A036826 -id:A036826 - cp's OEIS frontend

A054569 a(n) = 4n^2 - 6n + 3.

1, 7, 21, 43, 73, 111, 157, 211, 273, 343, 421, 507, 601, 703, 813, 931, 1057, 1191, 1333, 1483, 1641, 1807, 1981, 2163, 2353, 2551, 2757, 2971, 3193, 3423, 3661, 3907, 4161, 4423, 4693, 4971, 5257, 5551, 5853, 6163, 6481, 6807, 7141, 7483, 7833, 8191

Offset: 1

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

Move in 1-7 direction in a spiral organized like A068225 etc.
Third row of A082039. - Paul Barry, Apr 02 2003
Inverse binomial transform of A036826. - Paul Barry, Jun 11 2003
Equals the "middle sequence" T(2*n,n) of the Connell sequence A001614 as a triangle. - Johannes W. Meijer, May 20 2011
Ulam's spiral (SW spoke). - Robert G. Wilson v, Oct 31 2011

Harvey P. Dale, 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. A000384, A033951, A054552, A054554, A054556, A054567, A054568, A068225.
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.

List([1..50], n-> 4*n^2-6*n+3); # G. C. Greubel, Jul 04 2019

[4*n^2-6*n+3: n in [1..50]]; // G. C. Greubel, Jul 04 2019

f[n_]:= 4*n^2-6*n+3; Array[f, 50] (* Vladimir Joseph Stephan Orlovsky, Sep 02 2008 *)
LinearRecurrence[{3,-3,1},{1,7,21},50] (* Harvey P. Dale, Nov 17 2012 *)

a(n)=4*n^2-6*n+3 \\ Charles R Greathouse IV, Sep 24 2015

[4*n^2-6*n+3 for n in (1..50)] # G. C. Greubel, Jul 04 2019

a(n+1) = 4*n^2 + 2*n + 1. - Paul Barry, Apr 02 2003
a(n) = 4*n^2 - 6*n+3 - 3*0^n (with leading zero). - Paul Barry, Jun 11 2003
Binomial transform of [1, 6, 8, 0, 0, 0, ...]. - Gary W. Adamson, Dec 28 2007
a(n) = 8*n + a(n-1) - 10 (with a(1)=1). - Vincenzo Librandi, Aug 07 2010
From Colin Barker, Mar 23 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(1+x)*(1+3*x)/(1-x)^3. (End)
a(n) = A000384(n) + A000384(n-1). - Bruce J. Nicholson, May 07 2017
E.g.f.: -3 + (3 - 2*x + 4*x^2)*exp(x). - G. C. Greubel, Jul 04 2019
Sum_{n>=1} 1/a(n) = A339237. - R. J. Mathar, Jan 22 2021

Edited by Frank Ellermann, Feb 24 2002

A014477 Expansion of (1 + 2x)/(1 - 2x)^3.

Original entry on oeis.org

1, 8, 36, 128, 400, 1152, 3136, 8192, 20736, 51200, 123904, 294912, 692224, 1605632, 3686400, 8388608, 18939904, 42467328, 94633984, 209715200, 462422016, 1015021568, 2218786816, 4831838208, 10485760000, 22682796032, 48922361856, 105226698752, 225754218496

Offset: 0

N. J. A. Sloane

The sequence 0,1,8,... has a(n) = n^2*2^(n-1) and is the binomial transform of the hexagonal numbers A000384 (with leading 0). - Paul Barry, Jun 09 2003
As 0,1,8,... this is n^2*2^(n-1), the binomial transform of the hexagonal numbers A000384 (include the leading 0). Partial sums are A036826. - Paul Barry, Jun 10 2003
Sequence gives total value of all possible sums of distinct odd integers with maximum term less than 2n+1. E.g., for a(3) we can have the sums 1, 3, 5, 1+3, 1+5, 3+5, 1+3+5, which sum to 1+3+5+4+6+8+9 = 36. - Jon Perry, Feb 06 2004
Number of edges on a partially truncated (n+1)-cube (column 2 of A271316).

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

Cf. A007758, A036826, A118414, A355234.

