A218734 -id:A218734 - cp's OEIS frontend

A218747 a(n) = (44^n - 1)/43.

0, 1, 45, 1981, 87165, 3835261, 168751485, 7425065341, 326702875005, 14374926500221, 632496766009725, 27829857704427901, 1224513738994827645, 53878604515772416381, 2370658598693986320765, 104308978342535398113661, 4589595047071557517001085, 201942182071148530748047741

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 44 (A009988).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (45,-44).

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-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009988.

[n le 2 select n-1 else 45*Self(n-1) - 44*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{45, -44}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)
Join[{0},Accumulate[44^Range[0,20]]] (* Harvey P. Dale, Dec 28 2015 *)

A218747(n):=(44^n-1)/43$
makelist(A218747(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218747(n)=44^n\43
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-44*x)).
a(n) = 45*a(n-1) - 44*a(n-2).
a(n) = floor(44^n/43). (End)
E.g.f.: exp(x)*(exp(43*x) - 1)/43. - Elmo R. Oliveira, Aug 29 2024

A218748 a(n) = (45^n - 1)/44.

Original entry on oeis.org

0, 1, 46, 2071, 93196, 4193821, 188721946, 8492487571, 382161940696, 17197287331321, 773877929909446, 34824506845925071, 1567102808066628196, 70519626362998268821, 3173383186334922096946, 142802243385071494362571, 6426100952328217246315696

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 45 (A009989).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (46,-45).

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-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009989.

[n le 2 select n-1 else 46*Self(n-1) - 45*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 08 2012

LinearRecurrence[{46, -45}, {0, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)

A218748(n):=(45^n-1)/44$ makelist(A218748(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218748(n)=45^n\44
```

G.f.: x/((1-x)*(1-45*x)). - Vincenzo Librandi, Nov 08 2012
a(n) = 46*a(n-1) - 45*a(n-2) with a(0)=0, a(1)=1. - Vincenzo Librandi, Nov 08 2012
a(n) = 45*a(n-1) + 1 with a(0)=0. - Vincenzo Librandi, Nov 08 2012
a(n) = floor(45^n/44).  - Vincenzo Librandi, Nov 08 2012
E.g.f.: exp(23*x)*sinh(22*x)/22. - Elmo R. Oliveira, Aug 27 2024

A218749 a(n) = (46^n - 1)/45.

Original entry on oeis.org

0, 1, 47, 2163, 99499, 4576955, 210539931, 9684836827, 445502494043, 20493114725979, 942683277395035, 43363430760171611, 1994717814967894107, 91757019488523128923, 4220822896472063930459, 194157853237714940801115, 8931261248934887276851291, 410838017451004814735159387

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 46 (A009990).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (47,-46).

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-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009990.

[n le 2 select n-1 else 47*Self(n-1) - 46*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 08 2012

LinearRecurrence[{47, -46}, {0, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)
(46^Range[0,20]-1)/45 (* Harvey P. Dale, Aug 17 2017 *)

A218749(n):=(46^n-1)/45$ makelist(A218749(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218749(n)=46^n\45
```

From Vincenzo Librandi, Nov 08 2012: (Start)
G.f.: x/((1-x)*(1-46*x)).
a(n) = 47*a(n-1) - 46*a(n-2) with a(0)=0, a(1)=1.
a(n) = 46*a(n-1) + 1 with a(0)=0.
a(n) = floor(46^n/45). (End)
E.g.f.: exp(x)*(exp(45*x) - 1)/45. - Elmo R. Oliveira, Aug 29 2024

A218751 a(n) = (48^n - 1)/47.

Original entry on oeis.org

0, 1, 49, 2353, 112945, 5421361, 260225329, 12490815793, 599559158065, 28778839587121, 1381384300181809, 66306446408726833, 3182709427618887985, 152770052525706623281, 7332962521233917917489, 351982201019228060039473, 16895145648922946881894705, 810966991148301450330945841

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 48 (A009992).

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries related to partial sums.
Index entries for linear recurrences with constant coefficients, signature (49,-48).

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-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009992.

[n le 2 select n-1 else 49*Self(n-1)-48*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 08 2012

LinearRecurrence[{49, -48}, {0, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)

A218751(n):=floor((48^n-1)/47)$ makelist(A218751(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218751(n)=48^n\47
```

a(n) = floor(48^n/47).
From Vincenzo Librandi, Nov 08 2012: (Start)
G.f.: x/((1-x)*(1-48*x)).
a(n) = 49*a(n-1) - 48*a(n-2) with a(0)=0, a(1)=1.
a(n) = 48*a(n-1) + 1 with a(0)=0. (End)
E.g.f.: exp(x)*(exp(47*x) - 1)/47. - Elmo R. Oliveira, Aug 29 2024

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)).

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.

Original entry on oeis.org

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).

cp's OEIS Frontend

A218747 a(n) = (44^n - 1)/43.

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A218748 a(n) = (45^n - 1)/44.

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A218749 a(n) = (46^n - 1)/45.

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Magma

Mathematica

Maxima

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Formula

A218751 a(n) = (48^n - 1)/47.

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Author

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Comments

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Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula