A033428 -id:A033428 - cp's OEIS frontend

A174709 Partial sums of floor(n/6).

0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 21, 24, 27, 30, 33, 36, 40, 44, 48, 52, 56, 60, 65, 70, 75, 80, 85, 90, 96, 102, 108, 114, 120, 126, 133, 140, 147, 154, 161, 168, 176, 184, 192

Offset: 0

Mircea Merca, Nov 30 2010

Partial sums of A152467.

			a(7) = floor(0/6) + floor(1/6) + floor(2/6) + floor(3/6) + floor(4/6) + floor(5/6) + floor(6/6) + floor(7/6) = 0 + 0 + 0 + 0 + 0 + 0 + 1 + 1 = 2.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions, J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,1,-2,1).

Cf. A008724, A152467.

[Round(n*(n-4)/12): n in [0..60]]; // Vincenzo Librandi, Jun 22 2011

Maple
```
a(n):=round(n*(n-4)/12)
```

Table[Sum[Floor[k/6], {k,0,n}], {n,0,25}] (* G. C. Greubel, Dec 13 2016 *)

a(n)=(n-2)^2\12 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = round(n*(n-4)/12) = round((2*n^2 - 8*n - 1)/24).
a(n) = floor((n-2)^2/12).
a(n) = ceiling((n+1)*(n-5)/12).
a(n) = a(n-6) + n - 5, n > 5.
From R. J. Mathar, Nov 30 2010: (Start)
a(n) = 2*a(n-1) - a(n-2) + a(n-6) - 2*a(n-7) + a(n-8).
G.f.: -x^6 / ( (1+x)*(x^2-x+1)*(1+x+x^2)*(x-1)^3 ).
a(n) = -n/3 + 5/72 + n^2/12 + (-1)^n/24 + A057079(n+5)/6 + A061347(n)/18. (End)
a(6n) = A000567(n), a(6n+1) = 2*A000326(n), a(6n+2) = A033428(n), a(6n+3) = A049451(n), a(6n+4) = A045944(n), a(6n+5) = A028896(n). - Philippe Deléham, Mar 26 2013
a(n) = A008724(n-2). - R. J. Mathar, Jul 10 2015
Sum_{n>=6} 1/a(n) = Pi^2/18 - Pi/(2*sqrt(3)) + 49/12. - Amiram Eldar, Aug 13 2022

A244630 a(n) = 17*n^2.

Original entry on oeis.org

0, 17, 68, 153, 272, 425, 612, 833, 1088, 1377, 1700, 2057, 2448, 2873, 3332, 3825, 4352, 4913, 5508, 6137, 6800, 7497, 8228, 8993, 9792, 10625, 11492, 12393, 13328, 14297, 15300, 16337, 17408, 18513, 19652, 20825, 22032, 23273, 24548, 25857, 27200, 28577, 29988

Offset: 0

Vincenzo Librandi, Jul 03 2014

First bisection of A195047. - Bruno Berselli, Jul 03 2014

Norms of purely imaginary numbers in Z[sqrt(-17)] (for example, 3*sqrt(-17) has norm 153). - Alonso del Arte, Jun 23 2018

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

Cf. A008599, A195047.

Cf. similar sequences of the type k*n^2: A000290 (k = 1), A001105 (k = 2), A033428 (k = 3), A016742 (k = 4), A033429 (k = 5), A033581 (k = 6), A033582 (k = 7), A139098 (k = 8), A016766 (k = 9), A033583 (k = 10), A033584 (k = 11), A135453 (k = 12), A152742 (k = 13), A144555 (k = 14), A064761 (k = 15), A016802 (k = 16), this sequence (k = 17), A195321 (k = 18), A244631 (k = 19), A195322 (k = 20), A064762 (k = 21), A195323 (k = 22), A244632 (k = 23), A195824 (k = 24), A016850 (k = 25), A244633 (k = 26), A244634 (k = 27), A064763 (k = 28), A244635 (k = 29), A244636 (k = 30).

