A204674 -id:A204674 - cp's OEIS frontend

A054567 a(n) = 4n^2 - 7n + 4.

1, 6, 19, 40, 69, 106, 151, 204, 265, 334, 411, 496, 589, 690, 799, 916, 1041, 1174, 1315, 1464, 1621, 1786, 1959, 2140, 2329, 2526, 2731, 2944, 3165, 3394, 3631, 3876, 4129, 4390, 4659, 4936, 5221, 5514, 5815, 6124, 6441, 6766, 7099, 7440, 7789, 8146, 8511, 8884

Offset: 1

Enoch Haga, G. L. Honaker, Jr., Apr 10 2000

The number 1 is placed in the middle of a sheet of squared paper and the numbers 2, 3, 4, 5, 6, etc. are written in a clockwise spiral around 1, as in A068225 etc. This sequence is read off along one of the rays from 1.
Ulam's spiral (W spoke of A054552). - Robert G. Wilson v, Oct 31 2011
Also, numbers of the form m*(4*m+1)+1 for nonnegative m. - Bruno Berselli, Jan 06 2016
The sequence forms the 1x2 diagonal of the square maze arrangement in A081344. - Jarrod G. Sage, Jul 17 2024

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Robert G. Wilson v, Cover of the March 1964 issue of Scientific American
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A054566, A068225, A054552, A054554, A054556, A054569, A033951, A204674.
Cf. A266883: m*(4*m+1)+1 for m = 0,-1,1,-2,2,-3,3,...
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.

Table[4 n^2 - 7 n + 4, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Sep 01 2008 *)

Vec(-x*(4*x^2+3*x+1)/(x-1)^3 + O(x^100)) \\ Colin Barker, Oct 25 2014

a(n) = 8*n+a(n-1)-11 for n>1, a(1)=1. - Vincenzo Librandi, Aug 07 2010
a(n) = A204674(n-1) / n. - Reinhard Zumkeller, Jan 18 2012
From Colin Barker, Oct 25 2014: (Start)
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3).
G.f.: -x*(4*x^2+3*x+1) / (x-1)^3. (End)
E.g.f.: exp(x)*(4 - 3*x + 4*x^2) - 4. - Stefano Spezia, Apr 24 2024
a(n) = A016742(n-1) + n. - Jarrod G. Sage, Jul 17 2024

Edited by Frank Ellermann, Feb 24 2002

Typo fixed by Charles R Greathouse IV, Oct 28 2009

A226449 a(n) = n(5n^2-8*n+5)/2.

Original entry on oeis.org

0, 1, 9, 39, 106, 225, 411, 679, 1044, 1521, 2125, 2871, 3774, 4849, 6111, 7575, 9256, 11169, 13329, 15751, 18450, 21441, 24739, 28359, 32316, 36625, 41301, 46359, 51814, 57681, 63975, 70711, 77904, 85569, 93721, 102375, 111546, 121249, 131499, 142311, 153700

Offset: 0

Bruno Berselli, Jun 07 2013

Sequences of the type b(m)+m*b(m-1), where b is a polygonal number:
A006003(n) = A000217(n) + n*A000217(n-1)      (b = triangular numbers);
A069778(n) = A000290(n+1) + (n+1)*A000290(n)  (b = square numbers);
A143690(n) = A000326(n+1) + (n+1)*A000326(n)  (b = pentagonal numbers);
A212133(n) = A000384(n) + n*A000384(n-1)      (b = hexagonal numbers);
a(n)       = A000566(n) + n*A000566(n-1)      (b = heptagonal numbers);
A226450(n) = A000567(n) + n*A000567(n-1)      (b = octagonal numbers);
A226451(n) = A001106(n) + n*A001106(n-1)      (b = nonagonal numbers);
A204674(n) = A001107(n+1) + (n+1)*A001107(n)  (b = decagonal numbers).

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

Cf. (see the comment) A000566, A006003, A069778, A143690, A204674, A212133, A226450, A226451.

Magma
```
[n*(5*n^2-8*n+5)/2: n in [0..40]];
```

I:=[0,1,9,39]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi, Aug 18 2013

Table[n (5 n^2 - 8 n + 5)/2, {n, 0, 40}]
CoefficientList[Series[x (1 + 5 x + 9 x^2)/(1 - x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{4,-6,4,-1},{0,1,9,39},50] (* Harvey P. Dale, May 19 2017 *)

a(n)=n*(5*n^2-8*n+5)/2 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(1+5*x+9*x^2)/(1-x)^4.

a(n) - a(-n) = A008531(n) for n>0.

A033293 A Connell-like sequence: take 1 number = 1 (mod Q), 2 numbers = 2 (mod Q), 3 numbers = 3 (mod Q), etc., where Q = 8.

