A033292 -id:A033292 - cp's OEIS frontend

A143689 a(n) = (3*n^2 - n + 2)/2.

1, 2, 6, 13, 23, 36, 52, 71, 93, 118, 146, 177, 211, 248, 288, 331, 377, 426, 478, 533, 591, 652, 716, 783, 853, 926, 1002, 1081, 1163, 1248, 1336, 1427, 1521, 1618, 1718, 1821, 1927, 2036, 2148, 2263, 2381, 2502, 2626, 2753, 2883, 3016, 3152, 3291

Offset: 0

Gary W. Adamson, Aug 29 2008

Equals left border of triangle A033292.
Equals binomial transform of [1, 1, 3, 0, 0, 0, ...].
A242357(a(n)) = 1. - Reinhard Zumkeller, May 11 2014
These might be called "trisected pentagonal numbers": A figurate pentagonal number is composed of three triangles, of which the central one is the largest, and the removal of the triangular frame (3*n) of the central triangle trisects the figure. This is reflected in the formula a(n) = A000326(n+1) - 3*n. See illustration in links. - John Elias, May 27 2022

Michael De Vlieger, Table of n, a(n) for n = 0..10000
D. Bevan, D. Levin, P. Nugent, J. Pantone, and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510:08036 [math.CO], 2015-2016.
John Elias, Trisected Pentagonal Numbers
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

a(n) = A000326(n+1) - 3n. Third column of A107111.

Cf. A033292, A104249.

a143689 n = n*(3*n-1) `div` 2 + 1 -- Reinhard Zumkeller, May 11 2014

Table[(3n^2-n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,2,6},50] (* Harvey P. Dale, May 05 2014 *)

makelist(binomial(n, 2) + n^2 + 1, n, 0, 100); /* Franck Maminirina Ramaharo, Mar 01 2018 */

a(n)=(3*n^2-n+2)/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A000326(n+1) - 3*n. (A000326 are the pentagonal numbers.)
a(n) = (3*n^2 - n + 2)/2 = A027599(n+1)/2. - R. J. Mathar, Sep 03 2008
a(n) = a(n-1) + 3*n - 2 (with a(0)=1). - Vincenzo Librandi, Nov 25 2010
a(n) = 2*a(n-1) - a(n-2) + 3.
O.g.f.: (1-x+3*x^2)/((1-x)^3). - Eric Werley, Jun 27 2011
a(n) = A104249(-n). - Bruno Berselli, Jul 08 2015
a(n) = binomial(n,2) + n^2 + 1 = A152947(n+1) + A000290(n). - Franck Maminirina Ramaharo, Mar 01 2018
E.g.f.: exp(x)*(2 + 2*x + 3*x^2)/2. - Stefano Spezia, Apr 19 2025

Index of A000326 in definition, formula and example corrected by R. J. Mathar, Sep 03 2008

A049039 Geometric Connell sequence: 1 odd, 2 even, 4 odd, 8 even, ...

Original entry on oeis.org

1, 2, 4, 5, 7, 9, 11, 12, 14, 16, 18, 20, 22, 24, 26, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 121, 123, 125

Offset: 1

James Sellers

Reinhard Zumkeller, Rows n=1..13 of triangle, flattened
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Cf. A337300 (partial sums), A043529 (first differences).
Cf. A001614, A033292, A030196, A000295, A050487, A050488.
Cf. A160464, A160465 and A160473. - Johannes W. Meijer, May 24 2009

a049039 n k = a049039_tabl !! (n-1) !! (k-1)
a049039_row n = a049039_tabl !! (n-1)
a049039_tabl = f 1 1 [1..] where
   f k p xs = ys : f (2 * k) (1 - p) (dropWhile (<= last ys) xs) where
     ys  = take k $ filter ((== p) . (`mod` 2)) xs
-- Reinhard Zumkeller, Jan 18 2012, Jul 08 2011

