A023000 -id:A023000 - cp's OEIS frontend

A319076 Square array T(n,k) read by antidiagonal upwards in which column k lists the partial sums of the powers of the k-th prime, n >= 0, k >= 1.

1, 3, 1, 7, 4, 1, 15, 13, 6, 1, 31, 40, 31, 8, 1, 63, 121, 156, 57, 12, 1, 127, 364, 781, 400, 133, 14, 1, 255, 1093, 3906, 2801, 1464, 183, 18, 1, 511, 3280, 19531, 19608, 16105, 2380, 307, 20, 1, 1023, 9841, 97656, 137257, 177156, 30941, 5220, 381, 24, 1, 2047, 29524, 488281, 960800, 1948717

Offset: 0

Omar E. Pol, Sep 09 2018

T(n,k) is also the sum of the divisors of the n-th nonnegative power of the k-th prime, n >= 0, k >= 1.

			The corner of the square array is as follows:
         A126646 A003462 A003463  A023000    A016123    A091030     A091045
A000012        1,      1,      1,       1,         1,         1,          1, ...
A008864        3,      4,      6,       8,        12,        14,         18, ...
A060800        7,     13,     31,      57,       133,       183,        307, ...
A131991       15,     40,    156,     400,      1464,      2380,       5220, ...
A131992       31,    121,    781,    2801,     16105,     30941,      88741, ...
A131993       63,    364,   3906,   19608,    177156,    402234,    1508598, ...
.......      127,   1093,  19531,  137257,   1948717,   5229043,   25646167, ...
.......      255,   3280,  97656,  960800,  21435888,  67977560,  435984840, ...
.......      511,   9841, 488281, 6725601, 235794769, 883708281, 7411742281, ...
...

Rows 0-5: A000012, A008864, A060800, A131991, A131992, A131993.
Columns 1-15: A126646, A003462, A003463, A023000, A016123, A091030, A091045, A218722, A218726, A218732, A218734, A218740, A218744, A218746, A218750.
Main diagonal gives A319074.
Cf. A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Cf. A000040, A000203, A006093, A319075.

T(n, k) = sigma(prime(k)^n); \\ Michel Marcus, Sep 13 2018

T(n,k) = A000203(A000040(k)^n).
T(n,k) = Sum_{j=0..n} A000040(k)^j.
T(n,k) = Sum_{j=0..n} A319075(j,k).
T(n,k) = (A000040(k)^(n+1) - 1)/(A000040(k) - 1).
T(n,k) = (A000040(k)^(n+1) - 1)/A006093(k).

A361475 Array read by ascending antidiagonals: A(n, k) = (k^n - 1)/(k - 1), with k >= 2.

Original entry on oeis.org

0, 1, 0, 3, 1, 0, 7, 4, 1, 0, 15, 13, 5, 1, 0, 31, 40, 21, 6, 1, 0, 63, 121, 85, 31, 7, 1, 0, 127, 364, 341, 156, 43, 8, 1, 0, 255, 1093, 1365, 781, 259, 57, 9, 1, 0, 511, 3280, 5461, 3906, 1555, 400, 73, 10, 1, 0, 1023, 9841, 21845, 19531, 9331, 2801, 585, 91, 11, 1, 0

Offset: 0

Stefano Spezia, Mar 13 2023

			The array begins:
   0,  0,  0,   0,   0, ...
   1,  1,  1,   1,   1, ...
   3,  4,  5,   6,   7, ...
   7, 13, 21,  31,  43, ...
  15, 40, 85, 156, 259, ...
  ...

Cf. A003992, A361291 (k=2*n+1), A361476 (antidiagonal sums).

Cf. A000225 (k=2), A003462 (k=3), A002450 (k=4), A003463 (k=5), A003464 (k=6), A023000 (k=7), A023001 (k=8), A002452 (k=9), A002275 (k=10), A016123 (k=11).

A[n_,k_]:=(k^n-1)/(k-1); Flatten[Table[A[n-k+2,k],{n,0,10},{k,2,n+2}]]

E.g.f. of column k: exp(x)*(exp((k-1)*x) - 1)/(k - 1).
E.g.f. of column k: 2*exp((k+1)*x/2)*sinh((k-1)*x/2)/(k - 1).
A(n, k) = Sum_{i=0..n-1} k^i.

A096041 Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^7-M)/6, where M is the infinite lower Pascal's triangle matrix, 1<=k<=n.

