A176126 -id:A176126 - cp's OEIS frontend

A127276 Hankel transform of A127275.

1, 1, -2, -16, -64, -208, -608, -1664, -4352, -11008, -27136, -65536, -155648, -364544, -843776, -1933312, -4390912, -9895936, -22151168, -49283072, -109051904, -240123904, -526385152, -1149239296, -2499805184, -5419040768

Offset: 0

Paul Barry, Jan 10 2007

sign

The inverse binomial transform of this sequence yields 1, 0, -3, -8,..., which is 1 followed by the negated terms of A005563. [Paul Curtz, Dec 07 2010]

The smallest odd prime divisor of a(n) is >= 13. - Vladimir Shevelev, Feb 03 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A076616, A176126, A236999.

[2^n-n*(n+1)*2^(n-2): n in [0..30]]; // Vincenzo Librandi, Aug 11 2011

A127276:=n->2^n - n*(n + 1)*2^(n - 2); seq(A127276(n), n=0..26); # Wesley Ivan Hurt, Dec 02 2013

Table[2^n - n*(n + 1)*2^(n - 2), {n, 0, 26}] (* Wesley Ivan Hurt, Dec 02 2013 *)

Conjecture: G.f.: -(4*x-1)*(x-1) / ( (2*x-1)^3 ) and a(n) = 2^n-n*(n+1)*2^(n-2). - R. J. Mathar, Dec 11 2010

a(n) = A178987(n) - A178987(n+1). - Klaus Brockhaus, Jan 08 2011

A181407 a(n) = (n-4)(n-3)2^(n-2).

Original entry on oeis.org

3, 3, 2, 0, 0, 16, 96, 384, 1280, 3840, 10752, 28672, 73728, 184320, 450560, 1081344, 2555904, 5963776, 13762560, 31457280, 71303168, 160432128, 358612992, 796917760, 1761607680, 3875536896, 8489271296, 18522046464, 40265318400, 87241523200, 188441690112

Offset: 0

Paul Curtz, Jan 28 2011

Binomial transform of (3, 0, -1, followed by A005563).
The sequence and its successive differences are:
   3,  3,  2,   0,   0,   16,   96,  384,      a(n),
   0, -1, -2,   0,  16,   80,  288,  896,      A178987,
  -1, -1,  2,  16,  64,  208,  608, 2688,     -A127276,
   0,  3, 14,  48, 144,  400, 1056, 2688,      A176027,
   3, 11, 34,  96, 256,  656, 1632, 3968,      A084266(n+1)
   8, 23, 62, 160, 400,  976, 2336, 5504,
  15, 39, 98, 240, 576, 1360, 3168, 7296.
Division of the k-th column by abs(A174882(k)) gives
   3,  3,  1,  0,  0,  1,  3,  3,  5,  15,  21, 14,   A064038(n-3),
   0, -1, -1,  0,  1,  5,  9,  7, 10,  27,  35, 22,   A160050(n-3),
  -1, -1,  1,  2,  4, 13, 19, 13, 17,  43,  53, 32,   A176126,
   0,  3,  7,  6,  9, 25, 33, 21, 26,  63,  75, 44,   A178242,
   3, 11, 17, 12, 16, 41, 51, 31, 37,  87, 101, 58,
   8  23, 31, 20, 25, 61, 73, 43, 50, 115, 131, 74,
  15, 39, 49, 30, 36, 85, 99, 57, 65, 147, 165, 92.
Columns are (or from) A005563, A142463, A056220, A002378, A000290, A001844, A058331, A002061, A002522, A097080, A093328, A014206.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A176027, A181318.