Digits := 100: [seq(2*n-1-floor(evalf(log(n)/log(2))), n=1..100)];

a[0] = 0; a[n_?EvenQ] := a[n] = a[n/2]+n-1; a[n_?OddQ] := a[n] = a[(n-1)/2]+n; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 27 2011, after Ralf Stephan *)

a(n) = n<<1 - 1 - logint(n,2); \\ Kevin Ryde, Feb 12 2022

def A049039(n): return (n<<1)-n.bit_length() # Chai Wah Wu, Aug 01 2022

a(n) = 2n - 1 - floor(log_2(n)).
a(2^n-1) = 2^(n+1) - (n+2) = A000295(n+1), the Eulerian numbers.
a(0)=0, a(2n) = a(n) + 2n - 1, a(2n+1) = a(n) + 2n + 1. - Ralf Stephan, Oct 11 2003

Keyword tabf added by Reinhard Zumkeller, Jan 22 2012

A033291 A Connell-like sequence: take the first multiple of 1, the next 2 multiples of 2, the next 3 multiples of 3, etc.

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 16, 20, 24, 28, 30, 35, 40, 45, 50, 54, 60, 66, 72, 78, 84, 91, 98, 105, 112, 119, 126, 133, 136, 144, 152, 160, 168, 176, 184, 192, 198, 207, 216, 225, 234, 243, 252, 261, 270, 280, 290, 300, 310, 320, 330, 340, 350, 360, 370, 374, 385, 396, 407, 418, 429, 440, 451, 462

Offset: 1

Gary E. Stevens

Row sums are 0, 1, 6, 27, 88, 200, ... with g.f. -x*(1 + 4*x + 16*x^2 + 37*x^3 + 39*x^4 + 54*x^5 + 39*x^6 + 17*x^7 + 8*x^8 + x^9) / ( (1 + x + x^2)^3*(x-1)^5 ). - R. J. Mathar, Aug 10 2017

			Triangle begins
   1;
   2,  4;
   6,  9,  12;
  16, 20,  24,  28;
  30, 35,  40,  45,  50;
  54, 60,  66,  72,  78,  84;
  91, 98, 105, 112, 119, 126, 133; ...

Reinhard Zumkeller, Rows n = 1..100, flattened
Gary E. Stevens, A Connell-Like Sequence, J. Integer Sequences, 1 (1998), #98.1.4.

Cf. A192735 (left edge), A192736 (right edge).

Cf. A045975, A033292, A033293.

a033291 n k = a033291_tabl !! (n-1) !! (k-1)
a033291_row n = a033291_tabl !! (n-1)
a033291_tabl = f 1 [1..] where
   f k xs = ys : f (k+1) (dropWhile (<= last ys) xs) where
     ys  = take k $ filter ((== 0) . (`mod` k)) xs
a192735 n = head $ a033291_tabl !! (n-1)
a192736 n = last $ a033291_tabl !! (n-1)
-- Reinhard Zumkeller, Jan 18 2012, Jul 08 2011

A033291 := proc(n,k)
    A192735(n)+(k-1)*n ;
end proc:
seq(seq(A033291(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Aug 10 2017

Flatten[ Table[ n*(Floor[ (n-1)^2/3] + k), {n, 1, 12}, {k, 1, n}]] (* Jean-François Alcover, Sep 30 2011 *)

a(n)=my(q=(sqrtint(8*n-7)+1)\2); q*n-q*(q+1)\6*q \\ Charles R Greathouse IV, Jan 06 2016

a(n) = q(n)*n - q(n)*floor(q(n)*(q(n)+1)/6) with q(n) = ceiling((1/2)*(-1 + sqrt(1+8*(n)))).

Corrected and formula added by Johannes W. Meijer, Oct 07 2010

A136392 a(n) = 6n^2 - 10n + 5.

Original entry on oeis.org

1, 9, 29, 61, 105, 161, 229, 309, 401, 505, 621, 749, 889, 1041, 1205, 1381, 1569, 1769, 1981, 2205, 2441, 2689, 2949, 3221, 3505, 3801, 4109, 4429, 4761, 5105, 5461, 5829, 6209, 6601, 7005, 7421, 7849, 8289, 8741, 9205, 9681, 10169, 10669, 11181, 11705, 12241

