, (3*k + 1)*(n - k + 1)]; Flatten[ Table[ t[n, k], {n, 0, 11}, {k, 0, n}]] (* Robert G. Wilson v, Jan 21 2005 *)

T(n,k)=if(k>n,0,(n-k+1)*(3*k+1)) for(i=0,10, for(j=0,i,print1(T(i,j),", "));print())

A141431 Triangle T(n,k) = (k-1)(3n-k+1), read by rows.

Original entry on oeis.org

0, 0, 5, 0, 8, 14, 0, 11, 20, 27, 0, 14, 26, 36, 44, 0, 17, 32, 45, 56, 65, 0, 20, 38, 54, 68, 80, 90, 0, 23, 44, 63, 80, 95, 108, 119, 0, 26, 50, 72, 92, 110, 126, 140, 152, 0, 29, 56, 81, 104, 125, 144, 161, 176, 189, 0, 32, 62, 90, 116, 140, 162, 182, 200, 216, 230, 0, 35, 68, 99, 128, 155, 180, 203, 224, 243, 260, 275

Offset: 1

Roger L. Bagula and Gary W. Adamson, Aug 06 2008

			Triangle begins as:
  0;
  0,  5;
  0,  8, 14;
  0, 11, 20, 27;
  0, 14, 26, 36,  44;
  0, 17, 32, 45,  56,  65;
  0, 20, 38, 54,  68,  80,  90;
  0, 23, 44, 63,  80,  95, 108, 119;
  0, 26, 50, 72,  92, 110, 126, 140, 152;
  0, 29, 56, 81, 104, 125, 144, 161, 176, 189;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Columns: A016789 (k=2), A016933 (k=3), A008591 (k=4).

Cf. A245301 (row sums).

[(k-1)*(3*n-k+1): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 31 2021

A141431 := proc(n,k)
        (k-1)*(3*n-k+1) ;
end proc:
seq(seq(A141431(n,k),k=1..n),n=1..14) ; # R. J. Mathar, Nov 10 2011

Table[(k-1)*(3*n-k+1), {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 31 2021 *)

flatten([[(k-1)*(3*n-k+1) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 31 2021

G.f.: Sum_{n>=0} Sum_{k>=0} T(n,k)*x^n*y^k = y^2*x*(x*y-4*y+x+2)/((1-y)^3*(1-x)^2). - R. J. Mathar, Nov 27 2015. x and y swapped to align with standard, 19 Feb 2020

Sum_{k=1..n} T(n, k) = (n-1)*n*(7*n+1)/6 = A245301(n-1). - G. C. Greubel, Mar 31 2021

More terms added by G. C. Greubel, Mar 31 2021

A163979 a(n) = n*(n-1) + A144437(n+2).

Original entry on oeis.org

1, 3, 5, 7, 15, 23, 31, 45, 59, 73, 93, 113, 133, 159, 185, 211, 243, 275, 307, 345, 383, 421, 465, 509, 553, 603, 653, 703, 759, 815, 871, 933, 995, 1057, 1125, 1193, 1261, 1335, 1409, 1483, 1563, 1643, 1723, 1809, 1895, 1981, 2073, 2165, 2257, 2355, 2453, 2551

Offset: 0

Paul Curtz, Aug 07 2009

First differences are 2, 2, 2, 8, 8, 8, 14, 14, 14, 20, 20, 20,... (triplicated A016933).

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

LinearRecurrence[{2,-1,1,-2,1},{1,3,5,7,15},60]  (* or *) CoefficientList[ Series[-(1+x+5x^4-x^3)/((1+x+x^2)(x-1)^3), {x,0,60}],x]  (* Harvey P. Dale, Apr 20 2011 *)

x='x+O('x^50); Vec((1+x-x^3+5*x^4)/((1+x+x^2)*(1-x)^3)) \\ G. C. Greubel, Aug 24 2017

a(n) = A002378(n-1) + A144437(n+2).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
G.f.: (1 +x -x^3 +5*x^4)/( (1 +x +x^2)*(1 -x)^3 ).
E.g.f.: (1/3)*((7+3*x^2)*exp(x) - 4*exp(-x/2)*cos(sqrt(3)*x/2)). - G. C. Greubel, Aug 24 2017

Edited and extended by R. J. Mathar, Aug 12 2009

A215898 a(4n) = 1+4n, a(1+4n) = -2-6n, a(2+4n) = 4+6n, a(3+4n) = -3-4n.

