A009970 -id:A009970 - cp's OEIS frontend

A218729 a(n) = (26^n - 1)/25.

0, 1, 27, 703, 18279, 475255, 12356631, 321272407, 8353082583, 217180147159, 5646683826135, 146813779479511, 3817158266467287, 99246114928149463, 2580398988131886039, 67090373691429037015, 1744349715977154962391, 45353092615406029022167, 1179180408000556754576343

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 26 (A009970); q-integers for q=26.

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 (27,-26).

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. A009970.

[n le 2 select n-1 else 27*Self(n-1)-26*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{27, -26}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218729(n):=(26^n-1)/25$
makelist(A218729(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

PARI
```
A218729(n)=26^n\25
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1-x)*(1-26*x)).
a(n) = floor(26^n/25).
a(n) = 27*a(n-1) - 26*a(n-2). (End)
E.g.f.: exp(x)*(exp(25*x) - 1)/25. - Elmo R. Oliveira, Aug 29 2024

A106710 Number of words with n letters from an alphabet of size 26 with at least two equal consecutive letters.

Original entry on oeis.org

0, 26, 1326, 50726, 1725126, 55009526, 1684153926, 50135658326, 1462218522726, 41984966747126, 1190791264331526, 33440126095275926, 931432109043580326, 25766955599293244726, 708683864685628269126, 19394355959426432653526, 528467641885089690397926

Offset: 1

Luca Colucci, May 14 2005

			a(3) = 1326 because 26^3 - 26*(25^2) = 1326.

Colin Barker, Table of n, a(n) for n = 1..706
Index entries for linear recurrences with constant coefficients, signature (51,-650).

Cf. A009969, A009970,

Table[26*(26^(n-1) -25^(n-1)), {n, 25}] (* G. C. Greubel, Sep 10 2021 *)

a(n) = 26^n - 26*(25^(n - 1)); \\ Michel Marcus, Aug 14 2013

concat(0, Vec(26*x^2/((25*x-1)*(26*x-1)) + O(x^100))) \\ Colin Barker, Nov 05 2015

[26*(26^(n-1) - 25^(n-1)) for n in (1..25)] # G. C. Greubel, Sep 10 2021

a(n) = 26^n - 26*25^(n - 1).
From Colin Barker, Nov 05 2015: (Start)
a(n) = 51*a(n-1) - 650*a(n-2) for n>2.
G.f.: 26*x^2 / ((1-25*x)*(1-26*x)). (End)
From G. C. Greubel, Sep 10 2021: (Start)
a(n) = 26*(A009970(n-1) - A009969(n-1)).
E.g.f.: exp(26*x) - (26/25)*exp(25*x). (End)

More terms from Michel Marcus, Aug 14 2013

A165847 Totally multiplicative sequence with a(p) = 26.

Original entry on oeis.org

1, 26, 26, 676, 26, 676, 26, 17576, 676, 676, 26, 17576, 26, 676, 676, 456976, 26, 17576, 26, 17576, 676, 676, 26, 456976, 676, 676, 17576, 17576, 26, 17576, 26, 11881376, 676, 676, 676, 456976, 26, 676, 676, 456976, 26, 17576, 26, 17576, 17576

Offset: 1

Jaroslav Krizek, Sep 28 2009

G. C. Greubel, Table of n, a(n) for n = 1..10000

26^PrimeOmega[Range[100]] (* G. C. Greubel, Apr 13 2016 *)

a(n) = 26^bigomega(n); \\ Altug Alkan, Apr 13 2016

a(n) = A009970(A001222(n)) = 26bigomega(n) = 26^A001222(n).

A285511 Value of the n-th Roman number interpreted as Latin alphabetic number.

Original entry on oeis.org

9, 243, 6327, 256, 22, 581, 15115, 392999, 258, 24, 633, 16467, 428151, 16480, 646, 16805, 436939, 11360423, 16482, 648, 16857, 438291, 11395575, 438304, 16870, 438629, 11404363, 296513447, 438306, 16872, 438681, 11405715, 296548599, 11405728, 438694, 11406053, 296557387, 7710492071, 11405730, 636

Offset: 1

Martin Janecke, Apr 20 2017

Lists can be numbered using different counter styles, for example using the Latin alphabet A, B, C, ..., Z, AA, AB, ... or the Roman number system I, II, III, IV, V, VI, ... Both these counter styles are defined in CSS Counter Styles Level 3 as "upper-alpha" and "upper-roman". Roman number representations are defined for the range 1 to 3999 only. Roman numerals are a subset of Latin alphabet letters; for every Roman number there is exactly one alphabetic number that looks identical. Denote the n-th Roman number by R(n) and the m-th alphabetic number by L(m), then R(n) and L(a(n)) look identical.

			The number n = 1 is written "I" in the Roman number system. "I" being the ninth letter in the alphabet is also the ninth number in the alphabetic number system. Therefore a(1) = 9.
The number n = 2 is written "II" in the Roman number system. "II" is also the 243rd number in the alphabetic number system, because "I" is the ninth letter in the 26-letter alphabet and 9*26^1+9*26^0 = 243. Therefore a(2) = 243.
The number n = 3 is written "III" in the Roman number system. "III" is also the 6327th number in the alphabetic number system because "I" is the ninth letter in the 26-letter alphabet and 9*26^2+9*26^1+9*26^0 = 6327. Therefore a(3) = 6327.
The number n = 4 is written "IV" in the Roman number system. "IV" is also the 256th number in the alphabetic number system because "I" is the ninth letter in the 26-letter alphabet and "V" is the 22nd letter, therefore a(4) = 9*26^1 + 22 = 256.
The number n = 600 is written "DC" in the Roman number system. "DC" is also the 107th number in the alphabetic number system, because "D" and "C" are the fourth and third letters in the 26-letter alphabet and 4*26^1+3*26^0 = 107. Therefore a(600) = 107.

Martin Janecke, Table of n, a(n) for n = 1..3999
Tab Atkins Jr. et al., CSS Counter Styles Level 3 (W3C Candidate Recommendation), 11 June 2015
Martin Janecke, Roman numbers ↔ Latin alphabet counter, April 2017

Cf. A006968, A009970, A061493.

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A218729 a(n) = (26^n - 1)/25.

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A106710 Number of words with n letters from an alphabet of size 26 with at least two equal consecutive letters.

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A165847 Totally multiplicative sequence with a(p) = 26.

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A285511 Value of the n-th Roman number interpreted as Latin alphabetic number.

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