List([0..40], n-> (n-4)*(n-3)*2^(n-2)); # G. C. Greubel, Feb 21 2019

[(n-4)*(n-3)*2^(n-2): n in [0..40] ]; // Vincenzo Librandi, Feb 01 2011

Table[(n-4)*(n-3)*2^(n-2), {n,0,40}] (* G. C. Greubel, Feb 21 2019 *)

vector(40, n, n--; (n-4)*(n-3)*2^(n-2)) \\ G. C. Greubel, Feb 21 2019

[(n-4)*(n-3)*2^(n-2) for n in (0..40)] # G. C. Greubel, Feb 21 2019

a(n) = 16*A001788(n-4).
a(n+1) - a(n) = A178987(n).
G.f.: (3 - 15*x + 20*x^2) / (1-2*x)^3. - R. J. Mathar, Jan 30 2011
E.g.f.: (x^2 - 3*x + 3)*exp(2*x). - G. C. Greubel, Feb 21 2019

A182868 a(n) = -1 + n + 4*n^2.

Original entry on oeis.org

-1, 4, 17, 38, 67, 104, 149, 202, 263, 332, 409, 494, 587, 688, 797, 914, 1039, 1172, 1313, 1462, 1619, 1784, 1957, 2138, 2327, 2524, 2729, 2942, 3163, 3392, 3629, 3874, 4127, 4388, 4657, 4934, 5219, 5512, 5813, 6122, 6439, 6764, 7097, 7438, 7787, 8144, 8509, 8882, 9263, 9652

Offset: 0

Paul Curtz, Feb 01 2011

First quadrisection of A176126(n). Take clockwise (square) spiral from A023443(n)=n-1: a(n) is on the negative x-axis. Fourth quadrisection (-1-n+4*n^2) is on the negative y-axis.
Conjecture: the 4 quadrisections of (the family) A064038, A160050, A176126, A178242 (see A181407) come from square spiral.
a(n) mod 9 has period 9: 8,4,8,2,4,5,5,4,2. a(n) mod 10 has period 10: 9,4,7,8,7,4,9,2,3,2. Each polynomial modulo some constant c has a period of length c (and perhaps shorter ones). - Paul Curtz and Bruno Berselli, Feb 05 2011

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

[-1+n+4*n^2: n in [0..700] ] // Vincenzo Librandi, Feb 01 2011

f[n_]:=-1+n+4*n^2;f[Range[0,100]] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)

a(n)=-1+n+4*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A176126(4*n).
a(n) = 4*n^2 + n - 1.
a(n) = a(n-1) - 3 + 8*n.
a(n) = 2*a(n) - a(n-2) + 8.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -(1 - 7*x - 2*x^2)/(1-x)^3. - Bruno Berselli, Feb 05 2011

A185950 a(n) = 4*n^2 - n - 1.

Original entry on oeis.org

-1, 2, 13, 32, 59, 94, 137, 188, 247, 314, 389, 472, 563, 662, 769, 884, 1007, 1138, 1277, 1424, 1579, 1742, 1913, 2092, 2279, 2474, 2677, 2888, 3107, 3334, 3569, 3812, 4063, 4322, 4589, 4864, 5147, 5438, 5737, 6044, 6359, 6682, 7013, 7352, 7699, 8054, 8417, 8788, 9167, 9554, 9949, 10352, 10763, 11182, 11609

Offset: 0

Paul Curtz, Feb 07 2011

Write the sequence A023443 in a clockwise spiral. a(n) is on the y-axis.

a(n) mod 9 = period 9: repeat [8,2,4,5,5,4,2,8,4] = A182868(n+2) mod 9.