Original entry on oeis.org

1, 2, 10, 11, 19, 27, 28, 36, 44, 52, 53, 61, 69, 77, 85, 86, 94, 102, 110, 118, 126, 127, 135, 143, 151, 159, 167, 175, 176, 184, 192, 200, 208, 216, 224, 232, 233, 241, 249, 257, 265, 273, 281, 289, 297, 298, 306, 314, 322, 330, 338, 346, 354, 362, 370, 371, 379, 387, 395, 403, 411

Offset: 1

Gary E. Stevens

Reinhard Zumkeller, Rows n=1..120 of triangle, flattened
Gary E. Stevens, A Connell-Like Sequence, J. Integer Sequences, 1 (1998), #98.1.4.

Cf. A054552 (left edge), A001107 (right edge), A204674 (row sums), A204675 (central terms).

a033293 n k = a033293_tabl !! (n-1) !! (k-1)
a033293_row n = a033293_tabl !! (n-1)
a033293_tabl = f 1 [1..] where
   f k xs = ys : f (k+1) (dropWhile (<= last ys) xs) where
     ys  = take k $ filter ((== 0) . (`mod` 8) . (subtract k)) xs
-- Reinhard Zumkeller, Jan 18 2012 2011

row[1] = {1}; row[n_] := row[n] = Table[row[n-1][[-1]] + 8k + 1, {k, 0, n-1}]; Table[row[n], {n, 1, 11}] // Flatten (* Jean-François Alcover, Jan 25 2013 *)

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

Offset changed by Reinhard Zumkeller, Jan 18 2012

A226450 a(n) = n(3n^2 - 5*n + 3).

Original entry on oeis.org

0, 1, 10, 45, 124, 265, 486, 805, 1240, 1809, 2530, 3421, 4500, 5785, 7294, 9045, 11056, 13345, 15930, 18829, 22060, 25641, 29590, 33925, 38664, 43825, 49426, 55485, 62020, 69049, 76590, 84661, 93280, 102465, 112234, 122605, 133596, 145225, 157510, 170469

Offset: 0

Bruno Berselli, Jun 07 2013

See the comment in A226449.

For n >= 3, also the detour index of the n-barbell graph. - Eric W. Weisstein, Dec 20 2017

Bruno Berselli, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Barbell Graph
Eric Weisstein's World of Mathematics, Detour Index
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000567.

Similar sequences of the type b(m)+m*b(m-1), where b is a polygonal number: A006003, A069778, A143690, A204674, A212133, A226449, A226451.

Magma
```
[n*(3*n^2-5*n+3): n in [0..40]];
```

I:=[0,1,10,45]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..45]]; // Vincenzo Librandi, Aug 18 2013

Table[n (3 n^2 - 5 n + 3), {n, 0, 40}]
CoefficientList[Series[x (1 + 6 x + 11 x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 10, 45, 124}, {0, 20}] (* Eric W. Weisstein, Dec 20 2017 *)

a(n) = n*(3*n^2 - 5*n + 3); \\ Altug Alkan, Dec 20 2017

G.f.: x*(1+6*x+11*x^2)/(1-x)^4.

a(n) = A000567(n) + n*A000567(n-1).

A226451 a(n) = n(7n^2-12*n+7)/2.

Original entry on oeis.org

0, 1, 11, 51, 142, 305, 561, 931, 1436, 2097, 2935, 3971, 5226, 6721, 8477, 10515, 12856, 15521, 18531, 21907, 25670, 29841, 34441, 39491, 45012, 51025, 57551, 64611, 72226, 80417, 89205, 98611, 108656, 119361, 130747, 142835, 155646, 169201, 183521

Offset: 0

Bruno Berselli, Jun 07 2013

See the comment in A226449.

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

Cf. A001106.

Similar sequences of the type b(m)+m*b(m-1), where b is a polygonal number: A006003, A069778, A143690, A204674, A212133, A226449, A226450.

Magma
```
[n*(7*n^2-12*n+7)/2: n in [0..40]];
```

I:=[0,1,11,51]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Aug 18 2013

Table[n (7 n^2 - 12 n + 7)/2, {n, 0, 40}]
CoefficientList[Series[x (1 + 7 x + 13 x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

G.f.: x*(1+7*x+13*x^2)/(1-x)^4.
a(n) = A001106(n) + n*A001106(n-1).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. - Wesley Ivan Hurt, Oct 15 2023

cp's OEIS Frontend

A054567 a(n) = 4*n^2 - 7*n + 4.

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A226449 a(n) = n*(5*n^2-8*n+5)/2.

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A033293 A Connell-like sequence: take 1 number = 1 (mod Q), 2 numbers = 2 (mod Q), 3 numbers = 3 (mod Q), etc., where Q = 8.

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A226450 a(n) = n*(3*n^2 - 5*n + 3).

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Magma

Magma

Mathematica

PARI

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A226451 a(n) = n*(7*n^2-12*n+7)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Mathematica

Formula

A054567 a(n) = 4n^2 - 7n + 4.

A226449 a(n) = n(5n^2-8*n+5)/2.

A226450 a(n) = n(3n^2 - 5*n + 3).

A226451 a(n) = n(7n^2-12*n+7)/2.