A001022 -id:A001022 - cp's OEIS frontend

A085364 a(0)=1, for n>0: a(n) = 613^(n-1) - (1/2)Sum_{i=1..n-1} a(i)*a(n-i).

1, 6, 60, 654, 7458, 87378, 1042152, 12587730, 153479508, 1885010946, 23285957604, 289018502682, 3601315495050, 45023019250398, 564465885846216, 7094214579174558, 89351097367355826, 1127492973620753010

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jun 25 2003

From G. C. Greubel, May 23 2020: (Start)
This sequence is part of a class of sequences, for m >= 0, with the properties:
a(n) = 2*m*(4*m+1)^(n-1) - (1/2)*Sum_{k=1..n-1} a(k)*a(n-k).
a(n) = Sum_{k=0..n} m^k * binomial(n-1, n-k) * binomial(2*k, k).
n*a(n) = 2*((2*m+1)*n - (m+1))*a(n-1) - (4*m+1)*(n-2)*a(n-2).
a(n) = (2*m) * Hypergeometric2F1(-n+1, 3/2; 2; -4*m), for n > 0.
(4*m + 1)^n = Sum_{k=0..n} Sum_{j=0..k} a(j)*a(k-j).
G.f.: sqrt( (1 - t)/(1 - (4*m+1)*t) ).
This sequence is the case of m=3. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001022 (13^n), A085362, A085363.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt((1-x)/(1-13*x)) )); // G. C. Greubel, May 23 2020

seq(coeff(series( sqrt((1-x)/(1-13*x)) , x, n+1), x, n), n = 0..30); # G. C. Greubel, May 23 2020

CoefficientList[Series[Sqrt[(1-x)/(1-13x)], {x, 0, 25}], x]

my(x='x+O('x^66)); Vec(sqrt((1-x)/(1-13*x))) \\ Joerg Arndt, May 10 2013

def A085362_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( sqrt((1-x)/(1-13*x)) ).list()
A085362_list(30) # G. C. Greubel, May 23 2020

G.f.: sqrt((1-x)/(1-13*x))
Sum_{i=0..n} Sum_{j=0..i} a(j)*a(i-j) = 13^n.
D-finite with recurrence: n*a(n) = 2*(7*n-4)*a(n-1) - 13*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ 2*sqrt(3)*13^(n-1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 14 2012
a(0) = 1; a(n) = (6/n) * Sum_{k=0..n-1} (n+k) * a(k). - Seiichi Manyama, Mar 28 2023
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (1/4)^n * Sum_{k=0..n} 13^k * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)

A175434 (Digit sum of 2^n) mod n.

Original entry on oeis.org

0, 0, 2, 3, 0, 4, 4, 5, 8, 7, 3, 7, 7, 8, 11, 9, 14, 1, 10, 11, 5, 3, 18, 13, 4, 14, 8, 15, 12, 7, 16, 26, 29, 27, 24, 28, 19, 29, 32, 21, 9, 4, 13, 14, 17, 24, 21, 25, 16, 26, 29, 27, 24, 28, 37, 29, 23, 12, 18, 22, 13, 23, 26, 24, 21, 43, 43, 35, 20, 0, 15, 37, 37, 56, 50, 30, 27, 22, 31, 32, 26, 42, 39, 34, 43, 26, 20, 27, 24, 28, 55, 47, 32, 57, 45, 31, 40, 14, 8, 15

Offset: 1

N. J. A. Sloane, Dec 03 2010

			For n = 1,2,3,4,5,6, the digit-sum of 2^n is 2,4,8,7,5,10, so
a(1) through a(6) are 0,0,2,3,0,4. - _N. J. A. Sloane_, Aug 12 2014

Sum of digits of k^n mod n: (k=2) A000079, A001370, A175434, A175169; (k=3) A000244, A004166, A175435, A067862; (k=5) A000351, A066001, A175456; (k=6) A000400, A066002, A175457, A067864; (k=7) A000420, A066003, A175512, A067863; (k=8) A062933; (k=13) A001022, A175527, A175528, A175525; (k=21) A175589; (k=167) A175558, A175559, A175560, A175552.

Table[Mod[Total[IntegerDigits[2^n]],n],{n,100}] (* Harvey P. Dale, Aug 12 2014 *)

Offset changed to 1 at the suggestion of Harvey P. Dale, Aug 12 2014

A175528 (Digit sum of 13^n) mod n.

