A077616 -id:A077616 - cp's OEIS frontend

A094951 a(n) = A081038(n) + A077616(n).

6, 31, 144, 621, 2538, 9963, 37908, 140697, 511758, 1830519, 6456024, 22497669, 77590386, 265189059, 899198172, 3027619377, 10130328342, 33705582543, 111577100832, 367662044061, 1206427402746, 3943553157531, 12845313733284

Offset: 1

Gary W. Adamson, May 26 2004

Performing the same operation but using the multiplier [1 0 0] yields [3^n 2*A027471(n+1) A077616(n)]. Example: M^4 * [1 0 0] = [81 216 324] where 324 = A077616(4) and 216/2 = 108 = A027471(5).

			a(3) = 144 = 81 + 63 = A081038(3) + A077616(3).
a(4) = 621 = 297 + 324 = A081038(4) + A077616(4).
a(4) = 621 since M^4 * [1 1 1] = [81 297 621] = [3^4 A081038(4), a(4)].

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Cf. A081038, A077616, A027471.

List([1..30], n-> 3^(n-2)*(9+7*n+2*n^2)); # G. C. Greubel, Jun 06 2019

[3^(n-2)*(9+7*n+2*n^2): n in [1..30]]; // G. C. Greubel, Jun 06 2019

a[n_] := (MatrixPower[{{3, 0, 0}, {2, 3, 0}, {1, 2, 3}}, n].{{1}, {1}, {1}})[[3, 1]]; Table[ a[n], {n, 23}] (* Robert G. Wilson v, Jun 05 2004 *)
Table[3^(n-2)*(9+7*n+2*n^2), {n,1,30}] (* G. C. Greubel, Jun 06 2019 *)

vector(30, n, 3^(n-2)*(9+7*n+2*n^2)) \\ G. C. Greubel, Jun 06 2019

[3^(n-2)*(9+7*n+2*n^2) for n in (1..30)] # G. C. Greubel, Jun 06 2019

a(n) = A081038(n) + A077616(n).
Let M = the 3 X 3 matrix [3 0 0 / 2 3 0 / 1 2 3]; then M^n * [1 1 1] = [3^n A081038(n) a(n)], where a(n) - A081038(n) = A077616(n).
From Colin Barker, Nov 09 2012: (Start)
a(n) = 3^(n-2)*(9 + 7*n + 2*n^2).
a(n) = 9*a(n-1) - 27*a(n-2) + 27*a(n-3).
G.f.: x*(6 - 23*x + 27*x^2)/(1-3*x)^3. (End)
E.g.f.: -1 + (1 + 3*x + 2*x^2)*exp(3*x). - G. C. Greubel, Jun 06 2019

Edited and extended by Robert G. Wilson v, Jun 05 2004

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

A084901 a(n) = 4^(n-2)n(5*n+3)/2.

Original entry on oeis.org

0, 1, 13, 108, 736, 4480, 25344, 136192, 704512, 3538944, 17367040, 83623936, 396361728, 1853882368, 8573157376, 39258685440, 178241142784, 803158884352, 3594887626752, 15994458210304, 70781061038080, 311711546474496

Offset: 0

Paul Barry, Jun 10 2003

Binomial transform of A084900. Third binomial transform of heptagonal numbers A000566. Fourth binomial transform of (0,1,5,0,0,0,...).

Coefficients in the hypergeometric series identity 1 - 13*x/(x + 12) + 108*x*(x - 1)/((x + 12)*(x + 16)) - 736*x*(x - 1)*(x - 2)/((x + 12)*(x + 16)*(x + 20)) + ... = 0, valid in the half-plane Re(x) > 0. Cf. A276289 and A077616. - Peter Bala, May 30 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-48,64).

Cf. A084902, A077616, A276289.

List([0..30], n-> 2^(2*n-5)*n*(5*n+3)); # G. C. Greubel, Jun 06 2019

[2^(2*n-5)*n*(5*n+3): n in [0..30]]; // G. C. Greubel, Jun 06 2019

Table[2^(2*n-5)*n*(5*n+3), {n,0,30}] (* G. C. Greubel, Jun 06 2019 *)
LinearRecurrence[{12,-48,64},{0,1,13},30] (* or *) CoefficientList[ Series[-((x (1+x))/(-1+4 x)^3),{x,0,30}],x] (* Harvey P. Dale, Jul 14 2021 *)

vector(30, n, n--; 2^(2*n-5)*n*(5*n+3)) \\ G. C. Greubel, Jun 06 2019

[2^(2*n-5)*n*(5*n+3) for n in (0..30)] # G. C. Greubel, Jun 06 2019

G.f.: x*(1+x)/(1-4*x)^3.
E.g.f.: x*(2 + 5*x)*exp(4*x)/2. - G. C. Greubel, Jun 06 2019
a(n) = 12*a(n-1)-48*a(n-2)+64*a(n-3). - Wesley Ivan Hurt, May 28 2021

A276289 Expansion of x(1 + x)/(1 - 2x)^3.

