A214675 -id:A214675 - cp's OEIS frontend

A214661 Odd numbers obtained by transposing the left half of A176271 into rows of a triangle: T(n,k) = A176271(n - 1 + k, k), 1 <= k <= n.

1, 3, 9, 7, 15, 25, 13, 23, 35, 49, 21, 33, 47, 63, 81, 31, 45, 61, 79, 99, 121, 43, 59, 77, 97, 119, 143, 169, 57, 75, 95, 117, 141, 167, 195, 225, 73, 93, 115, 139, 165, 193, 223, 255, 289, 91, 113, 137, 163, 191, 221, 253, 287, 323, 361, 111, 135, 161, 189, 219, 251, 285, 321, 359, 399, 441

Offset: 1

Reinhard Zumkeller, Jul 25 2012

			.     Take the first n elements of the n-th diagonal (northwest to
.     southeast) of the triangle on the left side
.     and write this as n-th row on the triangle of the right side.
. 1:                1                    1
. 2:              3   _                  3  9
. 3:            7   9  __                7 15 25
. 4:         13  15  __  __             13 23 35 49
. 5:       21  23  25  __  __           21 33 47 63 ..
. 6:     31  33  35  __  __  __         31 45 61 .. .. ..
. 7:   43  45  47  49  __  __  __       43 59 .. .. .. .. ..
. 8: 57  59  61  63  __  __  __  __     57 .. .. .. .. .. .. .. .

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

Cf. A000290, A002061, A016754, A025581, A094727, A176271, A214604.

Cf. A051673 (row sums), A214675 (main diagonal).

import Data.List (transpose)
a214661 n k = a214661_tabl !! (n-1) !! (k-1)
a214661_row n = a214661_tabl !! (n-1)
a214661_tabl = zipWith take [1..] $ transpose $ map reverse a176271_tabl

[(n+k)^2-3*n-k+1: k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 10 2024

Table[(n+k)^2-3*n-k+1, {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 10 2024 *)

flatten([[(n+k)^2-3*n-k+1 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 10 2024

T(n, k) = (n+k)^2 - 3*n - k + 1.
T(n,k) = A176271(n+k-1, k).
T(n, k) = A214604(n,k) - 2*A025581(n,k).
T(n, k) = 2*A000290(A094727(n,k)) - A214604(n,k).
T(2*n-1, n) = A214675() (main diagonal).
T(n,1) = A002061(n).
T(n,n) = A016754(n-1).
Sum_{k=1..n} T(n, k) = A051673(n) (row sums).

A220083 a(n) = (15n^2 + 9n + 2)/2.

Original entry on oeis.org

1, 13, 40, 82, 139, 211, 298, 400, 517, 649, 796, 958, 1135, 1327, 1534, 1756, 1993, 2245, 2512, 2794, 3091, 3403, 3730, 4072, 4429, 4801, 5188, 5590, 6007, 6439, 6886, 7348, 7825, 8317, 8824, 9346, 9883, 10435, 11002, 11584, 12181, 12793, 13420, 14062

Offset: 0

Bruno Berselli, Dec 10 2012

Sequence related to the heptagonal numbers (A000566) by a(n) = n*A000566(n)-(n-1)*A000566(n-1).
Other similar sequences:
A005408(m) = (m+1)*A001477(m+1)-m*A001477(m), A001477 = nonn. integers;
A000326(m) = m*A000217(n)-(m-1)*A000217(m-1), A000217 = triangular numbers;
A003215(m) = (m+1)*A000290(m+1)-n*A000290(m), A000290 = square numbers;
A081267(m) = (m+1)*A000326(m+1)-n*A000326(m), A000326 = pentagonal numbers;
A080859(m) = (m+1)*A000384(m+1)-n*A000384(m), A000384 = hexagonal numbers;
A214675(m) = m*A000567(m)-(m-1)*A000567(m-1), A000567 = octagonal numbers.

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

Cf. A000326, A003215, A005408, A006597, A080859, A081267, A214675.

/* By first comment: */  A000566:=func; [n*A000566(n)-(n-1)*A000566(n-1): n in [1..45]];

[(15*n^2 + 9*n + 2)/2: n in [0..50]]; // Vincenzo Librandi, Aug 18 2013

A220083:=n->(15*n^2 + 9*n + 2)/2; seq(A220083(n), n=0..100); # Wesley Ivan Hurt, Nov 14 2013

Table[(15 n^2 + 9 n + 2)/2, {n, 0, 45}]
CoefficientList[Series[(1 + 10 x + 4 x^2) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)

A220083(n):=(15*n^2 + 9*n + 2)/2$ makelist(A220083(n),n,0,20); /* Martin Ettl, Dec 11 2012 */

a(n)=(15*n^2+9*n+2)/2 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: (1+10*x+4*x^2)/(1-x)^3.
Sum( a(i), i=0..n ) = A006597(n+1).
a(n) + a(-n) = A010005(n) for n>0.

A100176 Structured octagonal prism numbers.

Original entry on oeis.org

1, 16, 63, 160, 325, 576, 931, 1408, 2025, 2800, 3751, 4896, 6253, 7840, 9675, 11776, 14161, 16848, 19855, 23200, 26901, 30976, 35443, 40320, 45625, 51376, 57591, 64288, 71485, 79200, 87451, 96256, 105633, 115600, 126175, 137376, 149221, 161728, 174915, 188800

Offset: 1

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Number of divisors of 120^(n-1). - J. Lowell, Aug 30 2008

Partial sums of A214675. - J. M. Bergot, Jul 08 2013

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

Cf. A100177 (structured prisms), A100145 (for more on structured numbers).
Cf. similar sequences, with the formula (k*n - k + 2)*n^2/2, listed in A262000.
Cf. A000567, A016777, A214675.

