A003463 -id:A003463 - cp's OEIS frontend

A091030 Partial sums of powers of 13 (A001022).

1, 14, 183, 2380, 30941, 402234, 5229043, 67977560, 883708281, 11488207654, 149346699503, 1941507093540, 25239592216021, 328114698808274, 4265491084507563, 55451384098598320, 720867993281778161

Offset: 1

Wolfdieter Lang, Jan 23 2004

13^a(n) is highest power of 13 dividing (13^n)!.
For analogs with primes 2, 3, 5, 7 and 11 see A000225, A003462, A003463, A023000 and A016123 respectively.
Let A be the Hessenberg matrix of the order n, defined by: A[1,j]=1,A[i,i]:=13, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Feb 21 2010
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=14, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=(-1)^(n)*charpoly(A,1). - Milan Janjic, Feb 21 2010

			For n=6, a(6) = 1*6 + 12*15 + 144*20 + 1728*15 + 20736*6 + 248832*1 = 402234. - _Bruno Berselli_, Nov 12 2015

Delbert L. Johnson, Table of n, a(n) for n = 1..898
Index entries for linear recurrences with constant coefficients, signature (14,-13).

Cf. A000225, A003462, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125.

Cf. A001021, A135278.

a:=n->sum(13^(n-j),j=1..n): seq(a(n), n=1..17); # Zerinvary Lajos, Jan 04 2007

Table[13^n, {n, 0, 16}] // Accumulate (* Jean-François Alcover, Jul 05 2013 *)
LinearRecurrence[{14,-13},{1,14},20] (* Harvey P. Dale, Jan 19 2024 *)

A091030(n):=(13^n-1)/12$ makelist(A091030(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

a(n)=([0,1; -13,14]^(n-1)*[1;14])[1,1] \\ Charles R Greathouse IV, Sep 24 2015

[gaussian_binomial(n,1,13) for n in range(1,18)] # Zerinvary Lajos, May 28 2009

[(13^n-1)/12 for n in (1..30)] # Bruno Berselli, Nov 12 2015

G.f.: x/((1-13*x)*(1-x)) = (1/(1-13*x) - 1/(1-x))/12.
a(n) = Sum_{k=0..n-1} 13^k = (13^n-1)/12.
a(n) = 13*a(n-1)+1 for n>1, a(1)=1. - Vincenzo Librandi, Feb 05 2011
a(n) = Sum_{k=0...n-1} 12^k*binomial(n,n-1-k). - Bruno Berselli, Nov 12 2015
E.g.f.: exp(x)*(exp(12*x) - 1)/12. - Stefano Spezia, Mar 11 2023

A091045 Partial sums of powers of 17 (A001026).

Original entry on oeis.org

1, 18, 307, 5220, 88741, 1508598, 25646167, 435984840, 7411742281, 125999618778, 2141993519227, 36413889826860, 619036127056621, 10523614159962558, 178901440719363487, 3041324492229179280, 51702516367896047761

Offset: 1

Wolfdieter Lang, Jan 23 2004

17^a(n) is largest power of 17 dividing (17^n)!.

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=17, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Feb 21 2010

Robert Israel, Table of n, a(n) for n = 1..812
Index entries for linear recurrences with constant coefficients, signature (18, -17).

Cf. similar sequences of the form (k^n-1)/(k-1) with k prime: A000225 (k=2), A003462 (k=3), A003463 (k=5), A023000 (k=7), A016123 (k=11), A091030 (k=13), this sequence (k=17), A218722 (k=19), A218726 (k=23), A218732 (k=29), A218734 (k=31), A218740 (k=37), A218744 (k=41), A218746 (k=43), A218750 (k=47).

Cf. A001026.

[&+[17^i: i in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Feb 19 2018

ListTools:-PartialSums([seq(17^k,k=0..30)]); # Robert Israel, Feb 18 2018

Table[17^n, {n, 0, 16}] // Accumulate (* Jean-François Alcover, Jul 05 2013 *)

makelist(sum(17^k, k, 0, n), n, 0, 30); /* Martin Ettl, Nov 05 2012 */

[gaussian_binomial(n,1,17) for n in range(1,18)] # Zerinvary Lajos, May 28 2009

a(n) = Sum_{k=0..n-1} 17^k = (17^n - 1)/16.
G.f.: x/((1 - 17*x)*(1 - x))= (1/(1 - 17*x) - 1/(1 - x))/16.
a(n) = 17*a(n-1)+1 (with a(1)=1). - Vincenzo Librandi, Nov 16 2010
E.g.f.: exp(9*x)*sinh(8*x)/8. - Stefano Spezia, Mar 11 2023

A218722 a(n) = (19^n-1)/18.