List([0..45],n->17*n^2); # Muniru A Asiru, Jun 29 2018

Magma
```
[17*n^2: n in [0..40]];
```

seq(17*n^2,n=0..45); # Muniru A Asiru, Jun 29 2018

Mathematica
```
Table[17 n^2, {n, 0, 40}]
```

a(n)=17*n^2 \\ Charles R Greathouse IV, Oct 07 2015

for (i <- 0 to 50) yield 17 * (i * i) // Alonso del Arte, Jun 29 2018

G.f.: 17*x*(1 + x)/(1 - x)^3. [corrected by Bruno Berselli, Jul 03 2014]
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = 17*A000290(n). - Omar E. Pol, Jul 03 2014
a(n) = a(-n). - Muniru A Asiru, Jun 29 2018
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 17*x*(1 + x)*exp(x).
a(n) = n*A008599(n) = A195047(2*n). (End)

A004780 Binary expansion contains 2 adjacent 1's.

Original entry on oeis.org

3, 6, 7, 11, 12, 13, 14, 15, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 35, 38, 39, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 70, 71, 75, 76, 77, 78, 79, 83, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

N. J. A. Sloane

Complement of A003714. It appears that n is in the sequence if and only if C(3n,n) is even. - Benoit Cloitre, Mar 09 2003
Since the binary representation of these numbers contains two adjacent 1's, so for these values of n, we will have (n XOR 2n XOR 3n) != 0, and thus a two player Nim game with three heaps of (n, 2n, 3n) stones will be a winning configuration for the first player. - V. Raman, Sep 17 2012
A048728(a(n)) > 0. - Reinhard Zumkeller, May 13 2014
The set of numbers x such that Or(x,3*x) <> 3*x. - Gary Detlefs, Jun 04 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A005809, A048728, A242408.
Complement: A003714.
Subsequences (apart from any initial zero-term): A001196, A004755, A004767, A033428, A277335.

a004780 n = a004780_list !! (n-1)
a004780_list = filter ((> 1) . a048728) [1..]
-- Reinhard Zumkeller, May 13 2014

q:= n-> verify([1$2], Bits[Split](n), 'sublist'):
select(q, [$0..200])[];  # Alois P. Heinz, Oct 22 2021

is(n)=bitand(n,n+n)>0 \\ Charles R Greathouse IV, Sep 19 2012

from itertools import count, islice
def A004780_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:n&(n<<1), count(max(startvalue,1)))
A004780_list = list(islice(A004780_gen(),30)) # Chai Wah Wu, Jul 13 2022

a(n) ~ n. - Charles R Greathouse IV, Sep 19 2012

Offset corrected by Reinhard Zumkeller, Jul 28 2010

A226492 a(n) = n(11n-5)/2.

Original entry on oeis.org

0, 3, 17, 42, 78, 125, 183, 252, 332, 423, 525, 638, 762, 897, 1043, 1200, 1368, 1547, 1737, 1938, 2150, 2373, 2607, 2852, 3108, 3375, 3653, 3942, 4242, 4553, 4875, 5208, 5552, 5907, 6273, 6650, 7038, 7437, 7847, 8268, 8700, 9143, 9597, 10062, 10538, 11025, 11523