Original entry on oeis.org

1, -2, 4, -3, 5, -8, 10, -7, 9, -14, 16, -11, 13, -20, 22, -15, 17, -26, 28, -19, 21, -32, 34, -23, 25, -38, 40, -27, 29, -44, 46, -31, 33, -50, 52, -35, 37, -56, 58, -39, 41, -62, 64, -43, 45, -68, 70, -47, 49, -74, 76, -51, 53, -80, 82, -55, 57, -86, 88, -59

Offset: 0

Paul Curtz, Aug 25 2012

A permutation of A047253, numbers that are not divisible by 6.

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

Cf. A016813, A016933, A016957, A004767.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+3*x^2+3*x^4-x^5+x^6)/((1-x)*(1+x+x^2+x^3)^2))); // Bruno Berselli, Sep 06 2012

a[n_ /; Mod[n, 4] == 0] := n+1; a[n_ /; Mod[n, 4] == 1] := -(3n+1)/2; a[n_ /; Mod[n, 4] == 2] := (3n+2)/2; a[n_ /; Mod[n, 4] == 3] := -n; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Sep 03 2012 *)
LinearRecurrence[{-1,-1,-1,1,1,1,1},{1,-2,4,-3,5,-8,10},60] (* Harvey P. Dale, Mar 24 2023 *)

makelist(expand(1+(5-%i^(n*(n+1)))*((2*n+1)*(-1)^n-1)/8), n, 0, 60); /* Bruno Berselli, Sep 07 2012 */

a(n) = 2*a(n-4) - a(n-8).
a(2*n) + a(1+2*n) = -A109613(n)*(-1)^n.
a(3*n) + a(1+3*n) + a(2+3*n) = 3*a(n).
a(4*n) + a(1+4*n) + a(2+4*n) + a(3+4*n) = 0.
a(5*n) + a(1+5*n) + a(2+5*n) + a(3+5*n) + a(4+5*n) = 5*a(n).
From Bruno Berselli, Sep 07 2012: (Start)
G.f.: (1-x+3*x^2+3*x^4-x^5+x^6)/((1-x)*(1+x+x^2+x^3)^2).
a(n) = 1+(5-i^(n*(n+1)))*((2*n+1)*(-1)^n-1)/8, where i=sqrt(-1).
a(2*n) = 1+(5-(-1)^n)*n/2; a(2*n+1) = 1-(5+(-1)^n)*(n+1)/2.
a(n) = a(-n-1) = -a(n-1)-a(n-2)-a(n-3)+a(n-4)+a(n-5)+a(n-6)+a(n-7). (End)

A278814 a(n) = ceiling(sqrt(3n+1)).

Original entry on oeis.org

1, 2, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18

Offset: 0

Mohammad K. Azarian, Nov 28 2016

Cf. A002162, A016777, A016789, A016933, A017569, A032765, A058183, A131033, A007494, A051536, A007559.