Original entry on oeis.org

0, 1, 20, 381, 7240, 137561, 2613660, 49659541, 943531280, 17927094321, 340614792100, 6471681049901, 122961939948120, 2336276859014281, 44389260321271340, 843395946104155461, 16024522975978953760, 304465936543600121441

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 19 (A001029); q-integers for q=19: diagonal k=1 in triangle A022183.

Partial sums are in A014903. Also, the sequence is related to A014936 by A014936(n) = n*a(n)-sum(a(i), i=0..n-1) for n>0. - Bruno Berselli, Nov 06 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries related to partial sums
Index entries for linear recurrences with constant coefficients, signature (20,-19).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A001029, A014903, A014936, A022183.

[n le 2 select n-1 else 20*Self(n-1)-19*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{20, -19}, {0, 1}, 40] (* Vincenzo Librandi, Nov 07 2012 *)

A218722(n):=(19^n-1)/18$ makelist(A218722(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218722(n)=19^n\18
```

a(n) = floor(19^n/18).
G.f.: x/((1-x)*(1-19*x)). - Bruno Berselli, Nov 06 2012
a(n) = 20*a(n-1) - 19*a(n-2). - Vincenzo Librandi, Nov 07 2012
E.g.f.: exp(10*x)*sinh(9*x)/9. - Stefano Spezia, Mar 11 2023

A064108 a(n) = (20^n - 1)/19.

Original entry on oeis.org

0, 1, 21, 421, 8421, 168421, 3368421, 67368421, 1347368421, 26947368421, 538947368421, 10778947368421, 215578947368421, 4311578947368421, 86231578947368421, 1724631578947368421, 34492631578947368421, 689852631578947368421, 13797052631578947368421, 275941052631578947368421

Offset: 0

Jason Earls, Sep 17 2001

Partial sums of powers of 20 (A009964), q-integers for q=20: diagonal k=1 in triangle A022184.
Partial sums are in A014904. Also, the sequence is related to A014937 by A014937(n) = n*a(n)-Sum_{i=0..n-1} a(i), for n>0. - Bruno Berselli, Nov 06 2012
For n >= 1, a(n) is the total number of holes in a certain box fractal (start with 20 boxes, 1 hole) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 28 2015

			From _N. J. A. Sloane_, Nov 04 2014: Can also be obtained by writing powers of 2 in a staggered array and adding them (cf. A249604). For example, a(9) is:
..........1
.........2
........4
.......8
.....16
....32
...64
.128
256
-----------
26947368421

M. F. Hasler, Table of n, a(n) for n = 0..100
Kival Ngaokrajang, Illustration of initial terms
Index entries related to partial sums
Index entries related to q-numbers
Index entries for linear recurrences with constant coefficients, signature (21,-20).

Cf. A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218722, A064108, A218724, ..., A218733, ..., A218743, ..., A218752, A094028.
Cf. also A249604.
Cf. A009964, A014904, A014937, A022184.

a:=n->sum(20^(n-j), j=0..n): seq(a(n), n=0..15); # Zerinvary Lajos, Feb 11 2007

(20^Range[20]-1)/19 (* or *) NestList[20#+1&,1,20] (* Harvey P. Dale, Oct 04 2012 *)

A064108(n):=(20^n-1)/19$ makelist(A064108(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

for (n=0, 100, write("b064108.txt", n, " ", (20^n - 1)/19))  \\ Harry J. Smith, Sep 07 2009

A064108(n)=20^n\19  \\ M. F. Hasler, Nov 04 2012

[gaussian_binomial(n,1,20) for n in range(1,17)] # Zerinvary Lajos, May 29 2009

a(n) = 20*a(n-1) + 1, with a(0)=0. - Vincenzo Librandi, Aug 07 2010
a(0)=0, a(1)=1, a(n) = 21*a(n-1) - 20*a(n-2). - Harvey P. Dale, Oct 04 2012
a(n) = floor(20^n/19). - M. F. Hasler, Nov 04 2012
G.f.: x/((1 - x)*(1 - 20*x)). - Bruno Berselli, Nov 06 2012
E.g.f.: exp(x)*(exp(19*x) - 1)/19. - Stefano Spezia, Mar 23 2023

Edited and extended to offset 0 by M. F. Hasler, Nov 04 2012

A135518 Generalized repunits in base 15.

Original entry on oeis.org

1, 16, 241, 3616, 54241, 813616, 12204241, 183063616, 2745954241, 41189313616, 617839704241, 9267595563616, 139013933454241, 2085209001813616, 31278135027204241, 469172025408063616, 7037580381120954241, 105563705716814313616, 1583455585752214704241

Offset: 1

Julien Peter Benney (jpbenney(AT)gmail.com), Feb 19 2008

Primes in this sequence are given in A006033.
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=15, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Feb 21 2010
Partial sums are in A014898. Also, the sequence is related to A014930 by A014930(n) = n*a(n) - Sum_{i=1..n-1}( a(i) ). - Bruno Berselli, Nov 06 2012