Original entry on oeis.org

1, 8, 2, 57, 24, 3, 400, 228, 48, 4, 2801, 2000, 570, 80, 5, 19608, 16806, 6000, 1140, 120, 6, 137257, 137256, 58821, 14000, 1995, 168, 7, 960800, 1098056, 549024, 156856, 28000, 3192, 224, 8, 6725601, 8647200, 4941252, 1647072, 352926, 50400

Offset: 1

Gary W. Adamson, Jun 17 2004

			Triangle begins:
1
8 2
57 24 3
400 228 48 4
2801 2000 570 80 5
19608 16806 6000 1140 120 6

Cf. A007318. First column gives A023000. Row sums give A016131.

P:= proc(n) option remember; local M; M:= Matrix(n, (i, j)-> binomial(i-1, j-1)); (M^7-M)/6 end: T:= (n, k)-> P(n+1)[n+1, k]: seq(seq(T(n, k), k=1..n), n=1..11); # Alois P. Heinz, Oct 07 2009

P[n_] := P[n] = With[{M = Array[Binomial[#1-1, #2-1]&, {n, n}]}, (MatrixPower[M, 7] - M)/6]; T[n_, k_] := P[n+1][[n+1, k]]; Table[ Table[T[n, k], {k, 1, n}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

Edited with more terms by Alois P. Heinz, Oct 07 2009

A102303 a(n) = (1/6) * (7^(n+1) - 3*(-1)^n + 2).

Original entry on oeis.org

1, 9, 57, 401, 2801, 19609, 137257, 960801, 6725601, 47079209, 329554457, 2306881201, 16148168401, 113037178809, 791260251657, 5538821761601, 38771752331201, 271402266318409, 1899815864228857, 13298711049602001, 93090977347214001, 651636841430498009

Offset: 0

Roger L. Bagula, Mar 15 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,1,-7).

Cf. A000035, A023000.

[(7^(n+1)-3*(-1)^n+2)/6: n in [0..50]]; // G. C. Greubel, Feb 03 2024

Table[(7^(n+1) -3*(-1)^n +2)/6, {n, 0, 50}]

[(7^(n+1)-3*(-1)^n+2)/6 for n in range(51)] # G. C. Greubel, Feb 03 2024

From Chai Wah Wu, Mar 11 2021: (Start)
a(n) = 7*a(n-1) + a(n-2) - 7*a(n-3) for n > 2.
G.f.: -(1 + 2*x - 7*x^2)/((1 - x)*(1 + x)*(1 - 7*x)). (End)
From G. C. Greubel, Feb 03 2024: (Start)
a(n) = A023000(n+1) + A000035(n).
E.g.f.: (1/6)*(-3*exp(-x) + 2*exp(x) + 7*exp(7*x)). (End)

Edited by N. J. A. Sloane, May 29 2007

A102360 a(n) = sigma((7^n - 1)/6), where sigma(n) is the sum of positive divisors of n.

Original entry on oeis.org

1, 15, 80, 961, 2802, 52800, 142020, 2347506, 10512320, 96837120, 329849040, 8619170560, 16148168402, 221483030400, 1146781040640, 14442767744328, 38774519976900, 813898294809600, 1904350030969680, 35410174950039552

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 22 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..378

Cf. A000203, A023000.

a(n) = sigma((7^n - 1)/6); \\ Michel Marcus, Feb 21 2020

a(n) = A000203(A023000(n)). - Amiram Eldar, Feb 21 2020

A125823 Numbers whose base 7 representation is 4444....4.

Original entry on oeis.org

0, 4, 32, 228, 1600, 11204, 78432, 549028, 3843200, 26902404, 188316832, 1318217828, 9227524800, 64592673604, 452148715232, 3165041006628, 22155287046400, 155087009324804, 1085609065273632, 7599263456915428, 53194844198408000, 372363909388856004, 2606547365721992032

Offset: 1

Zerinvary Lajos, Feb 03 2007

			Base 7.................decimal
0.........................0
4.........................4
44.......................32
444.....................228
4444...................1600
44444.................11204
444444................78432
4444444..............549028
44444444............3843200
etc....................etc...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A023000.

