A062111 -id:A062111 - cp's OEIS frontend

A152920 Triangle read by rows: triangle A062111 reversed.

0, 1, 1, 2, 3, 4, 3, 5, 8, 12, 4, 7, 12, 20, 32, 5, 9, 16, 28, 48, 80, 6, 11, 20, 36, 64, 112, 192, 7, 13, 24, 44, 80, 144, 256, 448, 8, 15, 28, 52, 96, 176, 320, 576, 1024, 9, 17, 32, 60, 112, 208, 384, 704, 1280, 2304, 10, 19, 36, 68, 128, 240, 448, 832, 1536, 2816, 5120

Offset: 0

Paul Curtz, Dec 15 2008

			Triangle starts:
  0;
  1,  1;
  2,  3,  4;
  3,  5,  8, 12;
  4,  7, 12, 20, 32;
  ...

Alois P. Heinz, Rows n = 0..150, flattened

Cf. A053220, A058877 (row sums), A193844, A212697.

Columns and diagonals: A001787, A001792, A034007, A045623, A045891, A111297, A159694, A159695, A159696, A159697.

[2^k*(n-k/2): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 27 2022

A062111 := proc(n,k) (k+n)*2^(k-n-1) ; end: A152920 := proc(n,k) A062111(n-k,n) ; end: for n from 0 to 15 do for k from 0 to n do printf("%d,",A152920(n,k)) ; od: od: # R. J. Mathar, Jan 22 2009
# second Maple program:
T:= proc(n, k) option remember;
     `if`(k=0, n, T(n, k-1)+T(n-1, k-1))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Sep 12 2022

t[0, k_]:= k; t[n_, k_]:= t[n, k]= t[n-1, k] + t[n-1, k+1];
Table[t[n-k, k], {n,0,10}, {k,n,0,-1}]//Flatten (* Jean-François Alcover, Sep 11 2016 *)

flatten([[2^(k-1)*(2*n-k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Sep 27 2022

Row sums: (2^n-1)(n+1) = A058877(n). - R. J. Mathar, Jan 22 2009
T(2n, n) = 3*n*2^(n-1) = 3*A001787(n). - Philippe Deléham, Apr 20 2009
From Werner Schulte, Jul 31 2020: (Start)
T(n, k) = (2*n-k) * 2^(k-1) for 0 <= k <= n.
G.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = t*(1+x-3*x*t) / ((1-t)^2 * (1-2*x*t)^2).
Sum_{k=0..n} (-1)^k * binomial(n,k) * T(n,k) = 0 for n >= 0.
Sum_{k=0..n} binomial(n,k) * T(n,k) = 2*n * 3^(n-1) for n >= 0.
Define the array B(n,p) = (Sum_{k=0..n} binomial(p+k,p) * T(n,k))/(n+p+1) for n >= 0 and p >= 0. Then see the comment of Robert Coquereaux (2014) at A193844. Conjecture: B(n+1,p) = A(n,p). (End)
T(n, k) = T(n, k-1) + T(n-1, k-1) for k>=1, T(n,0) = n. - Alois P. Heinz, Sep 12 2022
From G. C. Greubel, Sep 27 2022: (Start)
T(n, n-1) = A001792(n).
T(2*n-1, n-1) = A053220(n).
T(2*n+1, n-1) = 3*A001792(n).
T(m*n, n) = (2*m-1)*A001787(n), for m >= 1. (End)

Edited by N. J. A. Sloane, Dec 19 2008

More terms from R. J. Mathar, Jan 22 2009

A159755 Triangle of C. A. Laisant for k<=0 (see A062111 and A152920).

Original entry on oeis.org

0, -1, 1, -2, -1, 4, -3, -3, 0, 12, -4, -5, -4, 4, 32, -5, -7, -8, -4, 16, 80, -6, -9, -12, -12, 0, 48, 192, -7, -11, -16, -20, -16, 16, 128, 448, -8, -13, -20, -28, -32, -16, 64, 320, 1024, -9, -15, -24, -36, -48, -48, 0, 192, 768, 2304

Offset: 0

Philippe Deléham, Apr 21 2009

			Triangle begins : 0 ; -1,1 ; -2,-1,4 ; -3,-3,0,12 ; -4,-5,-4,4,32 ; ...