			For n=4, a(4) = 15^3+15^2+15^1+1 = 3375+225+15+1 = 3616.
For n=6, a(6) = 1*6 + 14*15 + 14^2*20 + 14^3*15 + 14^4*6 + 14^5*1 = 813616. - _Bruno Berselli_, Nov 12 2015

G. C. Greubel, Table of n, a(n) for n = 1..250
Jon Grantham and Hester Graves, The abc Conjecture Implies That Only Finitely Many Cullen Numbers Are Repunits, arXiv:2009.04052 [math.NT], 2020.
Index entries for linear recurrences with constant coefficients, signature (16,-15).

Cf. A000225, A003462, A002450, A003463, A003464.
Cf. A023000, A023001, A002452, A016123, A016125.
Cf. A001023, A135278.

Table[FromDigits[PadRight[{},n,1],15],{n,20}] (* or *) LinearRecurrence[{16,-15},{1,16},20] (* Harvey P. Dale, Jul 08 2013 *)

A135518(n):=(15^n-1)/14$ makelist(A135518(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

a(n)=(15^n-1)/14 \\ Charles R Greathouse IV, Sep 24 2015

def a(n): return int('1'*n, 15)
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Jan 16 2021

[gaussian_binomial(n,1,15) for n in range(1,15)] # Zerinvary Lajos, May 28 2009

[(15^n-1)/14 for n in (1..30)] # Bruno Berselli, Nov 12 2015

a(n) = (15^n - 1)/14.
a(n) = 15*a(n-1) + 1 with n>1, a(1)=1. - Vincenzo Librandi, Aug 03 2010
G.f.: x/((1-x)*(1-15*x)). - Bruno Berselli, Nov 07 2012
a(1)=1, a(2)=16; for n>2, a(n) = 16*a(n-1) - 15*a(n-2). - Harvey P. Dale, Jul 08 2013
a(n) = Sum_{i=0...n-1} 14^i*binomial(n,n-1-i). - Bruno Berselli, Nov 12 2015
E.g.f.: (1/14)*(exp(15*x) - exp(x)). - G. C. Greubel, Oct 17 2016

A135519 Generalized repunits in base 14.

Original entry on oeis.org

1, 15, 211, 2955, 41371, 579195, 8108731, 113522235, 1589311291, 22250358075, 311505013051, 4361070182715, 61054982558011, 854769755812155, 11966776581370171, 167534872139182395, 2345488209948553531

Offset: 1

Julien Peter Benney (jpbenney(AT)gmail.com), Feb 19 2008

Primes are given in A006032.

Let A be the Hessenberg matrix of the order n, defined by: A[1,j]=1, A[i,i]:=14, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Feb 21 2010

			a(4) = 2955 because (14^4-1)/13 = 38416/13 = 2955.
For n=6, a(6) = 1*6 + 13*15 + 169*20 + 2197*15 + 28561*6 + 371293*1 = 579195. - _Bruno Berselli_, Nov 12 2015

Harvey P. Dale, Table of n, a(n) for n = 1..873
Index entries for linear recurrences with constant coefficients, signature (15,-14).

Cf. A000225, A001022, A002450, A002452, A003462, A003463, A003464, A016123, A016125, A023000, A023001, A135278.

Table[FromDigits[PadRight[{}, n, 1], 14], {n, 20}] (* or *) LinearRecurrence[{15, -14}, {1, 15}, 20] (* Harvey P. Dale, Aug 29 2016 *)

A135519(n):=(14^n-1)/13$ makelist(A135519(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

[gaussian_binomial(n,1,14) for n in range(1,15)] # Zerinvary Lajos, May 28 2009

[(14^n-1)/13 for n in (1..30)] # Bruno Berselli, Nov 12 2015

a(n) = (14^n - 1)/13.
a(n) = 14*a(n-1) + 1 for n>1, a(1)=1. - Vincenzo Librandi, Aug 03 2010
a(n) = Sum_{i=0..n-1} 13^i*binomial(n,n-1-i). - Bruno Berselli, Nov 12 2015
From G. C. Greubel, Oct 17 2016: (Start)
G.f.: x/((1-x)*(1-14*x)).
E.g.f.: (1/13)*(exp(14*x) - exp(x)). (End)

A024049 a(n) = 5^n - 1.

Original entry on oeis.org

0, 4, 24, 124, 624, 3124, 15624, 78124, 390624, 1953124, 9765624, 48828124, 244140624, 1220703124, 6103515624, 30517578124, 152587890624, 762939453124, 3814697265624, 19073486328124, 95367431640624

Offset: 0

N. J. A. Sloane

Numbers whose base 5 representation is 44444.......4. - Zerinvary Lajos, Feb 03 2007

For n > 0, a(n) is the sum of divisors of 3*5^(n-1). - Patrick J. McNab, May 27 2017

			For n = 5, a(5) = 4*5 + 16*10 + 64*10 + 256*5 + 1024*1 = 3124. - _Bruno Berselli_, Nov 11 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000351, A003463, A125831, A248722.

[5^n-1: n in [0..30]]; // Vincenzo Librandi, Jun 06 2011

5^Range[0,50]-1 (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)
LinearRecurrence[{6,-5},{0,4},30] (* Harvey P. Dale, Apr 06 2019 *)

a(n)=5^n-1 \\ Charles R Greathouse IV, Apr 17 2012

G.f.: 1/(1-5*x) - 1/(1-x) = 4*x/((1-5*x)*(1-x)). - Mohammad K. Azarian, Jan 14 2009
E.g.f.: exp(5*x) - exp(x). - Mohammad K. Azarian, Jan 14 2009
a(n+1) = 5*a(n) + 4. - Reinhard Zumkeller, Nov 22 2009
a(n) = Sum_{i=1..n} 4^i*binomial(n,n-i) for n>0, a(0)=0. - Bruno Berselli, Nov 11 2015
a(n) = A000351(n) - 1. - Sean A. Irvine, Jun 19 2019