List([1..30], n-> 2*(7^(n-1) -1)/3); # G. C. Greubel, Aug 03 2019

[2*(7^(n-1) -1)/3: n in [1..30]]; // G. C. Greubel, Aug 03 2019

Maple
```
seq(4*(7^n-1)/6, n=0..21);
```

2*(7^(Range[30]-1) -1)/3 (* G. C. Greubel, Aug 03 2019 *)

vector(30, n, 2*(7^(n-1) -1)/3) \\ G. C. Greubel, Aug 03 2019

[2*(7^(n-1) -1)/3 for n in (1..30)] # G. C. Greubel, Aug 03 2019

a(n) = 2*(7^(n-1) - 1)/3 = 4*A023000(n-1).
a(n) = 7*a(n-1) + 4, with a(1)=0. - Vincenzo Librandi, Sep 30 2010
From G. C. Greubel, Aug 03 2019: (Start)
G.f.: 4*x^2/((1-x)*(1-7*x)).
E.g.f.: 2*(exp(7*x) - exp(x))/3. (End)

Terms a(21) onward added by G. C. Greubel, Aug 03 2019

A272199 Expansion of 1/(1 - 2x + 13x^2).

Original entry on oeis.org

1, 2, -9, -44, 29, 630, 883, -6424, -24327, 34858, 385967, 318780, -4380011, -12904162, 31131819, 230017744, 55321841, -2879586990, -6478357913, 24477915044, 133174482957, -51863929658, -1834996137757, -2995761189960, 17863427410921, 74671750291322

Offset: 0

Wolfdieter Lang, Apr 27 2016

a(n) gives the coefficient c(13^n) of (eta(z^6))^4, a modular cusp form of weight 2, when expanded in powers of q = exp(2*Pi*i*z), Im(z) > 0, assuming alpha-multiplicativity (not valid for p = 2 and 3) with alpha(x) = x (weight 2) and input c(13) = +2. Eta is the Dedekind function. See the Apostol reference, p. 138, eq. (54) for alpha-multiplicativity and p. 130, eq. (39) with k=2. See also A000727 where a(n)=c(13^n) = A000727((13^n-1)/6)=A000727(2*A091030(n)), n >= 0. For the proof that alpha-multiplicativity leads to the recurrence involving Chebyshev's S polynomials see a comment on A168175, and the Apostol reference, Exercise 6., p. 139.

			a(2) = c(13^2) = A000727(2*A091030(2)) =
A000727(28) = -9.

Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, pp. 130, 138 - 139.

Indranil Ghosh, Table of n, a(n) for n = 0..1790
Index entries for linear recurrences with constant coefficients, signature (2,-13).
Index entries for sequences related to Chebyshev polynomials.

Cf. A023000, A049310, A168175.

I:=[1,2]; [n le 2 select I[n] else 2*Self(n-1)-13*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 25 2016

CoefficientList[Series[1/(1 - 2 x + 13 x^2), {x, 0, 25}], x] (* Michael De Vlieger, Apr 27 2016 *)
LinearRecurrence[{2, -13}, {1, 2}, 30] (* Vincenzo Librandi, Nov 25 2016 *)

Vec(1/(1-2*x+13*x^2) + O(x^99)) \\ Altug Alkan, Apr 28 2016

G.f.: 1/(1 - 2*x + 13*x^2).
a(n) = 2*a(n-1) - 13*a(n-2), a(-1) = 0, a(0) = 1.
a(n) = sqrt(13)^n * S(n, 2/sqrt(13)), n >= 0, with Chebyshev's S polynomials (A049310).
a(n) = (Ap^(n+1) - Am^(n+1))/(Ap - Am) with Ap:= 1 + 2*sqrt(3)*i and Am = 1 - 2*sqrt(3)*i, (Binet - de Moivre formula), where i is the imaginary unit.
E.g.f.: (sqrt(3)*sin(2*sqrt(3)*x) + 6*cos(2*sqrt(3)*x))*exp(x)/6. - Ilya Gutkovskiy, Apr 27 2016

A016198 Expansion of g.f. 1/((1-x)(1-2x)(1-5x)).

Original entry on oeis.org

1, 8, 47, 250, 1281, 6468, 32467, 162590, 813461, 4068328, 20343687, 101722530, 508620841, 2543120588, 12715635707, 63578244070, 317891351421, 1589457019248, 7947285620527, 39736429151210, 198682147853201, 993410743460308, 4967053725690147, 24835268645227950

Offset: 0

N. J. A. Sloane, Dec 11 1999

Index entries for linear recurrences with constant coefficients, signature (8,-17,10).

Cf. A000225, A000392, A002275, A002452, A003462, A003463, A003464, A016123, A016125, A016208, A016209, A016218, A016256, A023000, A023001.