Offset: 0

Bruno Berselli, Jun 11 2013

Sequences of numbers of the form n*(n*k - k + 6)/2:
. k from 0 to 10, respectively: A008585, A055998, A005563, A045943, A014105, A005475, A033428, A022264, A033991, A062741, A147874;
. k=11: a(n);
. k=12: A094159;
. k=13: 0, 3, 19, 48, 90, 145, 213, 294, 388, 495, 615, 748, 894, ...;
. k=14: 0, 3, 20, 51, 96, 155, 228, 315, 416, 531, 660, 803, 960, ...;
. k=15: A152773;
. k=16: A139272;
. k=17: 0, 3, 23, 60, 114, 185, 273, 378, 500, 639, 795, 968, ...;
. k=18: A152751;
. k=19: 0, 3, 25, 66, 126, 205, 303, 420, 556, 711, 885, 1078, ...;
. k=20: 0, 3, 26, 69, 132, 215, 318, 441, 584, 747, 930, 1133, ...;
. k=21: A152759;
. k=22: 0, 3, 28, 75, 144, 235, 348, 483, 640, 819, 1020, 1243, ...;
. k=23: 0, 3, 29, 78, 150, 245, 363, 504, 668, 855, 1065, 1298, ...;
. k=24: A152767;
. k=25: 0, 3, 31, 84, 162, 265, 393, 546, 724, 927, 1155, 1408, ...;
. k=26: 0, 3, 32, 87, 168, 275, 408, 567, 752, 963, 1200, 1463, ...;
. k=27: A153783;
. k=28: A195021;
. k=29: 0, 3, 35, 96, 186, 305, 453, 630, 836, 1071, 1335, 1628, ...;
. k=30: A153448;
. k=31: 0, 3, 37, 102, 198, 325, 483, 672, 892, 1143, 1425, 1738, ...;
. k=32: 0, 3, 38, 105, 204, 335, 498, 693, 920, 1179, 1470, 1793, ...;
. k=33: A153875.
Also:
a(n) - n         = A180223(n);
a(n) + n         = n*(11*n-3)/2 = 0, 4, 19, 45, 82, 130, 189, 259, ...;
a(n) - 2*n       = A051865(n);
a(n) + 2*n       = A022268(n);
a(n) - 3*n       = A152740(n-1);
a(n) + 3*n       = A022269(n);
a(n) - 4*n       = n*(11*n-13)/2 = 0, -1, 9, 30, 62, 105, 159, 224, ...;
a(n) + 4*n       = A254963(n);
a(n) - n*(n-1)/2 = A147874(n+1);
a(n) + n*(n-1)/2 = A094159(n) (case k=12);
a(n) - n*(n-1)   = A062741(n) (see above, this is the case k=9);
a(n) + n*(n-1)   = n*(13*n-7)/2 (case k=13);
a(n) - n*(n+1)/2 = A135706(n);
a(n) + n*(n+1)/2 = A033579(n);
a(n) - n*(n+1)   = A051682(n);
a(n) + n*(n+1)   = A186030(n);
a(n) - n^2       = A062708(n);
a(n) + n^2       = n*(13*n-5)/2 = 0, 4, 21, 51, 94, 150, 219, ..., etc.
Sum of reciprocals of a(n), for n > 0: 0.47118857003113149692081665034891...

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

Cf. sequences in Comments lines.
First differences are in A017425.
Cf. A033584, A180223.

Magma
```
[n*(11*n-5)/2: n in [0..50]];
```

I:=[0,3,17]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..46]]; // Vincenzo Librandi, Aug 18 2013

Table[n (11 n - 5)/2, {n, 0, 50}]
CoefficientList[Series[x (3 + 8 x) / (1 - x)^3, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{3,-3,1},{0,3,17},50] (* Harvey P. Dale, Jan 14 2019 *)

a(n)=n*(11*n-5)/2 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: x*(3+8*x)/(1-x)^3.
a(n) + a(-n) = A033584(n).
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: exp(x)*x*(6 + 11*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = n + A180223(n). (End)

A001479 Let p = A007645(n) be the n-th generalized cuban prime and write p = x^2 + 3*y^2; a(n) = x.

Original entry on oeis.org

0, 2, 1, 4, 2, 5, 4, 7, 8, 5, 2, 7, 10, 1, 10, 8, 2, 7, 4, 13, 1, 14, 8, 14, 11, 7, 14, 13, 16, 8, 11, 16, 17, 7, 2, 19, 4, 17, 19, 11, 1, 14, 5, 10, 22, 16, 4, 23, 20, 8, 23, 13, 10, 5, 16, 22, 20, 19, 25, 4, 11, 22, 25, 8, 26, 13, 1, 28, 28, 26, 23, 29, 28

Offset: 1

N. J. A. Sloane

A. J. C. Cunningham, Quadratic Partitions. Hodgson, London, 1904, p. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
B. van der Pol and P. Speziali, The primes in k(rho). Nederl. Akad. Wetensch. Proc. Ser. A. {54} = Indagationes Math. 13, (1951). 9-15 (1 plate).

