A154266 -id:A154266 - cp's OEIS frontend

A154262 a(n) = 9n^2 - 10n + 3.

3, 2, 19, 54, 107, 178, 267, 374, 499, 642, 803, 982, 1179, 1394, 1627, 1878, 2147, 2434, 2739, 3062, 3403, 3762, 4139, 4534, 4947, 5378, 5827, 6294, 6779, 7282, 7803, 8342, 8899, 9474, 10067, 10678, 11307, 11954, 12619, 13302, 14003, 14722, 15459, 16214, 16987

Offset: 0

Vincenzo Librandi, Jan 06 2009

The identity (81*n^2 + 72*n + 17)^2 - (9*n^2 + 8*n + 2)*(27*n + 12)^2 = 1 can be written as A154295(n+1)^2 - a(n+1)*A154266(n)^2 = 1. - Vincenzo Librandi, Feb 03 2012

For n >= 1, the continued fraction expansion of sqrt(a(n)) is [3n-2; {2, 1, 3n-3, 1, 2, 6n-4}]. For n=1, this collapses to [1; {2}]. - Magus K. Chu, Sep 05 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A154266, A154295.

I:=[2, 19, 54]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 02 2012

LinearRecurrence[{3, -3, 1}, {2, 19, 54}, 50] (* Vincenzo Librandi, Feb 02 2012 *)
Table[9n^2-10n+3,{n,0,50}] (* Harvey P. Dale, Feb 11 2023 *)

a(n)=9*n^2-10*n+3 \\ Charles R Greathouse IV, Dec 27 2011

From Vincenzo Librandi, Feb 02 2012: (Start)
G.f.: (3 - 7*x + 22*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
E.g.f.: (3 - x + 9*x^2)*exp(x). - Elmo R. Oliveira, Oct 31 2024

Edited by Charles R Greathouse IV, Jul 25 2010

A154295 a(n) = 81n^2 - 90n + 26.

Original entry on oeis.org

26, 17, 170, 485, 962, 1601, 2402, 3365, 4490, 5777, 7226, 8837, 10610, 12545, 14642, 16901, 19322, 21905, 24650, 27557, 30626, 33857, 37250, 40805, 44522, 48401, 52442, 56645, 61010, 65537, 70226, 75077, 80090, 85265, 90602, 96101, 101762

Offset: 0

Vincenzo Librandi, Jan 06 2009

The identity (81*n^2 + 72*n + 17)^2 - (9*n^2 + 8*n + 2)*(27*n + 12)^2 = 1 can be written as a(n+1)^2 - A154262(n+1)*A154266(n)^2 = 1. - Vincenzo Librandi, Feb 03 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A154262, A154266.

I:=[26, 17, 170]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 03 2012

LinearRecurrence[{3, -3, 1}, {26, 17, 170}, 40] (* Vincenzo Librandi, Feb 03 2012 *)
Table[81*n^2 - 90*n + 26,{n,0,25}] (* G. C. Greubel, Sep 10 2016 *)

for(n=0, 22, print1(81*n^2-90*n+26", ")); \\ Vincenzo Librandi, Feb 03 2012

x='x+O('x^99); Vec((26-61*x+197*x^2)/(1-x)^3) \\ Altug Alkan, Sep 10 2016

a(n) = A002522(|9n-5|). - R. J. Mathar, Jan 07 2009
G.f.: (26 - 61*x + 197*x^2)/(1 - x)^3. - Vincenzo Librandi, Feb 03 2012
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Vincenzo Librandi, Feb 03 2012
E.g.f.: (26 - 9*x + 81*x^2)*exp(x). - G. C. Greubel, Sep 10 2016

Corrected by Don Reble, Jun 16 2010

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)

cp's OEIS Frontend

A154262 a(n) = 9n^2 - 10n + 3.

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A154295 a(n) = 81n^2 - 90n + 26.

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

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cp's OEIS Frontend

A154262 a(n) = 9*n^2 - 10*n + 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A154295 a(n) = 81*n^2 - 90*n + 26.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A154262 a(n) = 9n^2 - 10n + 3.

A154295 a(n) = 81n^2 - 90n + 26.

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