Original entry on oeis.org

0, 0, 1, 2, 0, 1, 5, 2, 1, 7, 8, 7, 6, 13, 13, 5, 3, 1, 9, 15, 7, 11, 14, 13, 21, 3, 10, 9, 17, 13, 29, 9, 16, 23, 29, 28, 27, 26, 4, 33, 9, 16, 23, 37, 10, 17, 6, 16, 2, 32, 49, 32, 29, 46, 17, 44, 43, 11, 50, 58, 32, 56, 10, 45, 33, 61, 60, 18, 67, 66, 47, 1, 17, 15, 22, 69, 18, 61, 5, 11, 73, 63, 42, 40, 29, 18, 7, 57, 12, 46, 53, 53, 49, 11, 18, 40, 84, 39, 37, 35

Offset: 1

N. J. A. Sloane, Dec 03 2010

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A001022, A175527, A175525.

Array[Mod[Total[IntegerDigits[13^#]],#]&, 1000]

A038327 Triangle whose (i,j)-th entry is binomial(i,j)12^(i-j)1^j.

Original entry on oeis.org

1, 12, 1, 144, 24, 1, 1728, 432, 36, 1, 20736, 6912, 864, 48, 1, 248832, 103680, 17280, 1440, 60, 1, 2985984, 1492992, 311040, 34560, 2160, 72, 1, 35831808, 20901888, 5225472, 725760, 60480, 3024, 84, 1, 429981696, 286654464, 83607552

Offset: 0

N. J. A. Sloane

T(i,j) is the number of i-permutations of 13 objects a,b,c,d,e,f,g,h,i,j,k,l,m, with repetition allowed, containing j a's. - Zerinvary Lajos, Dec 21 2007

These are the rows of A013619 read right to left. Row sums are A001022(i). - R. J. Mathar, Mar 05 2008

			1
12, 1
144, 24, 1
1728, 432, 36, 1
20736, 6912, 864, 48, 1
248832, 103680, 17280, 1440, 60, 1
2985984, 1492992, 311040, 34560, 2160, 72, 1
35831808, 20901888, 5225472, 725760, 60480, 3024, 84, 1

B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

for i from 0 to 7 do seq(binomial(i, j)*12^(i-j), j = 0 .. i) od; # Zerinvary Lajos, Dec 21 2007

A013718 a(n) = 13^(2*n + 1).

Original entry on oeis.org

13, 2197, 371293, 62748517, 10604499373, 1792160394037, 302875106592253, 51185893014090757, 8650415919381337933, 1461920290375446110677, 247064529073450392704413, 41753905413413116367045797

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (169).

Bisection of A001022 (12^n).

Cf. A013708-A013717, A013719-A013729.

[13^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(13^(2*n+1),n=0..11); # Nathaniel Johnston, Jun 25 2011

a(n)=13^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 25 2008: (Start)
a(n) = 169*a(n-1); a(0)=13.
G.f.: 13/(1-169*x). (End)

A133371 Triangle read by rows: T(i,j) is the number of i-permutations of 14 objects a,b,c,d,e,f,g,h,i,j,k,l,m,n, with repetition allowed, containing j a's.

Original entry on oeis.org

1, 13, 1, 169, 26, 1, 2197, 507, 39, 1, 28561, 8788, 1014, 52, 1, 371293, 142805, 21970, 1690, 65, 1, 4826809, 2227758, 428415, 43940, 2535, 78, 1, 62748517, 33787663, 7797153, 999635, 76895, 3549, 91, 1

Offset: 0

Zerinvary Lajos, with help from Emeric Deutsch, Dec 21 2007

Mirror image of A123187. - Philippe Deléham, Dec 27 2007

			1
13, 1
169, 26, 1
2197, 507, 39, 1
28561, 8788, 1014, 52, 1
371293, 142805, 21970, 1690, 65, 1
4826809, 2227758, 428415, 43940, 2535, 78, 1
62748517, 33787663, 7797153, 999635, 76895, 3549, 91, 1

Cf. A001022, A008595.

for i from 0 to 7 do seq(binomial(i, j)*13^(i-j), j = 0 .. i) od;

Flatten[Table[Binomial[i,j] 13^(i-j),{i,0,7},{j,0,i}]] (* Harvey P. Dale, Nov 01 2011 *)

A013752 a(n) = 13^(3*n + 1).

Original entry on oeis.org

13, 28561, 62748517, 137858491849, 302875106592253, 665416609183179841, 1461920290375446110677, 3211838877954855105157369, 7056410014866816666030739693, 15502932802662396215269535105521, 34059943367449284484947168626829637, 74829695578286078013428929473144712489

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2197).

Subsequence of A001022.

Cf. A013753.

[13^(3*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 27 2011

NestList[2197*# &, 13, 15] (* Paolo Xausa, Aug 28 2024 *)

a(n)=13^(3*n+1) \\ Charles R Greathouse IV, Jun 27 2011

From Philippe Deléham, Nov 30 2008: (Start)
a(n) = 2197*a(n-1); a(0)=13.
G.f.: 13/(1-2197*x).
a(n) = A013753(n)/13. (End)
E.g.f.: 13*exp(13^3*x). - Stefano Spezia, Aug 29 2024

A013753 a(n) = 13^(3*n + 2).