Sum_{k=0..n}T(n,k)= A045618(n-2)for n>=2 . T(2n,n)=-A001787(n).

A058877 Number of labeled acyclic digraphs with n nodes containing exactly n-1 points of in-degree zero.

Original entry on oeis.org

0, 2, 9, 28, 75, 186, 441, 1016, 2295, 5110, 11253, 24564, 53235, 114674, 245745, 524272, 1114095, 2359278, 4980717, 10485740, 22020075, 46137322, 96468969, 201326568, 419430375, 872415206, 1811939301, 3758096356, 7784628195, 16106127330, 33285996513

Offset: 1

N. J. A. Sloane, Jan 07 2001

Convolution of 2^n+1 (A000051) and 2^n-1 (A000225). - Graeme McRae, Jun 07 2006
Let Q be a binary relation on the power set P(A) of a set A having n = |A| elements such that for all nonempty elements x,y of P(A), xRy if x is a proper subset of y and there are no z in P(A) such that x is a proper subset of z and z is a proper subset of y. Then a(n) = |Q|. - Ross La Haye, Feb 20 2008, Oct 21 2008
Also: convolution of A006589 with A000012 (i.e., partial sums of A006589). - R. J. Mathar, Jan 25 2009
The La Haye binary relation Q is more clearly stated as x is nonempty and y has one more element than x. If x is a k-set than the number of such pairs is binomial( n, k) * (n-k). - Michael Somos, Mar 29 2012
Select one of the n nodes of the digraph and select a nonempty subset of the rest to connect to the selected node. This can be done in n * (2^(n-1) - 1) ways. - Michael Somos, Mar 29 2012
Column 1 of A198204. - Peter Bala, Aug 01 2012
a(n) is the number of ternary sequences of length n that contain one 0 and at least one 1.  For example, a(3)=9 since the sequences are the 3 permutations of 011 and the 6 permutations of 012. - Enrique Navarrete, Apr 05 2021
a(n) is also the number of multiplications required to compute the permanent of general n X n matrices using canonical trellis method (see Theorem 5, p. 10 in Kiah et al.). - Stefano Spezia, Nov 02 2021

			G.f. = 2*x^2 + 9*x^3 + 28*x^4 + 75*x^5 + 186*x^6 + 441*x^7 + 1016*x^8 + ...

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 19, (1.6.4).
Gerta Rucker and Christoph Rucker, "Walk counts, Labyrinthicity and complexity of acyclic and cyclic graphs and molecules", J. Chem. Inf. Comput. Sci., 40 (2000), 99-106. See Table 1 on page 101. [From Parthasarathy Nambi, Sep 26 2008]

Vincenzo Librandi, Table of n, a(n) for n = 1..2001
Jia Huang and Erkko Lehtonen, Associative-commutative spectra for some varieties of groupoids, arXiv:2401.15786 [math.CO], 2024. See p. 12.
Han Mao Kiah, Alexander Vardy and Hanwen Yao, Computing Permanents on a Trellis, arXiv:2107.07377 [cs.IT], 2021.
Tamás Szakács, Convolution of second order linear recursive sequences. II. Commun. Math. 25, No. 2, 137-148 (2017), remark 3.
Tamás Szakács, Linear recursive sequences and factorials, Ph. D. Thesis, Univ. Debrecen (Hungary, 2024). See p. 36.
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Second column of A058876. Cf. A003025, A003026.
Column k=1 of A133399.
Cf. A198204.