T. D. Noe, Table of n, a(n) for n = 1..1000
A. J. C. Cunningham, Quadratic Partitions, Hodgson, London, 1904 [Annotated scans of selected pages]
S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).
B. van der Pol and P. Speziali, The primes in k(rho) (annotated and scanned copy)

Cf. A001480, A007645, A000196, A010052, A033428.

a001479 n = a000196 $ head $
   filter ((== 1) . a010052) $ map (a007645 n -) $ tail a033428_list
-- Reinhard Zumkeller, Jul 11 2013

nmax = 56; nextCuban[p_] := If[p1 = NextPrime[p]; Mod[p1, 3] > 1, nextCuban[p1], p1]; cubanPrimes = NestList[ nextCuban, 3, nmax ]; f[p_] := x /. ToRules[ Reduce[x > 0 && y > 0 && p == x^2 + 3*y^2, {x, y}, Integers]]; a[1] = 0; a[n_] := f[cubanPrimes[[n]]]; Table[ a[n] , {n, 1, nmax}] (* Jean-François Alcover, Oct 19 2011 *)

do(lim)=my(v=List(), q=Qfb(1,0,3)); forprime(p=2,lim, if(p%3==2,next); listput(v, qfbsolve(q,p)[1])); Vec(v) \\ Charles R Greathouse IV, Feb 07 2017

Definition revised by N. J. A. Sloane, Jan 29 2013

A140676 a(n) = n(3n + 4).

Original entry on oeis.org

0, 7, 20, 39, 64, 95, 132, 175, 224, 279, 340, 407, 480, 559, 644, 735, 832, 935, 1044, 1159, 1280, 1407, 1540, 1679, 1824, 1975, 2132, 2295, 2464, 2639, 2820, 3007, 3200, 3399, 3604, 3815, 4032, 4255, 4484, 4719, 4960, 5207, 5460, 5719, 5984, 6255, 6532, 6815

Offset: 0

Omar E. Pol, May 22 2008

The number of peers of a cell of an n^2 X n^2 sudoku is a(n-1). - Neven Sajko, Apr 20 2016

First differences are in A016921. - Wesley Ivan Hurt, Apr 21 2016

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. A000567, A016921, A033428, A045944, A067707, A067725, A140677, A140678, A140679, A140680, A140681, A140689.

[n*(3*n+4) : n in [0..80]]; // Wesley Ivan Hurt, Apr 21 2016

A140676:=n->n*(3*n+4): seq(A140676(n), n=0..100); # Wesley Ivan Hurt, Apr 21 2016

Table[n (3 n + 4), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 7, 20}, 50] (* Harvey P. Dale, May 04 2013 *)

a(n)=n*(3*n+4) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*n^2 + 4*n.
a(n) = 6*n + a(n-1) + 1 for n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
O.g.f.: x*(7 - x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - Harvey P. Dale, May 04 2013
E.g.f.: x*(7 + 3*x)*exp(x). - Ilya Gutkovskiy, Apr 20 2016
a(n) = A000567(n+1) - 1. - Neven Sajko, Apr 20 2016
From Amiram Eldar, Feb 26 2022: (Start)
Sum_{n>=1} 1/a(n) = 15/16 - Pi/(8*sqrt(3)) - 3*log(3)/8.
Sum_{n>=1} (-1)^(n+1)/a(n) = 9/16 - Pi/(4*sqrt(3)). (End)

A140681 a(n) = 3n(n+6).

Original entry on oeis.org

0, 21, 48, 81, 120, 165, 216, 273, 336, 405, 480, 561, 648, 741, 840, 945, 1056, 1173, 1296, 1425, 1560, 1701, 1848, 2001, 2160, 2325, 2496, 2673, 2856, 3045, 3240, 3441, 3648, 3861, 4080, 4305, 4536, 4773, 5016, 5265, 5520, 5781

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. A028560, A033428, A045944, A140676, A067725, A140677, A140678, A067707, A140679, A140680, A140689, A000217, A005563, A067728, A140091, A212331.