Original entry on oeis.org

169, 371293, 815730721, 1792160394037, 3937376385699289, 8650415919381337933, 19004963774880799438801, 41753905413413116367045797, 91733330193268616658399616009, 201538126434611150798503956371773, 442779263776840698304313192148785281, 972786042517719014174576083150881262357

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2197).

Subsequence of A001022.

[13^(3*n+2): n in [0..15]]; // Vincenzo Librandi, Jun 27 2011

13^(3*Range[0,20]+2) (* or *) NestList[2197*#&,169,20] (* Harvey P. Dale, Oct 10 2019 *)

a(n)=13^(3*n+2) \\ Charles R Greathouse IV, Jun 27 2011

From Philippe Deléham, Nov 30 2008: (Start)
a(n) = 2197*a(n-1); a(0)=169.
G.f.: 169/(1-2197*x).
a(n) = 13*A013752(n). (End)

A196792 a(n) = A047848(10, n).

Original entry on oeis.org

1, 2, 15, 184, 2381, 30942, 402235, 5229044, 67977561, 883708282, 11488207655, 149346699504, 1941507093541, 25239592216022, 328114698808275, 4265491084507564, 55451384098598321, 720867993281778162, 9371283912663116095, 121826690864620509224, 1583746981240066619901

Offset: 0

Vincenzo Librandi, Oct 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (14,-13).

Cf. A047848, A047849, A047850, A047851, A047852, A047853, A047854, A047855, A047856.

Cf. A001022 (first differences).

Magma
```
[(13^n+11)/12: n in [0..20]];
```

(13^Range[0,40] +11)/12 (* G. C. Greubel, Jan 17 2025 *)

def A196792(n): return (pow(13, n) + 11)//12
print([A196792(n) for n in range(41)]) # G. C. Greubel, Jan 17 2025

a(n) = (13^n + 11)/12.
a(n) = 13*a(n-1) - 11, with a(0) = 1.
G.f.: (1-12*x)/((1-x)*(1-13*x)). - Bruno Berselli, Oct 11 2011
From Elmo R. Oliveira, Aug 30 2024: (Start)
E.g.f.: exp(x)*(exp(12*x) + 11)/12.
a(n) = 14*a(n-1) - 13*a(n-2) for n > 1. (End)

A319074 a(n) is the sum of the first n nonnegative powers of the n-th prime.

Original entry on oeis.org

1, 4, 31, 400, 16105, 402234, 25646167, 943531280, 81870575521, 15025258332150, 846949229880161, 182859777940000980, 23127577557875340733, 1759175174860440565844, 262246703278703657363377, 74543635579202247026882160, 21930887362370823132822661921, 2279217547342466764922495586798

Offset: 1

Omar E. Pol, Sep 11 2018

nonn

			For n = 4 the 4th prime is 7 and the sum of the first four nonnegative powers of 7 is 7^0 + 7^1 + 7^2 + 7^3 = 1 + 7 + 49 + 343 = 400, so a(4) = 400.

Main diagonal of A319076.
Cf. A000040, A000203, A006093, A062457, A069459, A093360, A319075.
Cf. A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Cf. A126646, A003462, A003463, A023000, A016123, A091030, A091045, A218722, A218726, A218732, A218734, A218740, A218744, A218746, A218750.

a(n) = sum(k=0, n-1, prime(n)^k); \\ Michel Marcus, Sep 13 2018

a(n) = Sum_{k=0..n-1} A000040(n)^k.
a(n) = Sum_{k=0..n-1} A319075(k,n).
a(n) = (A000040(n)^n - 1)/(A000040(n) - 1).
a(n) = (A062457(n) - 1)/A006093(n).
a(n) = A069459(n)/A006093(n).
a(n) = A000203(A000040(n)^(n-1)).
a(n) = A000203(A093360(n)).

cp's OEIS Frontend

A085364 a(0)=1, for n>0: a(n) = 6*13^(n-1) - (1/2)*Sum_{i=1..n-1} a(i)*a(n-i).

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A175434 (Digit sum of 2^n) mod n.

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A175528 (Digit sum of 13^n) mod n.

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A038327 Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.

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A013718 a(n) = 13^(2*n + 1).

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A133371 Triangle read by rows: T(i,j) is the number of i-permutations of 14 objects a,b,c,d,e,f,g,h,i,j,k,l,m,n, with repetition allowed, containing j a's.

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A013752 a(n) = 13^(3*n + 1).

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A013753 a(n) = 13^(3*n + 2).

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A085364 a(0)=1, for n>0: a(n) = 613^(n-1) - (1/2)Sum_{i=1..n-1} a(i)*a(n-i).

A038327 Triangle whose (i,j)-th entry is binomial(i,j)12^(i-j)1^j.