[(n+1)*2^n-n-1: n in [0..30]]; // Vincenzo Librandi, Sep 26 2011

[seq (stirling2(n,2)*n,n=1..29)]; # Zerinvary Lajos, Dec 06 2006
a:=n->sum(k*binomial(n,k), k=2..n): seq(a(n), n=1..29); # Zerinvary Lajos, May 08 2007

Table[(n+1)*2^n-n-1, {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Jun 30 2011 *)
a[ n_] := Sum[ Binomial[ n, k] (n - k), {k, n-1}]; (* Michael Somos, Mar 29 2012 *)

{a(n) = if( n<1, 0, n * (2^(n-1) - 1))} /* Michael Somos, Mar 29 2012 */

[stirling_number2(i,2)*i for i in range(1,26)] # Zerinvary Lajos, Jun 27 2008

[(n+1)*gaussian_binomial(n,1,2) for n in range(0,29)] # Zerinvary Lajos, May 31 2009

a(n+1) = (n+1)*2^n - n - 1 = Sum_{j=0..n} (n+j)*2^(n-j-1) = A048493(n)-1 = Column sum of A062111. - Henry Bottomley, May 30 2001
From R. J. Mathar, Jan 25 2009: (Start)
G.f.: x^2*(2-3*x)/((1-2*x)*(1-x))^2.
a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4). (End)
a(n) = Sum_{k=1..n-1} binomial(n, k) * (n-k). - Michael Somos, Mar 29 2012
E.g.f: x*exp(x)*(exp(x)-1). - Enrique Navarrete, Apr 05 2021

More terms from Vladeta Jovovic, Apr 10 2001

A073371 Convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n >= 0, with itself.

Original entry on oeis.org

1, 2, 7, 16, 41, 94, 219, 492, 1101, 2426, 5311, 11528, 24881, 53398, 114083, 242724, 514581, 1087410, 2291335, 4815680, 10097401, 21126862, 44117867, 91963996, 191384541, 397682154, 825190479, 1710033272, 3539371201, 7317351686

Offset: 0

Wolfdieter Lang, Aug 02 2002

PSumSIGN transform of A045883(n-1). - Michael Somos, Jul 10 2003
Numbers of the form ((6*m+4)*2^m + (-1)^(m-1)*(3*m+4))/27. - Artur Jasinski, Feb 09 2007
With [0, 0, 0] prepended, this is an "autosequence" of the first kind, whose companion is [0, 0, 2, 3, 12, 25, 66, ...], that is A099429. - Jean-François Alcover, Jul 10 2022

G. C. Greubel, Table of n, a(n) for n = 0..1000
Wieb Bosma, Signed bits and fast exponentiation, J. Th. Nombres de Bordeaux, 13 no. 1 (2001), p. 27-41.
OEIS Wiki, Autosequence

Second (m=1) column of triangle A073370.

Cf. A001045, A062111, A099429, A127976.

[((5+3*n)*2^(n+2) + (-1)^n*(7+3*n))/27: n in [0..40]]; // G. C. Greubel, Sep 28 2022

Table[((6n+4)*2^n + (-1)^(n-1)(3n+4))/27, {n, 100}] (* Artur Jasinski, Feb 09 2007 *)

a(n) = if(n<-3, 0, ((5+3*n)*2^(n+2)+(7+3*n)*(-1)^n)/27)

def A073371(n): return ((5+3*n)*2^(n+2) + (-1)^n*(7+3*n))/27
[A073371(n) for n in range(40)] # G. C. Greubel, Sep 28 2022

a(n) = Sum_{k=0..n} b(k) * b(n-k), where b(k) = A001045(k+1).
a(n) = Sum_{k=0..floor(n/2)} (n-k+1) * binomial(n-k, k) * 2^k.
a(n) = ((n+1)*U(n+1) + 4*(n+2)*U(n))/9 with U(n) = A001045(n+1), n>=0.
G.f.: 1/(1 - (1+2*x)*x)^2.
G.f.: 1/((1+x)*(1-2*x))^2.
a(n) = ((5+3*n)*2^(n+2) + (7+3*n)*(-1)^n)/27.
a(n) = ((6*n+4)*2^(n) + (-1)^(n-1)*(3*n+4))/27. - Artur Jasinski, Feb 09 2007
E.g.f.: (1/27)*(4*(5+6*x)*exp(2*x) + (7-3*x)*exp(-x)). - G. C. Greubel, Sep 28 2022

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007

A159694 a(n) = 2*a(n-1) + 2^(n-1), for n > 0, with a(0)=6.