Original entry on oeis.org

0, 1, 7, 30, 104, 320, 912, 2464, 6400, 16128, 39680, 95744, 227328, 532480, 1232896, 2826240, 6422528, 14483456, 32440320, 72220672, 159907840, 352321536, 772800512, 1688207360, 3674210304, 7969177600, 17230200832, 37144756224, 79859548160, 171261820928, 366414397440

Offset: 0

Ilya Gutkovskiy, Aug 27 2016

Binomial transform of pentagonal numbers (A000326).
More generally, the binomial transform of k-gonal numbers is n*Hypergeometric2F1(k/(k-2),1-n;2/(k-2);-1), where Hypergeometric2F1(a,b;c;x) is the hypergeometric function.
Coefficients in the hypergeometric series identity 1 - 7*x/(x + 6) + 30*x*(x - 1)/((x + 6)*(x + 8)) - 104*x*(x - 1)*(x - 2)/((x + 6)*(x + 8)*(x + 10)) + ... = 0, valid in the half-plane Re(x) > 0. Cf. A077616 and A084901. - Peter Bala, May 30 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pentagonal Number
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A000326, A084901.

Cf. A001793 (binomial transform of triangular numbers), A001788 (binomial transform of squares), A084899 (binomial transform of heptagonal numbers).

List([0..40], n-> 2^(n-3)*n*(3*n+1)); # G. C. Greubel, Jun 02 2019

[2^(n-3)*n*(3*n+1): n in [0..40]]; // G. C. Greubel, Jun 02 2019

a:=series(x*(1+x)/(1-2*x)^3,x=0,31): seq(coeff(a,x,n),n=0..40); # Paolo P. Lava, Mar 27 2019

LinearRecurrence[{6, -12, 8}, {0, 1, 7}, 40]
Table[2^(n - 3) n (3 n + 1), {n, 0, 40}]

concat(0, Vec(x*(1+x)/(1-2*x)^3 + O(x^40))) \\ Altug Alkan, Aug 27 2016

[2^(n-3)*n*(3*n+1) for n in (0..40)] # G. C. Greubel, Jun 02 2019

O.g.f.: x*(1 + x)/(1 - 2*x)^3.
E.g.f.: x*(2 + 3*x)*exp(2*x)/2.
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3).
a(n) = Sum_{k = 0..n} binomial(n,k)*k*(3*k - 1)/2.
a(n) = 2^(n-3)*n*(3*n + 1).
Sum_{n>=1} 1/a(n) = 8*(-3*2^(1/3)*Hypergeometric2F1(1/3,1/3;4/3;-1) + 3 + log(2)) = 1.1906948190529335181687...

A062189 a(n) = 2 * 3^(n-2)n(1+2*n).

Original entry on oeis.org

0, 2, 20, 126, 648, 2970, 12636, 51030, 198288, 747954, 2755620, 9959598, 35429400, 124357194, 431530092, 1482720390, 5050815264, 17075199330, 57338232372, 191385721566, 635369601960, 2099044209402, 6903833113980

Offset: 0

Henry Bottomley, Jun 13 2001

Define a triangle with left (first) column T(n,0)=n^2 for n=0,1,2,3.. and the remaining terms T(r,c) = T(r-1,c-1) + 2*T(r,c-1). Then T(n,n) = a(n) on the diagonal. T(n,1) = A056105(n). - J. M. Bergot, Jan 26 2013

Harry J. Smith, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

List([0..30], n-> 2*3^(n-2)*n*(1+2*n)); # G. C. Greubel, Jun 06 2019

[2*3^(n-2)*n*(1+2*n): n in [0..30]]; // G. C. Greubel, Jun 06 2019

Table[2*3^(n-2)*n*(1+2*n), {n,0,30}] (* G. C. Greubel, Jun 06 2019 *)
LinearRecurrence[{9,-27,27},{0,2,20},30] (* Harvey P. Dale, Jun 08 2022 *)

{ for (n=0, 200, write("b062189.txt", n, " ", n*(4*n + 2)*3^(n - 2)) ) } \\ Harry J. Smith, Aug 02 2009

[2*3^(n-2)*n*(1+2*n) for n in (0..30)] # G. C. Greubel, Jun 06 2019

a(n) = A002943(n)*A000244(n-2). Binomial transform of A007758.
G.f.: 2*x*(1+x)/(1-3*x)^3. - Ralf Stephan, Mar 13 2003
a(n) = 2*A077616(n). - R. J. Mathar, Jan 29 2013
E.g.f.: 2*x*(1+2*x)*exp(3*x). - G. C. Greubel, Jun 06 2019

cp's OEIS Frontend

A094951 a(n) = A081038(n) + A077616(n).

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

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A084901 a(n) = 4^(n-2)*n*(5*n+3)/2.

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A276289 Expansion of x*(1 + x)/(1 - 2*x)^3.

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GAP

Magma

Maple

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A062189 a(n) = 2 * 3^(n-2)*n*(1+2*n).

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GAP

Magma

Mathematica

PARI

Sage

Formula

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

A084901 a(n) = 4^(n-2)n(5*n+3)/2.

A276289 Expansion of x(1 + x)/(1 - 2x)^3.

A062189 a(n) = 2 * 3^(n-2)n(1+2*n).