cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A218731 a(n) = (28^n - 1)/27.

Original entry on oeis.org

0, 1, 29, 813, 22765, 637421, 17847789, 499738093, 13992666605, 391794664941, 10970250618349, 307167017313773, 8600676484785645, 240818941573998061, 6742930364071945709, 188802050194014479853, 5286457405432405435885, 148020807352107352204781, 4144582605859005861733869
Offset: 0

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 28 (A009972).

Links

Crossrefs

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. A009972.

Programs

Formula

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-28*x)).
a(n) = floor(28^n/27).
a(n) = 29*a(n-1) - 28*a(n-2). (End)
E.g.f.: exp(x)*(exp(27*x) - 1)/27. - Elmo R. Oliveira, Aug 29 2024

A218739 a(n) = (36^n - 1)/35.

Original entry on oeis.org

0, 1, 37, 1333, 47989, 1727605, 62193781, 2238976117, 80603140213, 2901713047669, 104461669716085, 3760620109779061, 135382323952046197, 4873763662273663093, 175455491841851871349, 6316397706306667368565, 227390317427040025268341, 8186051427373440909660277
Offset: 0

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 36 (A009980).

Links

Crossrefs

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. A009980.

Programs

Formula

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1 - x)*(1 - 36*x)).
a(n) = 37*a(n-1) - 36*a(n-2).
a(n) = floor(36^n/35). (End)
E.g.f.: exp(x)*(exp(35*x) - 1)/35. - Stefano Spezia, Mar 28 2023

A218741 a(n) = (38^n - 1)/37.

Original entry on oeis.org

0, 1, 39, 1483, 56355, 2141491, 81376659, 3092313043, 117507895635, 4465300034131, 169681401296979, 6447893249285203, 245019943472837715, 9310757851967833171, 353808798374777660499, 13444734338241551098963, 510899904853178941760595, 19414196384420799786902611
Offset: 0

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 38 (A009982).

Links

Crossrefs

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. A009982.

Programs

Formula

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-38*x)).
a(n) = 39*a(n-1) - 38*a(n-2).
a(n) = floor(38^n/37). (End)
E.g.f.: exp(x)*(exp(37*x) - 1)/37. - Elmo R. Oliveira, Aug 29 2024

A218742 a(n) = (39^n - 1)/38.

Original entry on oeis.org

0, 1, 40, 1561, 60880, 2374321, 92598520, 3611342281, 140842348960, 5492851609441, 214221212768200, 8354627297959801, 325830464620432240, 12707388120196857361, 495588136687677437080, 19327937330819420046121, 753789555901957381798720, 29397792680176337890150081
Offset: 0

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 39 (A009983).

Links

Crossrefs

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. A009983.

Programs

Formula

a(n) = floor(39^n/38).
From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-39*x)).
a(n) = 40*a(n-1) - 39*a(n-2). (End)
E.g.f.: exp(20*x)*sinh(19*x)/19. - Elmo R. Oliveira, Aug 29 2024

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

Original entry on oeis.org

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

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 44 (A009988).

Links

Crossrefs

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.

Programs

Formula

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

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 45 (A009989).

Links

Crossrefs

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.

Programs

Formula

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

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 46 (A009990).

Links

Crossrefs

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.

Programs

Formula

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

Views

Author

M. F. Hasler, Nov 04 2012

Keywords

Comments

Partial sums of powers of 48 (A009992).

Links

Crossrefs

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.

Programs

Formula

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

A269948 Triangle read by rows, Stirling set numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+k^3*T(n-1, k), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 9, 1, 0, 1, 73, 36, 1, 0, 1, 585, 1045, 100, 1, 0, 1, 4681, 28800, 7445, 225, 1, 0, 1, 37449, 782281, 505280, 35570, 441, 1, 0, 1, 299593, 21159036, 33120201, 4951530, 130826, 784, 1
Offset: 0

Views

Author

Peter Luschny, Mar 22 2016

Keywords

Comments

Also called 3rd central factorial numbers.

Examples

			1,
0, 1,
0, 1, 1,
0, 1, 9,     1,
0, 1, 73,    36,     1,
0, 1, 585,   1045,   100,    1,
0, 1, 4681,  28800,  7445,   225,   1,
0, 1, 37449, 782281, 505280, 35570, 441, 1.

Crossrefs

Variant: A098436.
Cf. A007318 (order 0), A048993 (order 1), A269945 (order 2).
Cf. A000537, A023001, A098437.

Programs

  • Maple
    T := proc(n, k) option remember;
    `if`(n=k, 1,
    `if`(k<0 or k>n, 0,
     T(n-1, k-1) + k^3*T(n-1, k))) end:
for n from 0 to 9 do seq(T(n,k), k=0..n) od;
  • Mathematica
    T[n_, n_] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n - 1, k - 1] + k^3*T[n - 1, k]; T[, ] = 0;
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 25 2019 *)

Formula

T(n,2) = (8^(n-1)-1)/7 for n>=1 (cf. A023001).
T(n,n-1) = (n*(n-1)/2)^2 for n>=1 (cf. A000537).
Row sums: A098437.

A086930 Smallest b>1 such that in base b representation the n-th prime is a repunit.

Original entry on oeis.org

2, 4, 2, 10, 3, 16, 18, 22, 28, 2, 36, 40, 6, 46, 52, 58, 60, 66, 70, 8, 78, 82, 88, 96, 100, 102, 106, 108, 112, 2, 130, 136, 138, 148, 150, 12, 162, 166, 172, 178, 180, 190, 192, 196, 198, 14, 222, 226, 228, 232, 238, 15, 250, 256, 262, 268, 270, 276, 280, 282
Offset: 2

Views

Author

Reinhard Zumkeller, Sep 21 2003

Keywords

Comments

From Robert G. Wilson v, Mar 26 2014: (Start)
Obviously the first prime number, 2, can never become a repunit since it is even; therefore this sequence has the offset of 2.
Most terms, a(n), are one less than the n-th prime; e.g., for a(8) the eighth prime is 19_10 = 11_18. Therefore a(n) <= Pi(n)-1.
However there are some terms for which a(n) occurs before Pi(n)-1; e.g., for a(14) the fourteenth prime is 43_10 = 111_6.
Those indices, i, are: 4, 6, 11, 14, 21, 31, 37, 47, 53, 63, 82, 90, ..., . Prime(i) = A085104.
In those cases a(n) is a proper divisor of Prime(n)-1.
(End)

Examples

			n=6: A000040(6) = 13 = 1*3^2 + 1*3^1 + 1*3^0: ternary(13)='111' and binary(13)='1101', therefore a(6)=3.

Links

Crossrefs

Cf. A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A004022, A016123, A016125, A085104.

Programs

  • Mathematica
    f[n_] := Block[{i = 1, d, p = Prime@ n}, d = Rest@ Divisors[p - 1]; While[id = IntegerDigits[p, d[[i]]]; id != Reverse@ id || Union@ id != {1}, i++]; d[[i]]]; Array[f, 60, 2]
Previous Showing 61-70 of 83 results. Next

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