Original entry on oeis.org

6, 13, 28, 60, 128, 272, 576, 1216, 2560, 5376, 11264, 23552, 49152, 102400, 212992, 442368, 917504, 1900544, 3932160, 8126464, 16777216, 34603008, 71303168, 146800640, 301989888, 620756992, 1275068416, 2617245696, 5368709120

Offset: 0

Philippe Deléham, Apr 20 2009

Diagonal of triangles A062111, A152920.

			a(0) = 6,
a(1) = 2* 6 + 1 =  13,
a(2) = 2*13 + 2 =  28,
a(3) = 2*28 + 4 =  60,
a(4) = 2*60 + 8 = 128, ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjić and Boris Petković, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Milan Janjić and Boris Petković, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq., Vol. 17 (2014), Article 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A000079, A001787, A001792, A034007, A045623.
Cf. A045891, A062111, A111297, A152920.
Seventh row of triangle A062111. - Klaus Brockhaus, Sep 27 2009

[(12+n)*2^(n-1): n in [0..30]]; // G. C. Greubel, Sep 27 2022

Table[(6 + n/2)*2^n, {n, 0, 30}] (* Amiram Eldar, Jan 19 2021 *)

[(12+n)*2^(n-1) for n in range(30)] # G. C. Greubel, Sep 27 2022

a(n) = Sum_{k=0..n} (k+6)*binomial(n,k).
From Klaus Brockhaus, Sep 27 2009: (Start)
a(n) = (6 + n/2)*2^n.
G.f.: (6 - 11*x)/(1-2*x)^2. (End)
From Amiram Eldar, Jan 19 2021: (Start)
Sum_{n>=0} 1/a(n) = 8192*log(2) - 3934820/693.
Sum_{n>=0} (-1)^n/a(n) = 11509636/3465 - 8192*log(3/2). (End)
E.g.f.: (6 + x)*exp(2*x). - G. C. Greubel, Sep 27 2022

A159695 a(0)=7, a(n) = 2*a(n-1) + 2^(n-1) for n > 0.

Original entry on oeis.org

7, 15, 32, 68, 144, 304, 640, 1344, 2816, 5888, 12288, 25600, 53248, 110592, 229376, 475136, 983040, 2031616, 4194304, 8650752, 17825792, 36700160, 75497472, 155189248, 318767104, 654311424, 1342177280, 2751463424, 5637144576

Offset: 0

Philippe Deléham, Apr 20 2009

Diagonal of triangles A062111, A152920.

			a(0)=7, a(1) = 2*7 + 1 = 15, a(2) = 2*15 + 2 = 32, a(3) = 2*32 + 4 = 68, a(4) = 2*68 + 8 = 144, ...

G. C. Greubel, Table of n, a(n) for n = 0..3300
Index entries for linear recurrences with constant coefficients, signature (4, -4).

Cf. A000079, A001787, A001792, A045623, A045891, A034007, A111297, A159694.

[(14+n)*2^(n-1): n in [0..30]]; // G. C. Greubel, Jun 02 2018

LinearRecurrence[{4,-4}, {7,15}, 30] (* or *) Table[(14+n)*2^(n-1), {n, 0, 30}] (* G. C. Greubel, Jun 02 2018 *)
nxt[{n_,a_}]:={n+1,2a+2^n}; NestList[nxt,{0,7},30][[All,2]] (* Harvey P. Dale, Jan 01 2023 *)

for(n=0, 30, print1((14+n)*2^(n-1), ", ")) \\ G. C. Greubel, Jun 02 2018

a(n) = Sum_{k=0..n} (k+7)*binomial(n,k).
From R. J. Mathar, Apr 20 2009: (Start)
a(n) = (14+n)*2^(n-1).
a(n) = 4*a(n-1) - 4*a(n-2).
G.f.: (7-13*x)/(1-2x)^2. (End)
E.g.f.: (x+7)*exp(2*x). - G. C. Greubel, Jun 02 2018
From Amiram Eldar, Jan 19 2021: (Start)
Sum_{n>=0} 1/a(n) = 32768*log(2) - 204619418/9009.
Sum_{n>=0} (-1)^n/a(n) = 598484902/45045 - 32768*log(3/2). (End)

More terms from R. J. Mathar, Apr 20 2009

A130129 a(n) = (3n+1)2^n.