			  11--12--13--14--15
   |               |
  10   1---2---3  16
   |   |       |   |
   9   0-(-1)  4  17
   |           |   |
   8---7---6---5  18

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033951, A182868.

a185950 n = (4 * n - 1) * n - 1 -- Reinhard Zumkeller, Aug 14 2013

[-1-n+4*n^2: n in [0..80]]; // Vincenzo Librandi, Feb 08 2011

A185950:=n->4*n^2-n-1: seq(A185950(n), n=0..100); # Wesley Ivan Hurt, Jan 30 2017

Table[4n^2-n-1,{n,0,60}] (* or *) LinearRecurrence[{3,-3,1},{-1,2,13},60] (* Harvey P. Dale, May 22 2015 *)

a(n)=4*n^2-n-1 \\ Charles R Greathouse IV, Dec 21 2011

a(n) = A176126(4*n-1) = A054556(n+1) - 2 = A033991(n) - 1.
a(n) = a(n-1) + 8*n - 5.
a(n) = 2*a(n-1) - a(n-2) + 8.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: ( 1-5*x-4*x^2 ) / (x-1)^3. - R. J. Mathar, Feb 10 2011
E.g.f.: (4*x^2 + 3*x - 1)*exp(x). - G. C. Greubel, Jul 23 2017

A236999 Odd part of n*(n+3)/2-1 (A034856).

Original entry on oeis.org

1, 1, 1, 13, 19, 13, 17, 43, 53, 1, 19, 89, 103, 59, 67, 151, 169, 47, 13, 229, 251, 137, 149, 323, 349, 47, 101, 433, 463, 247, 263, 559, 593, 157, 83, 701, 739, 389, 409, 859, 901, 59, 247, 1033, 1079, 563, 587, 1223, 1273, 331, 43, 1429, 1483

Offset: 1

Vladimir Shevelev, Feb 02 2014

Also odd part of A176126(n-1) and of |A127276(n-1)|, n>=3.
Proof. By A127276 and A001788, we have odd part(A176126(n))=odd part(|A127276(n)|) = odd part(n*(n+1)-4), {odd part(A176126(n-1)), n>=3}={odd part((n+1)*(n+2)-4), n>=1}.
Let n=2^b*k, where k=k(n) is odd.
Then {odd part(A176126(n-1)), n>=3}={odd part((2^b*k+1)*(2^b*k+2)-4)}={odd part(2^(2*b)*k^2+3*2^b*k-2)}. Hence, if b>0, then {odd part(A176126(n-1), n>=3)= {odd part(2^(2*b-1)*k^2+3*2^(b-1)*k-1)}.
On the other hand, in this case odd part(a(n))=odd part(2^(b-1)*k*(2^b*k+3)-1)=odd part(2^(2*b-1)*k^2+3*2^(b-1)*k-1). It is left to consider the case of odd n. Setting n=2*m-1, m>=1, we easily find that for both expressions the odd part equals odd part(2*m^2+m-2).
The smallest prime divisor of a(n) is more than or equal to 13.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000265, A034856, A176126, A127276, A001788.

Map[#/2^IntegerExponent[#,2]&[(# (#+3)/2-1)]&,Range[100]] (* Peter J. C. Moses, Feb 02 2014 *)

a(n) = A000265(A034856(n)). - Michel Marcus, Feb 25 2025

A251091 a(n) = n^2 / gcd(n+2, 4).

Original entry on oeis.org

0, 1, 1, 9, 8, 25, 9, 49, 32, 81, 25, 121, 72, 169, 49, 225, 128, 289, 81, 361, 200, 441, 121, 529, 288, 625, 169, 729, 392, 841, 225, 961, 512, 1089, 289, 1225, 648, 1369, 361, 1521, 800, 1681, 441, 1849, 968, 2025, 529, 2209, 1152, 2401, 625, 2601, 1352

Offset: 0

Paul Curtz, May 08 2015

A061038(n), which appears in 4*a(n) formula, is a permutation of n^2.
Origin. In December 2010, I wrote in my 192-page Exercise Book no. 5, page 41, the array (difference table of the first row):
   1     0,   1/3,     1,   9/5,    8/3,   25/7,    9/2,   49/9, ...
  -1,  1/3,   2/3,   4/5, 13/15,  19/21,  13/14,  17/18,  43/45, ...
Numerators are listed in A176126, denominators are in A064038, and denominator - numerator = 2, 2, 1, 1,... (A014695).
  4/3, 1/3,  2/15,  1/15, 4/105,   1/42,   1/63,   1/90,  4/495, ...
  -1, -1/5, -1/15, -1/35, -1/70, -1/126, -1/210, -1/330, -1/495, ...
where the denominators of the second row are listed in A000332.
Also for those of the inverse binomial transform
  1, -1, 4/3, -1, 4/5, -2/3, 4/7, -1/2, 4/9, -2/5, 4/11, -1/3, ... ?
a(n) is the (n+1)-th term of the numerators of the first row.