PROG(y := [], n := 100, LOOP(IF(n = -1, RETURN y), y := ADJOIN(CEILING(SQRT(1 + 3·n)), y), n := n - 1))

seq(ceil(sqrt(3*k+1)), k=0..100); # Robert Israel, Nov 28 2016

Mathematica
```
Table[Ceiling[Sqrt[3n+1]],{n,0,100}]
```

a(n)=sqrtint(3*n)+1 \\ Charles R Greathouse IV, Nov 29 2016

from math import isqrt
def A278814(n): return 1+isqrt(3*n) # Chai Wah Wu, Jul 28 2022

a(n) = ceiling(sqrt(3n+1)).
From Robert Israel, Nov 28 2016: (Start)
G.f.: (1-x)^(-1)*Sum_{k>=0} (x^(3*k^2)+x^(3*k^2+2*k+1)+x^(3*k^2+4*k+2)).
a(n+1) = a(n)+1 if n is in A032765, otherwise a(n+1) = a(n). (End)
Sum_{n>=0} (-1)^n/a(n) = log(2) (A002162). - Amiram Eldar, Jun 18 2025

A340169 a(n) is the number of strings of length n over the alphabet {a,b,c} with the number of a's divisible by 3, and the number of b's and c's is at most 3.

Original entry on oeis.org

1, 2, 4, 9, 8, 40, 161, 14, 112, 673, 20, 220, 1761, 26, 364, 3641, 32, 544, 6529, 38, 760, 10641, 44, 1012, 16193, 50, 1300, 23401, 56, 1624, 32481, 62, 1984, 43649, 68, 2380, 57121, 74, 2812, 73113, 80, 3280, 91841, 86, 3784, 113521, 92, 4324, 138369, 98

Offset: 0

Diego Ramírez, Feb 25 2021

In regular languages, the empty string is considered and since it meets the conditions, a(0)=1.

			a(3)=9, because the strings are aaa, bbb, bbc, bcb, cbb, bcc, cbc, ccb, ccc.
a(4)=8, because the strings are aaab, aaba, abaa, baaa, aaac, aaca, acaa, caaa.

Rodrigo de Castro, Teoría de la computación, 2004, unilibros.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Diego Ramírez, Census Function for a Regular Language
Index entries for linear recurrences with constant coefficients, signature (0,0,4,0,0,-6,0,0,4,0,0,-1).

Cf. A016933.

L=[""]; for k=1:15 L=[L+"a",L+"b",L+"c"]; c=0; for n=1:length(L) if (mod(count(L(n),"a"),3)==0&& count(L(n),"b")+count(L(l),"c")<=3) c=c+1; end end disp(c) end %show the sequence from 1 to n

A340169[n_] := Switch[Mod[n, 3], 0, 4*n*(n-2)*(n-1)/3 + 1, 1, 2*n, 2, 2*n*(n-1)];
Array[A340169, 100, 0] (* Paolo Xausa, Jul 22 2024 *)

a(n) = 4*a(n-3) - 6*a(n-6) + 4*a(n-9) - a(n-12).
If n == 0 (mod 3), a(n) = 1 + (8/3)*n - 4*n^2 + (4/3)*n^3.
If n == 1 (mod 3), a(n) = 2*n.
If n == 2 (mod 3), a(n) = -2*n + 2*n^2.
G.f.: (1 + 2*x + 4*x^2 + 5*x^3 + 24*x^5 + 131*x^6 - 6*x^7 - 24*x^8 + 79*x^9 + 4*x^10 - 4x^11)/(1 - x^3)^4. - Stefano Spezia, Feb 28 2021

cp's OEIS Frontend

A304506 a(n) = 2*(3*n+1)*(9*n+8).

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A326728 A(n, k) = n*(k - 1)*k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.

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A016944 a(n) = (6*n + 2)^12.

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A078689 Continued fraction expansion of e^(1/3).

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A101468 Triangle read by rows: T(n,k)=(n+1-k)*(3*k+1).

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A141431 Triangle T(n,k) = (k-1)*(3*n-k+1), read by rows.

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A163979 a(n) = n*(n-1) + A144437(n+2).

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A304506 a(n) = 2(3n+1)(9n+8).

A326728 A(n, k) = n(k - 1)k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.

A101468 Triangle read by rows: T(n,k)=(n+1-k)(3k+1).

A141431 Triangle T(n,k) = (k-1)(3n-k+1), read by rows.