[(n+1)^2*2^n: n in [0..35]]; // Vincenzo Librandi, Aug 21 2011

a:=n->sum(binomial(n,j)*n*j,j=0..n): seq(a(n), n=0..25); # Zerinvary Lajos, Oct 19 2006
a:=n->sum(n*numbcomb(n)/2, j=1..n): seq(a(n), n=1..25); # Zerinvary Lajos, Apr 25 2007

f[n_]:=(n^2*2^n)/2;Table[f[n],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Dec 05 2009 *)

a(n)=(n+1)^2*2^n \\ Charles R Greathouse IV, Apr 07 2016

O.g.f.: (1 + 2*x)/(1 - 2*x)^3 (see the name).
a(n) = (n+1)^2*2^n = A007758(n+1)/2. - Henry Bottomley, Jun 13 2001
The binomial transform of 0, 1, 8, ... is A077616. - Paul Barry, Jul 24 2003
a(1)=1, a(n) = 2a(n-1) + (2n-1)*2^(n-1). - Jon Perry, Feb 06 2004
a(n) = sum of (n+1)-th row of the triangle in A118416. - Reinhard Zumkeller, Apr 27 2006
a(n) = Sum_{j=0..n} binomial(n,j)*n*j. - Zerinvary Lajos, Oct 19 2006
E.g.f.: exp(2*x)*(1 + 6*x + 8*x^2/2!). - Wolfdieter Lang, Jul 29 2017
Sum_{n>=0} 1/a(n) = Pi^2/6 - log(2)^2. - Daniel Suteu, Oct 31 2017
Sum_{n>=0} (-1)^n/a(n) = -2 *  Li_2(-1/2) = -2 * A355234. - Amiram Eldar, Oct 01 2022

A036800 a(n) = -6 + 2^(n+1)(3 - 2n + n^2).

Original entry on oeis.org

0, 2, 18, 90, 346, 1146, 3450, 9722, 26106, 67578, 169978, 417786, 1007610, 2392058, 5603322, 12976122, 29753338, 67633146, 152567802, 341835770, 761266170, 1686110202, 3716153338, 8153726970, 17817403386, 38788923386

Offset: 0

N. J. A. Sloane

This sequence is a part of a class of sequences of the type: a(n) = sum(i=0,n,(C^i)*(i^k)). This sequence has C=2, k=2. Sequence A036799 has C=2, k=1. Suppose C>=2, k>=1 are integers. What is the general closed form for a(n)? - Ctibor O. Zizka, Feb 07 2008

M. Petkovsek et al., A=B, Peters, 1996, p. 97.
Jolley, Summation of Series, Dover (1961), p. 6.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

Cf. A232599, A232600, A232601, A232602.

[-6+2^(n+1)*(3-2*n+n^2): n in [0..30]]; // Vincenzo Librandi, Oct 04 2011

A036800:= n-> 2^(n+1)*(3-2*n+n^2) -6; seq(A036800(n), n=0..30); # G. C. Greubel, Mar 31 2021

Table[ -6+2^(n+1)*(3-2*n+n^2),{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2010 *)
LinearRecurrence[{7,-18,20,-8},{0,2,18,90},30] (* Harvey P. Dale, Jun 13 2015 *)

a(n)=2^(n+1)*(3 - 2*n + n^2) - 6 \\ Charles R Greathouse IV, Jun 11 2015

[2^(n+1)*(3-2*n+n^2) -6 for n in (0..30)] # G. C. Greubel, Mar 31 2021

a(n) = Sum_{k=0..n} 2^k * k^2. - Benoit Cloitre, Jun 11 2003
From R. J. Mathar, Oct 03 2011: (Start)
G.f.: 2*x*(1+2*x) / ( (1-x)*(1-2*x)^3 ).
a(n) = 2*A036826(n). (End)
a(0)=0, a(1)=2, a(2)=18, a(3)=90, a(n)=7*a(n-1)-18*a(n-2)+ 20*a(n-3)- 8*a(n-4). - Harvey P. Dale, Jun 13 2015
a(n) = Sum_{k=0..n} Sum_{i=0..n} k^2 * C(k,i). - Wesley Ivan Hurt, Sep 21 2017
E.g.f.: 2*(3 -2*x +4*x^2)*exp(2*x) -6*exp(x). - G. C. Greubel, Mar 31 2021

A209359 a(n) = 2^n * (n^4 - 4n^3 + 18n^2 - 52*n + 75) - 75.

Original entry on oeis.org

0, 1, 33, 357, 2405, 12405, 53877, 207541, 731829, 2411445, 7531445, 22523829, 64991157, 181977013, 496680885, 1326120885, 3473604533, 8947236789, 22706651061, 56869519285, 140755599285, 344683708341, 835954147253, 2009692372917, 4792831180725, 11346431180725

Offset: 0

Bruno Berselli, Mar 07 2012

This sequence is related to A036828 by the transform a(n) = n*A036828(n) - sum(A036828(i), i=0..n-1).

