A136392 -id:A136392 - cp's OEIS frontend

A059722 a(n) = n(2n^2 - 2*n + 1).

0, 1, 10, 39, 100, 205, 366, 595, 904, 1305, 1810, 2431, 3180, 4069, 5110, 6315, 7696, 9265, 11034, 13015, 15220, 17661, 20350, 23299, 26520, 30025, 33826, 37935, 42364, 47125, 52230, 57691, 63520, 69729, 76330, 83335, 90756, 98605, 106894, 115635, 124840

Offset: 0

Henry Bottomley, Feb 07 2001

Mean of the first four nonnegative powers of 2n+1, i.e., ((2n+1)^0 + (2n+1)^1 + (2n+1)^2 + (2n+1)^3)/4. E.g., a(2) = (1 + 3 + 9 + 27)/4 = 10.
Equatorial structured meta-diamond numbers, the n-th number from an equatorial structured n-gonal diamond number sequence. There are no 1- or 2-gonal diamonds, so 1 and (n+2) are used as the first and second terms since all the sequences begin as such. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Starting with offset 1 = row sums of triangle A143803. - Gary W. Adamson, Sep 01 2008
Form an array from the antidiagonals containing the terms in A002061 to give antidiagonals 1; 3,3; 7,4,7; 13,8,8,13; 21,14,9,14,21; and so on. The difference between the sum of the terms in n+1 X n+1 matrices and those in n X n matrices is a(n) for n>0. - J. M. Bergot, Jul 08 2013
Sum of the numbers from (n-1)^2 to n^2. - Wesley Ivan Hurt, Sep 08 2014

Harry J. Smith, Table of n, a(n) for n = 0..1000
Bruno Berselli, Table of sequences with the formula m*(m+1)^2 - (k+2)*m^2 (table includes this sequence for k = 2-m, m >= 0).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A005900, A081436, A100178, A100179, A059722: "equatorial" structured diamonds; A000447: "polar" structured meta-diamond; A006484 for other structured meta numbers; and A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Cf. A000217, A000290, A001844, A002061, A002414, A053698, A059721, A059723, A136392, A143803.

[n*(2*n^2 - 2*n + 1) : n in [0..50]]; // Wesley Ivan Hurt, Sep 08 2014

A059722:=n->n*(2*n^2 - 2*n + 1): seq(A059722(n), n=0..50); # Wesley Ivan Hurt, Sep 08 2014

Table[n (2 n^2 - 2 n + 1), {n, 0, 50}] (* Wesley Ivan Hurt, Sep 08 2014 *)

a(n) = { n*(2*n^2 - 2*n + 1) } \\ Harry J. Smith, Jun 28 2009

a(n) = A053698(2*n-1)/4.
a(n) = Sum_{j=1..n} ((n+j-1)^2-j^2+1). - Zerinvary Lajos, Sep 13 2006
From R. J. Mathar, Sep 02 2008: (Start)
G.f.: x*(1 + x)*(1 + 5*x)/(1 - x)^4.
a(n) = A002414(n-1) + A002414(n).
a(n+1) - a(n) = A136392(n+1). (End)
a(n) = (A000290(n) + A000290(n+1)) * (A000217(n+1) - A000217(n)). - J. M. Bergot, Nov 02 2012
a(n) = n * A001844(n-1). - Doug Bell, Aug 18 2015
a(n) = A000217(n^2) - A000217(n^2-2*n). - Bruno Berselli, Jun 26 2018
E.g.f.: exp(x)*x*(1 + 4*x + 2*x^2). - Stefano Spezia, Jun 20 2021

Edited with new definition by N. J. A. Sloane, Aug 29 2008

A033292 A Connell-like sequence: take 1 number = 1 (mod Q), 2 numbers = 2 (mod Q), 3 numbers = 3 (mod Q), etc., where Q = 3.