Original entry on oeis.org

1, 8, 28, 80, 208, 512, 1216, 2816, 6400, 14336, 31744, 69632, 151552, 327680, 704512, 1507328, 3211264, 6815744, 14417920, 30408704, 63963136, 134217728, 281018368, 587202560, 1224736768, 2550136832, 5301600256, 11005853696, 22817013760

Offset: 0

Paul Curtz, Aug 01 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A016921 (inverse binomial transform).

Cf. A062111, A066373.

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

Table[2^n*(3*n+1), {n,0,30}] (* G. C. Greubel, Sep 28 2022 *)

[2^n*(3*n+1) for n in range(30)] # G. C. Greubel, Sep 28 2022

a(n+1) - a(n) = A066373(n+3). - J. M. Bergot, Apr 21 2013
G.f.: ( 1+4*x ) / (1-2*x)^2. - R. J. Mathar, Apr 25 2013
E.g.f.: (1+6*x)*exp(2*x). - G. C. Greubel, Sep 28 2022

A159696 a(0)=8, a(n) = 2*a(n-1) + 2^(n-1) for n > 0.

Original entry on oeis.org

8, 17, 36, 76, 160, 336, 704, 1472, 3072, 6400, 13312, 27648, 57344, 118784, 245760, 507904, 1048576, 2162688, 4456448, 9175040, 18874368, 38797312, 79691776, 163577856, 335544320, 687865856, 1409286144, 2885681152, 5905580032

Offset: 0

Philippe Deléham, Apr 20 2009

Diagonal of triangles A062111, A152920.

			a(0)=8, a(1) = 2*8 + 1 = 17, a(2) = 2*17 + 2 = 36, a(3) = 2*36 + 4 = 76, a(4) = 2*76 + 8 = 160, ...

G. C. Greubel, Table of n, a(n) for n = 0..3300
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A000079, A001787, A001792, A045623, A045891, A034007, A111297, A159694, A159695.

[(16+n)*2^(n-1): n in [0..30]]; // G. C. Greubel, Jun 02 2018

LinearRecurrence[{4,-4}, {8,17}, 30] (* or *) Table[(16+n)*2^(n-1), {n,0,30}] (* G. C. Greubel, Jun 02 2018 *)

for(n=0, 30, print1((16+n)*2^(n-1), ", ")) \\ G. C. Greubel, Jun 02 2018

a(n) = Sum_{k=0..n} (k+8)*binomial(n,k).
From R. J. Mathar, Apr 20 2009: (Start)
a(n) = (16+n)*2^(n-1).
a(n) = 4*a(n-1) - 4*a(n-2).
G.f.: (8-15*x)/(1-2*x)^2. (End)
E.g.f.: (x+8)*exp(2*x). - G. C. Greubel, Jun 02 2018

More terms from R. J. Mathar, Apr 20 2009

A159697 a(0)=9, a(n) = 2*a(n-1) + 2^(n-1) for n > 0.

Original entry on oeis.org

9, 19, 40, 84, 176, 368, 768, 1600, 3328, 6912, 14336, 29696, 61440, 126976, 262144, 540672, 1114112, 2293760, 4718592, 9699328, 19922944, 40894464, 83886080, 171966464, 352321536, 721420288, 1476395008, 3019898880

Offset: 0

Philippe Deléham, Apr 20 2009

Diagonal of triangles A062111, A152920.

			a(0)=9, a(1) = 2*9 + 1 = 19, a(2) = 2*19 + 2 = 40, a(3) = 2*40 + 4 = 84, a(4) = 2*84 + 8 = 176, ...

