A017209 -id:A017209 - cp's OEIS frontend

A304507 a(n) = 5(n+1)(9*n+4).

20, 130, 330, 620, 1000, 1470, 2030, 2680, 3420, 4250, 5170, 6180, 7280, 8470, 9750, 11120, 12580, 14130, 15770, 17500, 19320, 21230, 23230, 25320, 27500, 29770, 32130, 34580, 37120, 39750, 42470, 45280, 48180, 51170, 54250, 57420, 60680, 64030, 67470, 71000, 74620

Offset: 0

Emeric Deutsch, May 14 2018

The first Zagreb index of the single-defect 5-gonal nanocone CNC(5,n) (see definition in the Doslic et al. reference, p. 27).
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
The M-polynomial of CNC(5,n) is M(CNC(5,n); x,y) = 5*x^2*y^2 + 10*n*x^2*y^3 + 5*n*(3*n+1)*x^3*y^3/2.
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.

Colin Barker, Table of n, a(n) for n = 0..1000
Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. 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. A008587, A011862, A017209, A062708, A108579, A304508.

List([0..50], n -> 5*(n+1)*(9*n+4)); # Muniru A Asiru, May 15 2018

Maple
```
seq((5*(n+1))*(9*n+4), n = 0 .. 40);
```

Array[5 (# + 1) (9 # + 4) &, 41, 0] (* or *)
LinearRecurrence[{3, -3, 1}, {20, 130, 330}, 41] (* or *)
CoefficientList[Series[10 (2 + 7 x)/(1 - x)^3, {x, 0, 40}], x] (* Michael De Vlieger, May 14 2018 *)

a(n) = 5*(n+1)*(9*n+4); \\ Altug Alkan, May 14 2018

Vec(10*(2 + 7*x) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 14 2018

From Colin Barker, May 14 2018: (Start)
G.f.: 10*(2 + 7*x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
a(n) = 10*A062708(n+1) for n >= 0. - Robert G. Wilson v, May 14 2018
a(n) = 5*A011862(9*n+7) = 5*A108579(6*n+7). - Bruno Berselli, May 15 2018
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: 5*exp(x)*(4 + 22*x + 9*x^2).
a(n) = 5*A017209(n)*A008587(n+1). (End)

A358509 Sum of decimal digits of (3^n - 1)/2 (A003462).

Original entry on oeis.org

0, 1, 4, 4, 4, 4, 13, 13, 13, 22, 22, 31, 22, 31, 31, 31, 22, 31, 31, 31, 31, 31, 49, 49, 40, 40, 49, 67, 58, 58, 58, 76, 58, 76, 85, 85, 85, 85, 94, 85, 85, 94, 103, 103, 85, 94, 103, 112, 103, 112, 130, 130, 94, 121, 112, 112, 121, 103, 103, 121, 112, 121, 121, 139, 121, 148, 121, 157, 157, 157, 157, 175, 157, 157

Offset: 0

Paul Curtz, Nov 20 2022

a(n) == 4 (mod 9) for n >= 2. - Robert Israel, Nov 21 2022

			For n=5, (3^n - 1)/2 = 121 so that a(5) = 1+2+1 = 4.

Michel Marcus, Table of n, a(n) for n = 0..5000

Cf. A003462, A007953, A017209.

Cf. A004166 (of 3^n).

seq(convert(convert((3^n-1)/2,base,10),`+`),n=0..100); # Robert Israel, Nov 21 2022

a[n_] := Total[IntegerDigits[(3^n - 1)/2]]; Array[a, 100, 0] (* Amiram Eldar, Nov 20 2022 *)

a(n) = sumdigits((3^n - 1)/2); \\ Michel Marcus, Nov 20 2022

def A358509(n): return sum(map(int,str((3**n-1)>>1))) # Chai Wah Wu, Nov 21 2022

a(n) = A007953(A003462(n)).

A360962 Square array T(n,k) = k((3+6n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.

Original entry on oeis.org

0, 0, 1, 0, 4, 5, 0, 7, 17, 12, 0, 10, 29, 39, 22, 0, 13, 41, 66, 70, 35, 0, 16, 53, 93, 118, 110, 51, 0, 19, 65, 120, 166, 185, 159, 70, 0, 22, 77, 147, 214, 260, 267, 217, 92, 0, 25, 89, 174, 262, 335, 375, 364, 284, 117, 0, 28, 101, 201, 310, 410, 483, 511, 476, 360, 145

Offset: 0

Paul Curtz, Feb 27 2023

The main diagonal is A024394.

The antidiagonals sums are A000537.