Bruno Berselli, Table of n, a(n) for n = 0..1000
B. Berselli, A description of the transform in Comments lines: website Matem@ticamente (in Italian).
Index entries for linear recurrences with constant coefficients, signature (11,-50,120,-160,112,-32).

Cf. A000079, A000337, A036826, A036828.

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((1+2*x)*(1+20*x+4*x^2)/((1-x)*(1-2*x)^5)));

LinearRecurrence[{11, -50, 120, -160, 112, -32}, {0, 1, 33, 357, 2405, 12405}, 26]
Table[2^n(n^4-4n^3+18n^2-52n+75)-75,{n,0,30}] (* Harvey P. Dale, Mar 08 2023 *)

for(n=0, 25, print1(2^n*(n^4-4*n^3+18*n^2-52*n+75)-75", "));

G.f.: x*(1+2*x)*(1+20*x+4*x^2)/((1-x)*(1-2*x)^5).

a(n) = (1/2) * Sum_{k=0..n} Sum_{i=0..n} k^4 * C(k,i). - Wesley Ivan Hurt, Sep 21 2017

A036828 A036827/2.

Original entry on oeis.org

0, 1, 17, 125, 637, 2637, 9549, 31501, 97037, 283661, 795661, 2158605, 5697549, 14696461, 37175309, 92471309, 226689037, 548667405, 1313079309, 3111125005, 7305429005, 17016291341, 39346765837, 90378862605, 206342979597, 468486979597, 1058239676429

Offset: 0

N. J. A. Sloane

This sequence is related to A036826 by a(n) = n*A036826(n) - Sum_{i=0..n-1} A036826(i). - Bruno Berselli, Mar 06 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-32,56,-48,16).

Cf. A036826, A209359.

m:=26; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(-(4*x^2+8*x+1)/((x-1)*(2*x-1)^4))); // Bruno Berselli, Mar 06 2012

LinearRecurrence[{9, -32, 56, -48, 16}, {0, 1, 17, 125, 637}, 27] (* Bruno Berselli, Mar 06 2012 *)

a(n) = 2^n*(n^3-3*n^2+9*n-13)+13 \\ Bruno Berselli, Mar 06 2012

G.f.: -x*(4*x^2+8*x+1)/((x-1)*(2*x-1)^4). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 13 2009

a(n) = 2^n*(n^3-3*n^2+9*n-13)+13. - Bruno Berselli, Mar 06 2012

Typo in definition corrected by R. J. Mathar, Sep 16 2009

A248830 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x-2k)^k for 0 <= k <= n.

Original entry on oeis.org

1, 3, 1, 3, 9, 1, 3, 45, 19, 1, 3, 173, 211, 33, 1, 3, 573, 1811, 633, 51, 1, 3, 1725, 13331, 9273, 1491, 73, 1, 3, 4861, 88595, 115113, 32851, 3013, 99, 1, 3, 13053, 547347, 1276329, 606291, 92613, 5475, 129, 1, 3, 33789, 3201555, 13033641, 9896019, 2360613, 223203, 9201, 163, 1, 3, 84989, 17947155, 125008041, 147521619, 52760613, 7480803, 479601, 14563, 201, 1

Offset: 0

Derek Orr, Oct 15 2014

Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-0)^0 + A_1*(x-2)^1 + A_2*(x-4)^2 + ... + A_n*(x-2n)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

			1;
3,     1;
3,     9,       1;
3,    45,      19,        1;
3,   173,     211,       33,       1;
3,   573,    1811,      633,      51,       1;
3,  1725,   13331,     9273,    1491,      73,      1;
3,  4861,   88595,   115113,   32851,    3013,     99,    1;
3, 13053,  547347,  1276329,  606291,   92613,   5475,  129,   1;
3, 33789, 3201555, 13033641, 9896019, 2360613, 223203, 9201, 163, 1;

Cf. A036826, A058331, A248829, A242598.

for(n=0,10,for(k=0,n,if(!k,if(n,print1(3,", "));if(!n,print1(1,", ")));if(k,print1(sum(i=1,n,((2*k)^(i-k)*i*binomial(i,k)))/k,", "))))

T(n,n-1) = 2*n^2 + 1 for n > 0.

T(n,1) = 2^n*(n^2-2*n+3)-3 for n > 0.

A236538 Triangle read by rows: T(n,k) = (n+1)2^(n-2)+(k-1)2^(n-1) for 1 <= k <= n.