Offset: 1

Gary W. Adamson, Dec 28 2007

Binomial transform of [1, 8, 12, 0, 0, 0, ...].
Numbers k such that 6*k - 5 is the square of a number of the form 6*k - 5, contained in A199859. - Eleonora Echeverri-Toro, Nov 29 2011
Central terms of the triangle A033292. - Reinhard Zumkeller, Feb 06 2012
Sequence found by reading the line from 1, in the direction 1, 9, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Jul 18 2012

Kelvin Voskuijl, Table of n, a(n) for n = 1..10000
John Elias, Illustration: Centered nesting cube frames.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000567, A001318, A016777, A033292, A033580, A059722, A199859, A201279, A204675.

a136392 n = 2 * n * (3*n - 5) + 5 -- Reinhard Zumkeller, Feb 06 2012

Table[6n^2-10n+5,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,9,29},50] (* Harvey P. Dale, Mar 05 2023 *)

a(n)=6*n^2-10*n+5 \\ Charles R Greathouse IV, Nov 29 2011

a(n) = n*(3*n - 2) + (n-1)*(3*n - 5), n > 1.
a(n) = n*A016777(n-1) + (n-1)*A016777(n-2).
a(n) = a(n-1) + 12*n - 16 (with a(1)=1). - Vincenzo Librandi, Nov 24 2010
G.f.: x*(1+x)*(1+5*x)/(1-x)^3. - Colin Barker, Jan 09 2012
a(n) = 1 + A033580(n-1). - Omar E. Pol, Jul 18 2012
a(n) = A059722(n) - A059722(n-1). - J. M. Bergot, Nov 02 2012
a(n) = A000567(n-1) + A000567(n). - Charlie Marion, May 29 2024
From Elmo R. Oliveira, Oct 31 2024: (Start)
E.g.f.: exp(x)*(2*x*(3*x - 2) + 5) - 5.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A143690 a(n) = A007318 * [1, 6, 14, 9, 0, 0, 0, ...].

Original entry on oeis.org

1, 7, 27, 70, 145, 261, 427, 652, 945, 1315, 1771, 2322, 2977, 3745, 4635, 5656, 6817, 8127, 9595, 11230, 13041, 15037, 17227, 19620, 22225, 25051, 28107, 31402, 34945, 38745, 42811, 47152, 51777, 56695, 61915, 67446, 73297, 79477, 85995, 92860, 100081, 107667

Offset: 0

Gary W. Adamson, Aug 29 2008

Binomial transform of [1, 6, 14, 9, 0, 0, 0,...].

Row sums of triangle A033292.

			a(3) = 70 = (1, 3, 3, 1) dot (1, 6, 14, 9) = (1 + 18 + 42 + 9). a(3) = 70 = sum of row 3 terms of triangle A033292: (13 + 16 + 19, + 22).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A000326, A002412, A033292.

Cf. A226449. - Bruno Berselli, Jun 09 2013

Table[(n+1)*(3*n^2+2*n+2)/2, {n,0,50}] (* G. C. Greubel, May 30 2021 *)

[(n+1)*(3*n^2+2*n+2)/2 for n in (0..50)] # G. C. Greubel, May 30 2021

From R. J. Mathar, Aug 29 2008: (Start)
G.f.: (1 +3*x +5*x^2)/(1-x)^4.
a(n) = A002412(n+1) + 5*A000292(n-1). (End)
a(n) = A000326(n+1) + (n+1)*A000326(n). - Bruno Berselli, Jun 07 2013
From G. C. Greubel, May 30 2021: (Start)
a(n) = (n+1)*(3*n^2 +2*n +2)/2.
E.g.f.: (1/2)*(2 +12*x +14*x^2 +3*x^3)*exp(x). (End)

Extended beyond a(14) by R. J. Mathar, Aug 29 2008

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A143689 a(n) = (3*n^2 - n + 2)/2.

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A049039 Geometric Connell sequence: 1 odd, 2 even, 4 odd, 8 even, ...

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A033291 A Connell-like sequence: take the first multiple of 1, the next 2 multiples of 2, the next 3 multiples of 3, etc.

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A136392 a(n) = 6*n^2 - 10*n + 5.

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A143690 a(n) = A007318 * [1, 6, 14, 9, 0, 0, 0, ...].

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A136392 a(n) = 6n^2 - 10n + 5.