[3*n^3-2*n^2: n in [1..50] ]; // Vincenzo Librandi, Aug 02 2011

[seq(3*n^3-2*n^2,n=1..47)]; # Zerinvary Lajos, Jun 29 2006

f[n_] := 3 n^3 - 2 n^2; Table[f[n], {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Apr 27 2010 *)

a(n)=3*n^3-2*n^2 \\ Charles R Greathouse IV, Aug 10 2017

a(n) = 3*n^3 - 2*n^2.
G.f.: x*(1+12*x+5*x^2)/(1-x)^4. - Colin Barker, Jun 08 2012
a(n) = Sum_{i=0..n-1} n*(6*i+1). - Bruno Berselli, Sep 08 2015
Sum_{n>=1} 1/a(n) = sqrt(3)*Pi/8 - Pi^2/12 + 9*log(3)/8 = 1.0936465529153418... . - Vaclav Kotesovec, Oct 04 2016
a(n) = n*A000567(n) = n^2 * A016777(n-1). - Bruce J. Nicholson, Aug 10 2017
From Elmo R. Oliveira, Aug 06 2025: (Start)
E.g.f.: exp(x)*x*(1 + 7*x + 3*x^2).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

More terms from Zerinvary Lajos, Jun 29 2006

A214659 a(n) = n(7n^2 - 3*n - 1)/3.

Original entry on oeis.org

0, 1, 14, 53, 132, 265, 466, 749, 1128, 1617, 2230, 2981, 3884, 4953, 6202, 7645, 9296, 11169, 13278, 15637, 18260, 21161, 24354, 27853, 31672, 35825, 40326, 45189, 50428, 56057, 62090, 68541, 75424, 82753, 90542, 98805, 107556, 116809, 126578, 136877

Offset: 0

Reinhard Zumkeller, Jul 25 2012

a(n) = the sum of the n X n matrices of A204008. For example, for n = 3, the sum of the 9 elements of the 3 X 3 submatrix of A204008 is 1 + 4 + 7 + 4 + 5 + 8 + 7 + 8 + 9 = 53. - J. M. Bergot, Jul 15 2013

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. A002378, A051673, A204008, A214604, A214675.

a214659 n = ((7 * n - 3) * n - 1) * n `div` 3

[(7*n^3-3*n^2-n)/3 : n in [0..50]]; // Wesley Ivan Hurt, Apr 11 2015

A214659:=n->(7*n^3-3*n^2-n)/3: seq(A214659(n), n=0..50); # Wesley Ivan Hurt, Apr 11 2015

Table[(7 n^3 -3 n^2 -n)/3, {n,0,50}] (* Wesley Ivan Hurt, Apr 11 2015 *)
LinearRecurrence[{4,-6,4,-1}, {0,1,14,53}, 51] (* G. C. Greubel, Mar 09 2024 *)

[(7*n^3-3*n^2-n)/3 for n in range(51)] # G. C. Greubel, Mar 09 2024

a(n) = Sum_{k=0..n} A214604(n, k) for n > 0 (row sums).
a(n) = A002378(n) + A051673(n).
From Wesley Ivan Hurt, Apr 11 2015: (Start)
a(n) = (7*n^3 - 3*n^2 - n)/3.
G.f.: x*(1+10*x+3*x^2)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
E.g.f.: (x/3)*(3 + 18*x + 7*x^2)*exp(x). - G. C. Greubel, Mar 09 2024

A214660 a(n) = 9n^2 - 11n + 3.

Original entry on oeis.org

1, 17, 51, 103, 173, 261, 367, 491, 633, 793, 971, 1167, 1381, 1613, 1863, 2131, 2417, 2721, 3043, 3383, 3741, 4117, 4511, 4923, 5353, 5801, 6267, 6751, 7253, 7773, 8311, 8867, 9441, 10033, 10643, 11271, 11917, 12581, 13263, 13963, 14681, 15417, 16171, 16943

Offset: 1

Reinhard Zumkeller, Jul 25 2012

Central terms of triangle A214604.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A214604, A214675.

Haskell
```
a214660 n = (9 * n - 11) * n + 3
```

[9*n^2-11*n+3: n in [1..60]]; // G. C. Greubel, Mar 09 2024

Table[9n^2-11n+3,{n,60}] (* or *) LinearRecurrence[{3,-3,1},{1,17,51},60] (* Harvey P. Dale, Aug 29 2021 *)

a(n)=9*n^2-11*n+3 \\ Charles R Greathouse IV, Jun 17 2017

[9*n^2-11*n+3 for n in range(1,61)] # G. C. Greubel, Mar 09 2024

G.f.: (1+14*x+3*x^2)/(1-x)^3. - Harvey P. Dale, Aug 29 2021

E.g.f.: -3 + (3 - 2*x + 9*x^2)*exp(x). - G. C. Greubel, Mar 09 2024

cp's OEIS Frontend

A214661 Odd numbers obtained by transposing the left half of A176271 into rows of a triangle: T(n,k) = A176271(n - 1 + k, k), 1 <= k <= n.

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A220083 a(n) = (15*n^2 + 9*n + 2)/2.

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A100176 Structured octagonal prism numbers.

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Magma

Maple

Mathematica

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A214659 a(n) = n*(7*n^2 - 3*n - 1)/3.

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Author

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Programs

Haskell

Magma

Maple

Mathematica

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Formula

A214660 a(n) = 9*n^2 - 11*n + 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

SageMath

Formula

A220083 a(n) = (15n^2 + 9n + 2)/2.

A214659 a(n) = n(7n^2 - 3*n - 1)/3.

A214660 a(n) = 9n^2 - 11n + 3.