Original entry on oeis.org

1, 2, 5, 6, 9, 12, 13, 16, 19, 22, 23, 26, 29, 32, 35, 36, 39, 42, 45, 48, 51, 52, 55, 58, 61, 64, 67, 70, 71, 74, 77, 80, 83, 86, 89, 92, 93, 96, 99, 102, 105, 108, 111, 114, 117, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 146, 149, 152, 155, 158, 161, 164, 167, 170, 173, 176, 177, 180

Offset: 1

Gary E. Stevens

Left border of the triangle (1, 2, 6, 13, 23, 36, ...) = A143689 = A000326(n) - 3n, where A000326 = the pentagonal numbers, right border. - Gary W. Adamson, Aug 29 2008
Row sums = A143690: (1, 7, 27, 70, 145, 261, 427, 652, ...). - Gary W. Adamson, Aug 29 2008
Central terms = A136392. - Reinhard Zumkeller, Jan 18 2012

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.
Gary E. Stevens, A Connell-Like Sequence, J. Integer Sequences, 1 (1998), #98.1.4.

a033292 n k = a033292_tabl !! (n-1) !! (k-1)
a033292_row n = a033292_tabl !! (n-1)
a033292_tabl = f 1 [1..] where
   f k xs = ys : f (k+1) (dropWhile (<= last ys) xs) where
     ys  = take k $ filter ((== 0) . (`mod` 3) . (subtract k)) xs
-- Reinhard Zumkeller, Jan 18 2012 2011

row[n_] := n*(3*n-1)/2 + Range[1, 3*n+1, 3]; Flatten[ Table[ row[n], {n, 0, 11}]] (* Jean-François Alcover, Aug 03 2012 *)

More terms from jeroen.lahousse(AT)icl.com

Offset changed by Reinhard Zumkeller, Jan 18 2012

A201279 a(n) = 6n^2 + 10n + 5.

Original entry on oeis.org

5, 21, 49, 89, 141, 205, 281, 369, 469, 581, 705, 841, 989, 1149, 1321, 1505, 1701, 1909, 2129, 2361, 2605, 2861, 3129, 3409, 3701, 4005, 4321, 4649, 4989, 5341, 5705, 6081, 6469, 6869, 7281, 7705, 8141, 8589, 9049, 9521, 10005, 10501, 11009, 11529, 12061

Offset: 0

Eleonora Echeverri-Toro, Nov 29 2011

Numbers n where 6n-5 is a square of a number type 6n-1.
Also sequence found by reading the line from 5, in the direction 5, 21,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Jul 18 2012
The spiral mentioned above naturally appears on a "graphene" like lattice (planar net 6^3). The opposite diagonal is A080859. - Yuriy Sibirmovsky, Oct 04 2016
First differences of A048395. - Leo Tavares, Nov 24 2021 [Corrected by Omar E. Pol, Dec 26 2021]

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Yuriy Sibirmovsky, Diagonals of a 'graphene' number spiral.
Leo Tavares, Illustration: Diamond Frame Stars
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A136392, A080859.

Cf. A003154, A022144, A016754, A046092, A048395.

[6*n^2 + 10*n + 5: n in [0..60]]; // Vincenzo Librandi, Dec 01 2011

LinearRecurrence[{3,-3,1},{5,21,49},50] (* Vincenzo Librandi, Dec 01 2011 *)
Table[6 n^2 + 10 n + 5, {n, 0, 44}] (* or *)
CoefficientList[Series[(1 + x) (5 + x)/(1 - x)^3, {x, 0, 44}], x] (* Michael De Vlieger, Oct 04 2016 *)

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

G.f.: (1+x)*(5+x)/(1-x)^3. - Colin Barker, Jan 09 2012
a(n) = 1 + A033579(n+1). - Omar E. Pol, Jul 18 2012
a(n) = (n+1)*A001844(n+1)-n*A001844(n). [Bruno Berselli, Jan 15 2013]
From Leo Tavares, Nov 24 2021: (Start)
a(n) = A003154(n+2) - A022144(n+1). See Diamond Frame Stars illustration.
a(n) = A016754(n) + A046092(n+1). (End)

A319384 a(n) = a(n-1) + 2a(n-2) - 2a(n-3) - a(n-4) + a(n-5), a(0)=1, a(1)=5, a(2)=9, a(3)=21, a(4)=29.