			a(0) = 0/2, a(1) = 1/1, a(2) = 4/4, a(3) = 9/1.

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

Cf. A000290, A000332, A016754, A026741, A060819, A061038, A109008, A139098, A168077, A176895, A181900.

[(1-(1/16)*(1+(-1)^n)*(5-(-1)^(n div 2)) )*n^2: n in [0..60]]; // Vincenzo Librandi, Jun 12 2015

seq(seq((4*i+j-1)^2/[2,1,4,1][j],j=1..4),i=0..30); # Robert Israel, May 14 2015

f[n_] := Switch[ Mod[n, 4], 0, n^2/2, 1, n^2, 2, n^2/4, 3, n^2]; Array[f, 50, 0] (* or *) Table[(4 i + j - 1)^2/{2, 1, 4, 1}[[j]], {i, 0, 12}, {j, 4}] // Flatten (* after Robert Israel *) (* or *) LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {0, 1, 1, 9, 8, 25, 9, 49, 32, 81, 25, 121}, 53] (* or *) CoefficientList[ Series[-((x (1 + x (1 + x (9 + x (8 + x (22 + x (6 + x (22 + x (8 + x (9 + x + x^2))))))))))/(-1 + x^4)^3), {x, 0, 52}], x] (* Robert G. Wilson v, May 19 2015 *)

concat(0, Vec(-x*(x^10 + x^9 + 9*x^8 + 8*x^7 + 22*x^6 + 6*x^5 + 22*x^4 + 8*x^3 + 9*x^2 + x + 1) / ((x-1)^3*(x+1)^3*(x^2+1)^3) + O(x^100))) \\ Colin Barker, May 14 2015

a(n) = n^2/(period 4: repeat 2, 1, 4, 1).
a(4n) = 8*n^2, a(2n+1) = a(4n+2) = (2*n+1)^2.
a(n+4) = a(n) + 8*A060819(n).
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12), n>11.
4*a(n) = (period 4: repeat 2, 1, 4, 1) * A061038(n).
G.f.: -x*(x^10+x^9+9*x^8+8*x^7+22*x^6+6*x^5+22*x^4+8*x^3+9*x^2+x+1) / ((x-1)^3*(x+1)^3*(x^2+1)^3). - Colin Barker, May 14 2015
a(2n) = A181900(n), a(2n+1) = A016754(n). [Bruno Berselli, May 14 2015]
a(n) = ( 1 - (1/16)*(1+(-1)^n)*(5-(-1)^(n/2)) )*n^2. - Bruno Berselli, May 14 2015
Sum_{n>=1} 1/a(n) = 13*Pi^2/48. - Amiram Eldar, Aug 12 2022

Missing term (1521) inserted in the sequence by Colin Barker, May 14 2015
Definition uses a formula by Jean-François Alcover, Jul 01 2015
Keyword:mult added by Andrew Howroyd, Aug 06 2018

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A127276 Hankel transform of A127275.

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A181407 a(n) = (n-4)*(n-3)*2^(n-2).

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A182868 a(n) = -1 + n + 4*n^2.

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A185950 a(n) = 4*n^2 - n - 1.

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A236999 Odd part of n*(n+3)/2-1 (A034856).

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A251091 a(n) = n^2 / gcd(n+2, 4).

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A181407 a(n) = (n-4)(n-3)2^(n-2).