Sum_{n>=1} 1/a(n) = A248722. - Amiram Eldar, Nov 13 2020
a(n) = 2*A125831(n) = 4*A003463(n). - Elmo R. Oliveira, Dec 10 2023

A190958 a(n) = 2a(n-1) - 10a(n-2), with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 2, -6, -32, -4, 312, 664, -1792, -10224, -2528, 97184, 219648, -532544, -3261568, -1197696, 30220288, 72417536, -157367808, -1038910976, -504143872, 9380822016, 23803082752, -46202054656, -330434936832, -198849327104, 2906650714112, 7801794699264

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

For the difference equation a(n) = c*a(n-1) - d*a(n-2), with a(0) = 0, a(1) = 1, the solution is a(n) = d^((n-1)/2) * ChebyshevU(n-1, c/(2*sqrt(d))) and has the alternate form a(n) = ( ((c + sqrt(c^2 - 4*d))/2)^n - ((c - sqrt(c^2 - 4*d))/2)^n )/sqrt(c^2 - 4*d). In the case c^2 = 4*d then the solution is a(n) = n*d^((n-1)/2). The generating function is x/(1 - c*x + d^2) and the exponential generating function takes the form (2/sqrt(c^2 - 4*d))*exp(c*x/2)*sinh(sqrt(c^2 - 4*d)*x/2) for c^2 > 4*d, (2/sqrt(4*d - c^2))*exp(c*x/2)*sin(sqrt(4*d - c^2)*x/2) for 4*d > c^2, and x*exp(sqrt(d)*x) if c^2 = 4*d. - G. C. Greubel, Jun 10 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..190
Index entries for linear recurrences with constant coefficients, signature (2,-10).

Sequences of the form a(n) = c*a(n-1) - d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
.A128834,A107920,A106852,A106853,A106854,A145934,A145976,A145978,A146078,A146080
.A000027,A009545,A088137,A088138,A045873,A088139,A089181,A090591,A127357,A190958
.A001906,A000225,A057083,A049072,A190959,A190960,A190961,A190962,A190963,A190964
.A001353,A007070,A003462,A001787,A099456,A190965,A168175,A190966,A190967,A190968
.A004254,A107839,A116415,A002450,A030191,A001047,A099450,A190969,A190970,A190971
.A001109,A154244,A138395,A084326,A003463,A030192,A081179,A006516,A027471,A106392
.A004187,A186446,A190972,A190973,A190974,A003464,A030240,A152268,A099459,A016127
.A001090,A190975,A190976,A099156,A190977,A190978,A023000,A057084,A154245,A154235
.A018913,A190979,A190980,A190981,A190982,A190983,A190984,A023001,A057085,A178869
A004189,A190869,A190985,A190986,A190987,A190988,A190989,A190990,A002452,A057086

I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1)-10*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 17 2011

Mathematica
```
LinearRecurrence[{2,-10}, {0,1}, 50]
```

a(n)=([0,1; -10,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

[lucas_number1(n,2,10) for n in (0..50)] # G. C. Greubel, Jun 10 2022

G.f.: x / ( 1 - 2*x + 10*x^2 ). - R. J. Mathar, Jun 01 2011
E.g.f.: (1/3)*exp(x)*sin(3*x). - Franck Maminirina Ramaharo, Nov 13 2018
a(n) = 10^((n-1)/2) * ChebyshevU(n-1, 1/sqrt(10)). - G. C. Greubel, Jun 10 2022
a(n) = (1/3)*10^(n/2)*sin(n*arctan(3)) = Sum_{k=0..floor(n/2)} (-1)^k*3^(2*k)*binomial(n,2*k+1). - Gerry Martens, Oct 15 2022

A218724 a(n) = (21^n - 1)/20.

Original entry on oeis.org

0, 1, 22, 463, 9724, 204205, 4288306, 90054427, 1891142968, 39714002329, 833994048910, 17513875027111, 367791375569332, 7723618886955973, 162195996626075434, 3406115929147584115, 71528434512099266416, 1502097124754084594737, 31544039619835776489478

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 21 (A009965); q-integers for q=21: diagonal k=1 in triangle A022185.
Partial sums are in A014905. Also, the sequence is related to A014938 by A014938(n) = n*a(n) - Sum_{i=0..n-1} a(i) for n > 0. - Bruno Berselli, Nov 06 2012
For n >= 1, 4*a(n) is the total number of holes in a certain box fractal (start with 21 boxes, 4 holes) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 27 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Kival Ngaokrajang, Illustration of initial terms
Index entries related to partial sums
Index entries for linear recurrences with constant coefficients, signature (22,-21).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218725-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009965, A014905, A014938, A022185.

[n le 2 select n-1 else 22*Self(n-1) - 21*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{22, -21}, {0, 1}, 40] (* Vincenzo Librandi, Nov 07 2012 *)

A218724(n):=(21^n-1)/20$ makelist(A218724(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218724(n)=21^n\20
```