G. C. Greubel, Table of n, a(n) for n = 0..3300
Index entries for linear recurrences with constant coefficients, signature (4, -4).

Cf. A000079, A001787, A001792, A045623, A045891, A034007, A111297, A159694, A159695, A159696.

I:=[9,19]; [n le 2 select I[n] else 4*Self(n-1) - 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Jun 02 2018

RecurrenceTable[{a[0]==9,a[n]==2a[n-1]+2^(n-1)},a,{n,30}] (* or *) LinearRecurrence[{4,-4},{9,19},30] (* Harvey P. Dale, Mar 24 2013 *)

Vec((9-17*x)/(1-2*x)^2 + O(x^40)) \\ Michel Marcus, Sep 29 2015

a(n) = Sum_{k=0..n} (k+9)*binomial(n,k).
From R. J. Mathar, Apr 20 2009: (Start)
a(n) = (18+n)*2^(n-1).
a(n) = 4*a(n-1) - 4*a(n-2).
G.f.: (9-17*x)/(1-2*x)^2. (End)
a(0)=9, a(1)=19, a(n) = 4*a(n-1) - 4*a(n-2). - Harvey P. Dale, Mar 24 2013
a(n) = 2*A079862(n-10). - Michel Marcus, Sep 29 2015
E.g.f.: (x+9)*exp(2*x). - G. C. Greubel, Jun 02 2018

More terms from Vincenzo Librandi, Apr 30 2009

A053219 Reverse of triangle A053218, read by rows.

Original entry on oeis.org

1, 3, 2, 8, 5, 3, 20, 12, 7, 4, 48, 28, 16, 9, 5, 112, 64, 36, 20, 11, 6, 256, 144, 80, 44, 24, 13, 7, 576, 320, 176, 96, 52, 28, 15, 8, 1280, 704, 384, 208, 112, 60, 32, 17, 9, 2816, 1536, 832, 448, 240, 128, 68, 36, 19, 10, 6144, 3328, 1792, 960, 512, 272, 144, 76, 40

Offset: 1

Asher Auel, Jan 01 2000

First element in each row gives A001792. Difference between center element of row 2n-1 and row sum of row n (A053220(n+4) - A053221(n+4)) gives A045618(n).
Subtriangle of triangle in A062111. - Philippe Deléham, Nov 21 2011
Can be seen as the transform of 1, 2, 3, 4, 5, ... by a variant of the boustrophedon algorithm (see the Sage implementation). - Peter Luschny, Oct 30 2014

			Triangle begins:
1
3, 2
8, 5, 3
20, 12, 7, 4
48, 28, 16, 9, 5 ...

Cf. A053218 (reverse of this triangle), A053220 (center elements), A053221 (row sums), A001792, A045618, A062111.

Map[Reverse,NestList[FoldList[Plus,#[[1]]+1,#]&,{1},10]]//Grid (* Geoffrey Critzer, Jun 27 2013 *)

def u():
    for n in PositiveIntegers():
        yield n
def bous_variant(f):
    k = 0
    am = next(f)
    a = [am]
    while True:
        yield list(a)
        am = next(f)
        a.append(am)
        for m in range(k,-1,-1):
            am += a[m]
            a[m] = am
        k += 1
b = bous_variant(u())
[next(b) for  in range(8)] # _Peter Luschny, Oct 30 2014

cp's OEIS Frontend

A152920 Triangle read by rows: triangle A062111 reversed.

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A159755 Triangle of C. A. Laisant for k<=0 (see A062111 and A152920).

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A058877 Number of labeled acyclic digraphs with n nodes containing exactly n-1 points of in-degree zero.

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A073371 Convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n >= 0, with itself.

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A159694 a(n) = 2*a(n-1) + 2^(n-1), for n > 0, with a(0)=6.

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A159695 a(0)=7, a(n) = 2*a(n-1) + 2^(n-1) for n > 0.

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Magma

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PARI

Formula

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A130129 a(n) = (3n+1)2^n.