A140681:=n->3*n*(n+6); seq(A140681(n), n=0..100); # Wesley Ivan Hurt, Dec 10 2013

Table[3n(n+6), {n,0,100}] (* Wesley Ivan Hurt, Dec 10 2013 *)
LinearRecurrence[{3,-3,1},{0,21,48},50] (* Harvey P. Dale, May 03 2023 *)

a(n)=3*n*(n+6) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A028560(n)*3 = 3*n^2 + 18*n = n*(3*n+18).
a(n) = 6*n + a(n-1) + 15 with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
from G. C. Greubel, Jul 20 2017: (Start)
G.f.: 3*x*(7 - 5*x)/(1-x)^3.
E.g.f.: 3*x*(x+7)*exp(x). (End)
From Amiram Eldar, Feb 26 2022: (Start)
Sum_{n>=1} 1/a(n) = 49/360.
Sum_{n>=1} (-1)^(n+1)/a(n) = 37/1080. (End)

A152759 3 times 9-gonal (or nonagonal) numbers: a(n) = 3n(7*n-5)/2.

Original entry on oeis.org

0, 3, 27, 72, 138, 225, 333, 462, 612, 783, 975, 1188, 1422, 1677, 1953, 2250, 2568, 2907, 3267, 3648, 4050, 4473, 4917, 5382, 5868, 6375, 6903, 7452, 8022, 8613, 9225, 9858, 10512, 11187, 11883, 12600, 13338, 14097, 14877, 15678, 16500, 17343, 18207, 19092, 19998

Offset: 0

Omar E. Pol, Dec 14 2008

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

Cf. A001106, A139268, A226491.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152767, A153783, A153448, A153875.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=21: see Comments lines of A226492.

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,3,6!,21}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
CoefficientList[Series[3 x (1 + 6 x) / (1 - x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 05 2013 *)
LinearRecurrence[{3,-3,1},{0,3,27},40] (* Harvey P. Dale, May 26 2015 *)

a(n)=3*n*(7*n-5)/2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = (21*n^2 - 15*n)/2 = 3*A001106(n).
a(n) = a(n-1) + 21*n - 18 with n > 0, a(0)=0. - Vincenzo Librandi, Nov 26 2010
G.f.: 3*x*(1+6*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
a(n) = n + A226491(n). - Bruno Berselli, Jun 11 2013
From Elmo R. Oliveira, Dec 15 2024: (Start)
E.g.f.: 3*exp(x)*x*(2 + 7*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A132111 Triangle read by rows: T(n,k) = n^2 + k*n + k^2, 0 <= k <= n.

Original entry on oeis.org

0, 1, 3, 4, 7, 12, 9, 13, 19, 27, 16, 21, 28, 37, 48, 25, 31, 39, 49, 61, 75, 36, 43, 52, 63, 76, 91, 108, 49, 57, 67, 79, 93, 109, 127, 147, 64, 73, 84, 97, 112, 129, 148, 169, 192, 81, 91, 103, 117, 133, 151, 171, 193, 217, 243, 100, 111, 124, 139, 156, 175, 196, 219

Offset: 0

Reinhard Zumkeller, Aug 10 2007

Permutation of A003136, the Loeschian numbers. [This is false - some terms are repeated, the first being 49. - Joerg Arndt, Dec 18 2015]
Row sums give A132112.
Central terms give A033582.
T(n,k+1) = T(n,k) + n + 2*k + 1;
T(n+1,k) = T(n,k) + 2*n + k + 1;
T(n+1,k+1) = T(n,k) + 3*(n+k+1);
T(n,0) = A000290(n);
T(n,1) = A002061(n+1) for n>0;
T(n,2) = A117950(n+1) for n>1;
T(n,n-2) = A056107(n-1) for n>1;
T(n,n-1) = A003215(n-1) for n>0;
T(n,n) = A033428(n).
T(n,k) is the norm N(alpha) of the integer alpha = n*1 - k*omega, where omega = exp(2*Pi*i/3) = (-1 + i*sqrt(3))/2 in the imaginary quadratic number field Q(sqrt(-3)): N = |alpha|^2 = (n + k/2)^2 + (3/4)*k^2  = n^2 + n*k + k^2 = T(n,k), with n >= 0, and k <= n. See also triangle A073254 for T(n,-k). - Wolfdieter Lang, Jun 13 2021