Cf. A016127.

a:=n->sum((5^(n-j)-2^(n-j))/3,j=0..n): seq(a(n), n=1..20); # Zerinvary Lajos, Jan 04 2007

Join[{a=1,b=8},Table[c=7*b-10*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2011 *)

a(n) = (25*5^n - 16*2^n + 3)/12. - Bruno Berselli, Feb 09 2011
a(n) = [(5^0-2^0) + (5^1-2^1) + ... + (5^n-2^n)]/3. - r22lou(AT)cox.net, Nov 14 2005
a(0)=1, a(n) = 5*a(n-1) + 2^(n+1) - 1. - Vincenzo Librandi, Feb 07 2011
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(x)*(25*exp(4*x) - 16*exp(x) + 3)/12.
a(n) = 8*a(n-1) - 17*a(n-2) + 10*a(n-3).
a(n) = A016127(n+1) - A003463(n+2). (End)

More terms from Wesley Ivan Hurt, May 05 2014

A016201 Expansion of g.f. 1/((1-x)(1-2x)(1-7x)).

Original entry on oeis.org

1, 10, 77, 554, 3909, 27426, 192109, 1345018, 9415637, 65910482, 461375421, 3229632042, 22607432485, 158252043778, 1107764339213, 7754350440026, 54280453211253, 379963172740914, 2659742209710685, 18618195469023370, 130327368285260741, 912291578001019490, 6386041046015525037

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (10,-23,14).

Cf. A016130, A023000.

a:=n->sum((7^(n-j+1)-2^(n-j+1))/5, j=0..n+1): seq(a(n), n=0..19); # Zerinvary Lajos, Jan 15 2007

CoefficientList[Series[1/((1-x)(1-2x)(1-7x)),{x,0,20}],x](* or *) LinearRecurrence[{10,-23,14},{1,10,77},20] (* Harvey P. Dale, Mar 06 2019 *)

a(n) = (49*7^n - 24*2^n + 5)/30. - Bruno Berselli, Feb 09 2011
a(0)=1, a(n) = 7*a(n-1) + 2^(n+1) - 1. - Vincenzo Librandi, Feb 09 2011
From Elmo R. Oliveira, Mar 27 2025: (Start)
E.g.f.: exp(x)*(49*exp(6*x) - 24*exp(x) + 5)/30.
a(n) = 10*a(n-1) - 23*a(n-2) + 14*a(n-3).
a(n) = A016130(n+1) - A023000(n+2). (End)

More terms from Elmo R. Oliveira, Mar 27 2025

A125729 Numbers whose base 7 representation is 555....5.

Original entry on oeis.org

0, 5, 40, 285, 2000, 14005, 98040, 686285, 4804000, 33628005, 235396040, 1647772285, 11534406000, 80740842005, 565185894040, 3956301258285, 27694108808000, 193858761656005, 1357011331592040, 9499079321144285

Offset: 1

Zerinvary Lajos, Feb 02 2007

			Base 7.............decimal
0........................0
5........................5
55......................40
555....................285
5555..................2000
55555................14005
555555...............98040
5555555.............686285
55555555...........4804000
555555555.........33628005
etc....................etc...

Index entries for linear recurrences with constant coefficients, signature (8,-7).

Maple
```
seq(5*(7^n-1)/6, n=0..21);
```

Table[FromDigits[Table[5,n],7],{n,0,19}] (* James C. McMahon, Dec 21 2024 *)

a(n) = 5*(7^(n-1)-1)/6 = 5*A023000(n-1).
a(n) = 7*a(n-1) + 5 for n > 1. - Vincenzo Librandi, Sep 30 2010
G.f.: 5*x^2 / ( (7*x-1)*(x-1) ). - R. J. Mathar, Sep 30 2013

cp's OEIS Frontend

A319076 Square array T(n,k) read by antidiagonal upwards in which column k lists the partial sums of the powers of the k-th prime, n >= 0, k >= 1.

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A361475 Array read by ascending antidiagonals: A(n, k) = (k^n - 1)/(k - 1), with k >= 2.

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A096041 Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^7-M)/6, where M is the infinite lower Pascal's triangle matrix, 1<=k<=n.

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A102303 a(n) = (1/6) * (7^(n+1) - 3*(-1)^n + 2).

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A102360 a(n) = sigma((7^n - 1)/6), where sigma(n) is the sum of positive divisors of n.

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A125823 Numbers whose base 7 representation is 4444....4.

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A272199 Expansion of 1/(1 - 2*x + 13*x^2).

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A272199 Expansion of 1/(1 - 2x + 13x^2).

A016198 Expansion of g.f. 1/((1-x)(1-2x)(1-5x)).

A016201 Expansion of g.f. 1/((1-x)(1-2x)(1-7x)).