			The rows are:
  0  1  5  12  22  35  51  70 ... = A000326
  0  4 17  39  70 110 159 217 ... = A022266
  0  7 29  66 118 185 267 364 ... = A022272
  0 10 41  93 166 260 375 511 ... = A022278
  0 13 53 120 214 335 483 658 ... = A022284
  ... .
Columns: A000004, A016777, A017581, A154266=3*A017209, 2*A348845, 5*A161447, 3*A158057(n+1), ... (coefficients from A026741).
Difference between two consecutive rows are: A033428.
This square array read by antidiagonals leads to the triangle
  0
  0  1
  0  4  5
  0  7 17 12
  0 10 29 39  22
  0 13 41 66  70  35
  0 16 53 93 118 110 51
  ... .

Cf. A000326, A000537, A022266, A022272, A022278, A022284, A024394.
Cf. A033428.
Cf. A000004, A016777, A017209, A017581, A026741, A154266, A158057, A161447, A348845.

T:= (n,k)-> k*(k*(3+6*n)-1)/2:
seq(seq(T(d-k,k), k=0..d), d=0..10);  # Alois P. Heinz, Feb 28 2023

T[n_, k_] := ((6*n + 3)*k - 1)*k/2; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 27 2023 *)

T(n,k) = k*((3+6*n)*k-1)/2; \\ Michel Marcus, Feb 27 2023

Take successively sequences n*(3*n-1)/2, n*(9*n-1)/2, n*(15*n-1)/2, n*(21*n-1)/2, ... listed in the EXAMPLE section.
From Stefano Spezia, Feb 21 2024: (Start)
G.f.: y*(1 + 2*y + x*(2 + y))/((1 - x)^2*(1 - y)^3).
E.g.f.: exp(x+y)*y*(2 + 3*y + 6*x*(1 + y))/2. (End)

A017211 a(n) = (9*n + 4)^3.

Original entry on oeis.org

64, 2197, 10648, 29791, 64000, 117649, 195112, 300763, 438976, 614125, 830584, 1092727, 1404928, 1771561, 2197000, 2685619, 3241792, 3869893, 4574296, 5359375, 6229504, 7189057, 8242408

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578 (n^3), A017209 (9*n+4).

[(9*n+4)^3: n in [0..35]]; // Vincenzo Librandi, Jul 25 2011

A017211:=n->(9*n+4)^3; seq(A017211(n), n=0..35); # Wesley Ivan Hurt, Jan 22 2014

Table[(9n+4)^3, {n, 0, 35}] (* Wesley Ivan Hurt, Jan 22 2014 *)
LinearRecurrence[{4,-6,4,-1},{64,2197,10648,29791},30] (* Harvey P. Dale, Jan 21 2017 *)

a(n) = A017209(n)^3 = A000578(A017209(n)). - Wesley Ivan Hurt, Jan 22 2014
G.f.: (64+1941*x+2244*x^2+125*x^3)/(x-1)^4. - R. J. Mathar, Jul 14 2016
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Wesley Ivan Hurt, May 26 2024

A017220 a(n) = (9*n + 4)^12.

Original entry on oeis.org

16777216, 23298085122481, 12855002631049216, 787662783788549761, 16777216000000000000, 191581231380566414401, 1449225352009601191936, 8182718904632857144561, 37133262473195501387776, 142241757136172119140625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Cf. A008456 (n^12), A017209 (9*n+4).

[(9*n+4)^12: n in [0..15]]; // Vincenzo Librandi, Jul 24 2011

A168401 a(n) = 4 + 9*floor(n/2).

Original entry on oeis.org

4, 13, 13, 22, 22, 31, 31, 40, 40, 49, 49, 58, 58, 67, 67, 76, 76, 85, 85, 94, 94, 103, 103, 112, 112, 121, 121, 130, 130, 139, 139, 148, 148, 157, 157, 166, 166, 175, 175, 184, 184, 193, 193, 202, 202, 211, 211, 220, 220, 229, 229, 238, 238, 247, 247, 256, 256

Offset: 1

Vincenzo Librandi, Nov 25 2009

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

Cf. A017209.

[4+9*Floor(n/2): n in [1..60]]; // Bruno Berselli, Sep 18 2013

Table[4 + 9 Floor[n/2], {n, 60}] (* Vincenzo Librandi, Sep 18 2013 *)

a(n)=n\2*9 + 4 \\ Charles R Greathouse IV, Jul 21 2016

a(n) = 9*n - a(n-1) - 1, with n > 1, a(1)=4.
G.f.: x*(4 + 9*x - 4*x^2)/((1+x)*(x-1)^2). - Vincenzo Librandi, Sep 18 2013
a(n) = a(n-1) + a(n-2) - a(n-3). - Vincenzo Librandi, Sep 18 2013
E.g.f.: (1/4)*(9 - 16*exp(x) + (7 + 18*x)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 21 2016

New definition by Vincenzo Librandi Sep 18 2013

A301622 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 4.