			From _Philippe Deléham_, Apr 16 2014: (Start)
Triangle begins:
   0;
   1,  3;
   4,  7,  12;
   9, 13,  19,  27;
  16, 21,  28,  37,  48;
  25, 31,  39,  49,  61,  75;
  36, 43,  52,  63,  76,  91, 108;
  49, 57,  67,  79,  93, 109, 127, 147;
  64, 73,  84,  97, 112, 129, 148, 169, 192;
  81, 91, 103, 117, 133, 151, 171, 193, 217, 243;
  ...
(End)

Cf. A073254.

Flatten[Table[n^2+k*n+k^2,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jun 10 2013 *)

A152767 3 times 10-gonal (or decagonal) numbers: a(n) = 3n(4*n-3).

Original entry on oeis.org

0, 3, 30, 81, 156, 255, 378, 525, 696, 891, 1110, 1353, 1620, 1911, 2226, 2565, 2928, 3315, 3726, 4161, 4620, 5103, 5610, 6141, 6696, 7275, 7878, 8505, 9156, 9831, 10530, 11253, 12000, 12771, 13566, 14385, 15228, 16095, 16986, 17901, 18840, 19803, 20790, 21801

Offset: 0

Omar E. Pol, Dec 15 2008

3*A172078(n) = n*a(n) - Sum_{k=0..n-1} a(k). - Bruno Berselli, Dec 12 2010

			For n=8, a(8) = (1*3 + 5*7 + 9*11 +..+ 29*31) - (2*4 + 6*8 + 10*12 +..+ 26*28) = 696 (see Problem 1052 in References). - _Bruno Berselli_, Dec 12 2010

"Supplemento al Periodico di Matematica", Raffaello Giusti Editore (Livorno), Jan. 1910 p. 47 (Problem 1052).

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. A001107, A001539, A002939, A139271, A153794, A172078.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A153783, A153448, A153875.
Cf. numbers of the form n*(n*k - k + 6)/2, this sequence is the case k=24: see Comments lines of A226492.

Table[3n(4n-3),{n,0,40}] (* Harvey P. Dale, May 26 2012 *)

a(n)=3*n*(4*n-3) \\ Charles R Greathouse IV, Sep 28 2015

a(n) = 12*n^2 - 9*n = 3*A001107(n).
a(n) = a(n-1) + 24*n - 21, n > 0. - Vincenzo Librandi, Nov 26 2010
a(n) = Sum_{k=0..n-1} A001539(k) - Sum_{k=0..n-1} 4*A002939(k) if n > 0 (see References, Problem 1052). - Bruno Berselli, Dec 08 2010 - Jan 21 2011
G.f.: -3*x*(1+7*x)/(x-1)^3.
a(0)=0, a(1)=3, a(2)=30, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 26 2012
From Elmo R. Oliveira, Dec 15 2024: (Start)
E.g.f.: 3*exp(x)*x*(1 + 4*x).
a(n) = A153794(n) - n. (End)

cp's OEIS Frontend

A174709 Partial sums of floor(n/6).

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A244630 a(n) = 17*n^2.

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A004780 Binary expansion contains 2 adjacent 1's.

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A226492 a(n) = n*(11*n-5)/2.

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A001479 Let p = A007645(n) be the n-th generalized cuban prime and write p = x^2 + 3*y^2; a(n) = x.

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A140676 a(n) = n*(3*n + 4).

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A226492 a(n) = n(11n-5)/2.

A140676 a(n) = n(3n + 4).

A140681 a(n) = 3n(n+6).

A152759 3 times 9-gonal (or nonagonal) numbers: a(n) = 3n(7*n-5)/2.

A152767 3 times 10-gonal (or decagonal) numbers: a(n) = 3n(4*n-3).