a(n) = floor(21^n/20).
G.f.: x/((1-x)*(1-21*x)). - Bruno Berselli, Nov 06 2012
a(n) = 22*a(n-1) - 21*a(n-2). - Vincenzo Librandi, Nov 07 2012
a(n) = 21*a(n-1) + 1. - Kival Ngaokrajang, Jan 27 2015
a(n) = a(n-1) + 21^(n-1), n >= 1, a(0) = 0.  - Wolfdieter Lang, Feb 02 2015
E.g.f.: exp(11*x)*sinh(10*x)/10. - Elmo R. Oliveira, Aug 29 2024

A218734 a(n) = (31^n - 1)/30.

Original entry on oeis.org

0, 1, 32, 993, 30784, 954305, 29583456, 917087137, 28429701248, 881320738689, 27320942899360, 846949229880161, 26255426126284992, 813918209914834753, 25231464507359877344, 782175399728156197665, 24247437391572842127616, 751670559138758105956097

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 31 (A009975).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See pp. 2, 17.
Index entries related to partial sums.
Index entries for linear recurrences with constant coefficients, signature (32,-31).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218733, A132469, A218736-A218753, A133853, A094028, A218723.

[n le 2 select n-1 else 32*Self(n-1)-31*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{32, -31}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218734(n):=(31^n-1)/30$
makelist(A218734(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
a(n)=31^n\30
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1 - x)*(1 - 31*x)).
a(n) = 32*a(n-1) - 31*a(n-2) for n > 1.
a(n) = floor(31^n/30). (End)
E.g.f.: exp(16*x)*sinh(15*x)/15. - Stefano Spezia, Mar 11 2023

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