Original entry on oeis.org

13, 31, 49, 67, 103, 121, 139, 157, 193, 211, 229, 247, 283, 301, 319, 337, 373, 391, 409, 427, 463, 481, 499, 517, 553, 571, 589, 607, 643, 661, 679, 697, 733, 751, 769, 787, 823, 841, 859, 877, 913, 931, 949, 967, 1003, 1021, 1039, 1057, 1093, 1111

Offset: 1

Gary Croft, Mar 24 2018

Numbers == {13, 31, 49, 67} mod 90 with additive sum sequence 13{+18+18+18+36} {repeat ...}. Includes all prime numbers > 5 with digital root 4.

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

Intersection of A007775 and A017209.

Filtered(Filtered([1..1200],n->n mod 2 <> 0 and n mod 3 <> 0 and n mod 5 <> 0),i->i-9*Int((i-1)/9)=4); # Muniru A Asiru, Apr 22 2018

Rest@ CoefficientList[Series[x (13 + 18 x + 18 x^2 + 18 x^3 + 23 x^4)/((1 - x)^2*(1 + x) (1 + x^2)), {x, 0, 50}], x] (* Michael De Vlieger, Apr 21 2018 *)
LinearRecurrence[{1,0,0,1,-1},{13,31,49,67,103},50] (* Harvey P. Dale, May 11 2019 *)

Vec(x*(13 + 18*x + 18*x^2 + 18*x^3 + 23*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^60)) \\ Colin Barker, Mar 25 2018

Numbers == {13, 31, 49, 67} mod 90.
From Colin Barker, Mar 25 2018: (Start)
G.f.: x*(13 + 18*x + 18*x^2 + 18*x^3 + 23*x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
(End)

Last term corrected by Colin Barker, Mar 25 2018

A350521 a(n) = 18*n + 4.

Original entry on oeis.org

4, 22, 40, 58, 76, 94, 112, 130, 148, 166, 184, 202, 220, 238, 256, 274, 292, 310, 328, 346, 364, 382, 400, 418, 436, 454, 472, 490, 508, 526, 544, 562, 580, 598, 616, 634, 652, 670, 688, 706, 724, 742, 760, 778, 796, 814, 832, 850, 868, 886, 904, 922, 940, 958

Offset: 0

Omar E. Pol, Jan 03 2022

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

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

Bisection of A017209.

Cf. A006370, A008600, A016921, A017185, A242215, A298035.

GAP
```
List([0..53], n-> 18*n+4)
```
Magma
```
[18*n+4: n in [0..53]];
```
Maple
```
seq(18*n+4, n=0..53);
```
Mathematica
```
Table[18n+4, {n, 0, 53}]
```
Maxima
```
makelist(18*n+4, n, 0, 53);
```
PARI
```
a(n)=18*n+4
```
Python
```
[18*n+4 for n in range(53)]
```

a(n) = A242215(n) - 1.
a(n) = A298035(n+1) + 1.
From Elmo R. Oliveira, Apr 08 2024: (Start)
G.f.: 2*(2+7*x)/(1-x)^2.
E.g.f.: 2*exp(x)*(2 + 9*x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 2*A017185(n) = A006370(A016921(n)). (End)

A358736 a(n) is the number of appearances of (9*n + 4) in A358509.

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Nov 30 2022

nonn

Is this sequence well-defined? - Joerg Arndt, Dec 12 2022

Cf. A017209, A003462, A358509.

More terms from Michel Marcus, Nov 30 2022

cp's OEIS Frontend

A304507 a(n) = 5*(n+1)*(9*n+4).

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A358509 Sum of decimal digits of (3^n - 1)/2 (A003462).

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A360962 Square array T(n,k) = k*((3+6*n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.

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Maple

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A017211 a(n) = (9*n + 4)^3.

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Magma

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A017220 a(n) = (9*n + 4)^12.

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Magma

A168401 a(n) = 4 + 9*floor(n/2).

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Magma

Mathematica

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Extensions

A301622 Numbers not divisible by 2, 3 or 5 (A007775) with digital root 4.

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A304507 a(n) = 5(n+1)(9*n+4).

A360962 Square array T(n,k) = k((3+6n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.