Original entry on oeis.org

1, 3, 5, 8, 12, 16, 20, 28, 36, 44, 48, 64, 80, 96, 112, 112, 144, 176, 208, 240, 272, 256, 320, 384, 448, 512, 576, 640, 576, 704, 832, 960, 1088, 1216, 1344, 1472, 1280, 1536, 1792, 2048, 2304, 2560, 2816, 3072, 3328, 2816, 3328, 3840, 4352, 4864, 5376

Offset: 1

Fedor Igumnov, Jan 28 2014

1, 9, 45, 161, 497, 1409, ... is the sequence of perimeters (sum of border elements) of the triangle.
1, 5, 80, 3520, 394240, 107233280, 68629299200, ... is the sequence of determinants of the triangle.
Only the first three terms are odd.

			Triangle begins:
================================================
\k |    1     2     3     4     5     6     7
n\ |
================================================
1  |    1;
2  |    3,    5;
3  |    8,   12,   16;
4  |   20,   28,   36,   44;
5  |   48,   64,   80,   96,  112;
6  |  112,  144,  176,  208,  240,  272;
7  |  256,  320,  384,  448,  512,  576,  640;
...

Fedor Igumnov, T(n,k) for n = 1..26

Cf. A001792 (column 1), A053220 (right border). Also:
A014477, row sums;
A036826, partial sums;
A058962, central elements in odd rows;
A045623, second column;
A045891, third column;
A034007, fourth column;
A167667, subdiagonal;
A130129, second subdiagonal.

int a(int n, int k) {return (n+1)*pow(2,n-2)+(k-1)*pow(2,n-1);}

/* As triangle: */ [[(n+1)*2^(n-2)+(k-1)*2^(n-1): k in [1..n]]: n in [1..10]]; // Bruno Berselli, Jan 28 2014

t[n_, k_] := (n + 1)*2^(n - 2) + (k - 1)*2^(n - 1); Table[t[n, k], {n, 10}, {k, n}] // Flatten (* Bruno Berselli, Jan 28 2014 *)

T(n,k) = T(n-1,k) + T(n-1,k+1).

Sum_{k=1..n} T(n,k) = n^2*2^(n-1) = A014477(n-1).

More terms from Bruno Berselli, Jan 28 2014

A337631 a(n) is the sum of the squares of diameters of all nonempty subsets of {1,2,...,n}.

Original entry on oeis.org

0, 1, 10, 55, 228, 801, 2526, 7387, 20440, 54229, 139218, 348111, 851916, 2047945, 4849606, 11337667, 26214336, 60030909, 136314810, 307232695, 687865780, 1530920881, 3388997550, 7465861035, 16374562728, 35769024421, 77846282146, 168845901727

Offset: 1

Enrique Navarrete, Sep 20 2020

Partial sums of A036826.

For the sum of diameters of subsets of {1,2,...,n} see A045618.

			For n = 3, the nonempty subsets of {1,2,3} are {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}; the diameters of these sets are 0,0,0,1,1,2,2 and the sum of the squares of these numbers is 10.

Index entries for linear recurrences with constant coefficients, signature (8,-25,38,-28,8).

Cf. A036826, A045618.

From Stefano Spezia, Sep 21 2020: (Start)
G.f.: x*(1 + 2*x)/((1 - x)^2*(1 - 2*x)^3).
a(n) = 8*a(n-1) - 25*a(n-2) + 38*a(n-3) - 28*a(n-4) + 8*a(n-5) for n > 4.
a(n) = 2^(n+1)*(n^2 - 4*n + 8) - 3*n - 16. (End)

cp's OEIS Frontend

A054569 a(n) = 4*n^2 - 6*n + 3.

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A014477 Expansion of (1 + 2*x)/(1 - 2*x)^3.

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A036800 a(n) = -6 + 2^(n+1)*(3 - 2*n + n^2).

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A209359 a(n) = 2^n * (n^4 - 4*n^3 + 18*n^2 - 52*n + 75) - 75.

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A036828 A036827/2.

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A248830 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x-2k)^k for 0 <= k <= n.

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A236538 Triangle read by rows: T(n,k) = (n+1)*2^(n-2)+(k-1)*2^(n-1) for 1 <= k <= n.

A054569 a(n) = 4n^2 - 6n + 3.

A014477 Expansion of (1 + 2x)/(1 - 2x)^3.

A036800 a(n) = -6 + 2^(n+1)(3 - 2n + n^2).

A209359 a(n) = 2^n * (n^4 - 4n^3 + 18n^2 - 52*n + 75) - 75.

A236538 Triangle read by rows: T(n,k) = (n+1)2^(n-2)+(k-1)2^(n-1) for 1 <= k <= n.