Original entry on oeis.org

1, 5, 9, 21, 29, 49, 61, 89, 105, 141, 161, 205, 229, 281, 309, 369, 401, 469, 505, 581, 621, 705, 749, 841, 889, 989, 1041, 1149, 1205, 1321, 1381, 1505, 1569, 1701, 1769, 1909, 1981, 2129, 2205, 2361, 2441, 2605, 2689, 2861, 2949, 3129, 3221, 3409, 3505, 3701, 3801, 4005, 4109, 4321, 4429, 4649, 4761, 4989, 5105, 5341, 5461

Offset: 0

Paul Curtz, Sep 18 2018

The two bisections A136392(n+1)=1,9,29,61, ... and A201279(n)=5,21,49, ... are in the hexagonal spiral based on 2*n+1:
.
                    67--65--63--61
                    /             \
                  69  33--31--29  59
                  /   /         \   \
                71  35  11---9  27  57
                /   /   /     \   \   \
              73  37  13   1   7  25  55
                  /   /   /   /   /   /
                39  15   3---5  23  53
                  \   \         /   /
                  41  17--19--21  51
                    \             /
                    43--45--47--49
.
A201279(n) - A136892(n) = 20*n.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A001844, A032766, A104585.
In the spiral: A003154(n+1), A080859, A126587, A136392, A201279, A227776.
Partial sums of A382154.

[(6*n^2 + 6*n + 5 - (2*n + 1)*(-1)^n)/4 : n in [0..50]]; // Wesley Ivan Hurt, Jan 19 2021

Table[(6 n^2 + 6 n + 5 - (2 n + 1)*(-1)^n)/4, {n, 0, 80}] (* Wesley Ivan Hurt, Jan 07 2021 *)

Vec((1 + x^2)*(1 + 4*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^50)) \\ Colin Barker, Jun 05 2019

def A319384(n): return (n*(3*n+4)+3 if n&1 else n*(3*n+2)+2)>>1 # Chai Wah Wu, Mar 25 2025

a(2*n) = A136392(n+1), a(2*n+1) = A201279(n).
a(-n) = a(n).
a(2*n) + a(2*n+1) = 6*A001844(n).
a(n) = (6*n^2 + 6*n + 5 - (2*n + 1)*(-1)^n)/4. - Wesley Ivan Hurt, Oct 04 2018
G.f.: (1 + x^2)*(1 + 4*x + x^2) / ((1 - x)^3*(1 + x)^2). - Colin Barker, Jun 05 2019
a(n) = A104585(n) + A032766(n+1). - Alex W. Nowak, Jan 08 2021

More terms from N. J. A. Sloane, Mar 23 2025

A204675 a(n) = 16n^2 + 2n + 1.

Original entry on oeis.org

1, 19, 69, 151, 265, 411, 589, 799, 1041, 1315, 1621, 1959, 2329, 2731, 3165, 3631, 4129, 4659, 5221, 5815, 6441, 7099, 7789, 8511, 9265, 10051, 10869, 11719, 12601, 13515, 14461, 15439, 16449, 17491, 18565, 19671, 20809, 21979, 23181, 24415, 25681, 26979

Offset: 0

Reinhard Zumkeller, Jan 18 2012

Central terms of the triangle A033293.

Also sequence found by reading the line from 1, in the direction 1, 19, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012

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

Cf. A017077, A033293, A074377, A136392.

Haskell
```
a204675 n = 2 * n * (8 * n + 1) + 1
```

I:=[1, 19, 69]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Mar 19 2012

CoefficientList[Series[(1+x)*(1+15*x)/(1-x)^3,{x,0,50}],x] (* or *) LinearRecurrence[{3, -3, 1}, {1, 19, 69}, 50] (* Vincenzo Librandi, Mar 19 2012 *)

a(n)=16*n^2+2*n+1 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (1+x)*(1+15*x)/(1-x)^3. - Bruno Berselli, Jan 18 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Jun 09 2023
E.g.f.: exp(x)*(1 + 2*x*(9 + 8*x)). - Elmo R. Oliveira, Oct 18 2024

cp's OEIS Frontend

A059722 a(n) = n*(2*n^2 - 2*n + 1).

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A033292 A Connell-like sequence: take 1 number = 1 (mod Q), 2 numbers = 2 (mod Q), 3 numbers = 3 (mod Q), etc., where Q = 3.

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Haskell

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Extensions

A201279 a(n) = 6n^2 + 10n + 5.

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Author

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Comments

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Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A319384 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), a(0)=1, a(1)=5, a(2)=9, a(3)=21, a(4)=29.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

Extensions

A204675 a(n) = 16*n^2 + 2*n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Formula

A059722 a(n) = n(2n^2 - 2*n + 1).

A319384 a(n) = a(n-1) + 2a(n-2) - 2a(n-3) - a(n-4) + a(n-5), a(0)=1, a(1)=5, a(2)=9, a(3)=21, a(4)=29.

A204675 a(n) = 16n^2 + 2n + 1.