A003462 -id:A003462 - cp's OEIS frontend

A132469 a(n) = (2^(5*n) - 1)/31.

0, 1, 33, 1057, 33825, 1082401, 34636833, 1108378657, 35468117025, 1134979744801, 36319351833633, 1162219258676257, 37191016277640225, 1190112520884487201, 38083600668303590433, 1218675221385714893857, 38997607084342876603425, 1247923426698972051309601

Offset: 0

A.K. Devaraj, Aug 22 2007

Partial sums of powers of 32 (A009976), a.k.a. q-numbers for q=32. - M. F. Hasler, Nov 05 2012

A. K. Devaraj, "Minimum Universal Exponent Generalisation of Fermat's Theorem", in ISSN #1550-3747, Proceedings of Hawaii Intl Conference on Statistics, Mathematics & Related Fields, 2004.

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. See Table 1.
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (33,-32).

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.

[n le 2 select n-1 else 33*Self(n-1) - 32*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

Table[(2^(5 n) - 1)/31, {n, 16}] (* Robert G. Wilson v *)
LinearRecurrence[{33, -32}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A132469(n):=(32^n-1)/31$
makelist(A132469(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

A132469(n)=32^n\31  \\ M. F. Hasler, Nov 07 2012

[gaussian_binomial(5*n,1,2)/31 for n in range(1,17)] # Zerinvary Lajos, May 28 2009

a(n) = (32^n - 1)/31 = floor(32^n/31) = Sum_{k=0..n} 32^k. - M. F. Hasler, Nov 05 2012
G.f.: x/((1 - x)*(1 - 32*x)). - Bruno Berselli, Nov 06 2012
E.g.f.: exp(x)*(exp(31*x) - 1)/31. - Stefano Spezia, Mar 23 2023

Edited and extended by Robert G. Wilson v, Aug 22 2007

Edited and extended to offset 0 by M. F. Hasler, Nov 05 2012

A218721 a(n) = (18^n-1)/17.

Original entry on oeis.org

0, 1, 19, 343, 6175, 111151, 2000719, 36012943, 648232975, 11668193551, 210027483919, 3780494710543, 68048904789775, 1224880286215951, 22047845151887119, 396861212733968143, 7143501829211426575, 128583032925805678351

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 18 (A001027), q-integers for q=18: diagonal k=1 in triangle A022182.
Partial sums are in A014901. Also, the sequence is related to A014935 by A014935(n) = n*a(n) - Sum_{i=0..n-1} a(i), for n>0. - Bruno Berselli, Nov 06 2012
From Bernard Schott, May 06 2017: (Start)
Except for 0, 1 and 19, all terms are Brazilian repunits numbers in base 18, and so belong to A125134. From n = 3 to n = 8286, all terms are composite. See link "Generalized repunit primes".
As explained in the extensions of A128164, a(25667) = (18^25667 - 1)/17 would be (is) the smallest prime in base 18. (End)

			a(3) = (18^3 - 1)/17 = 343 = 7 * 49; a(6) = (18^6 - 1)/17 = 2000719 = 931 * 2149. - _Bernard Schott_, May 01 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..800
Harvey Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Index entries related to partial sums
Index entries for linear recurrences with constant coefficients, signature (19,-18).

Cf. A000225, A001027, A002275, A002450, A002452, A003462, A003463, A003464, A014901, A014935, A016123, A016125, A022182, A023000, A023001, A064108, A091030, A091045, A094028, A125134, A128164, A131865, A135518, A135519, A218722, A218724, A218733, A218743, A218752.

[n le 2 select n-1 else 19*Self(n-1)-18*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{19, -18}, {0, 1}, 40] (* Vincenzo Librandi, Nov 07 2012 *)
Join[{0},Accumulate[18^Range[0,20]]] (* Harvey P. Dale, Nov 08 2012 *)

A218721(n):=(18^n-1)/17$ makelist(A218721(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218721(n)=18^n\17
```

a(n) = floor(18^n/17).
G.f.: x/((1-x)*(1-18*x)). - Bruno Berselli, Nov 06 2012
a(n) = 19*a(n-1) - 18*a(n-2). - Vincenzo Librandi, Nov 07 2012
E.g.f.: exp(x)*(exp(17*x) - 1)/17. - Stefano Spezia, Mar 11 2023

A218753 a(n) = (49^n - 1)/48.

Original entry on oeis.org

0, 1, 50, 2451, 120100, 5884901, 288360150, 14129647351, 692352720200, 33925283289801, 1662338881200250, 81454605178812251, 3991275653761800300, 195572507034328214701, 9583052844682082520350, 469569589389422043497151, 23008909880081680131360400

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 49 (A087752).

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 (50,-49)

Cf. A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218752, A133853, A094028, A218723.

Cf. A087752.

[n le 2 select n-1 else 50*Self(n-1) - 49*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 08 2012

LinearRecurrence[{50, -49}, {0, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)
Join[{0},Accumulate[49^Range[0,20]]] (* Harvey P. Dale, Apr 14 2023 *)

A218753(n):=floor(49^n/48)$ makelist(A218753(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218753(n)=49^n\48
```

G.f.: x/((1-x)*(1-49*x)). - Vincenzo Librandi, Nov 08 2012
a(n) = 50*a(n-1) - 49*a(n-2) with a(0)=0, a(1)=1. - Vincenzo Librandi, Nov 08 2012
a(n) = 49*a(n-1) + 1 with a(0)=0. - Vincenzo Librandi, Nov 08 2012
a(n) = floor(49^n/48). - Vincenzo Librandi, Nov 08 2012
E.g.f.: exp(25*x)*sinh(24*x)/24. - Elmo R. Oliveira, Aug 27 2024

A160870 Array read by antidiagonals: T(n,k) is the number of sublattices of index n in generic k-dimensional lattice (n >= 1, k >= 1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 7, 1, 1, 7, 13, 15, 1, 1, 6, 35, 40, 31, 1, 1, 12, 31, 155, 121, 63, 1, 1, 8, 91, 156, 651, 364, 127, 1, 1, 15, 57, 600, 781, 2667, 1093, 255, 1, 1, 13, 155, 400, 3751, 3906, 10795, 3280, 511, 1, 1, 18, 130, 1395, 2801, 22932, 19531, 43435, 9841, 1023, 1

Offset: 1

N. J. A. Sloane, Nov 19 2009

			Array begins:
  1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
  1,3,7,15,31,63,127,255,511,1023,2047,4095,8191,16383,32767,65535,...
  1,4,13,40,121,364,1093,3280,9841,29524,88573,265720,797161,2391484,...
  1,7,35,155,651,2667,10795,43435,174251,698027,2794155,11180715,...
  1,6,31,156,781,3906,19531,97656,488281,2441406,12207031,61035156,...
  ...

Günter Scheja, Uwe Storch, Lehrbuch der Algebra, Teil 2. BG Teubner, Stuttgart, 1988. [§63, Aufg. 13]

Álvar Ibeas, First 100 antidiagonals, flattened
Michael Baake, Solution of the coincidence problem in dimensions d≤4, arXiv:math/0605222 [math.MG], 2006. [Appx. A]
B. Gruber, Alternative formulas for the number of sublattices, Acta Cryst. A53 (1997) 807-808.
J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.
Yi Ming Zou, Gaussian binomials and the number of sublattices, arXiv:math/0610684 [math.CO], 2006.
Yi Ming Zou, Gaussian binomials and the number of sublattices, Acta Cryst. A62 (2006) 409-410.

Columns: A000203, A001001, A038991, A038992, A038993, A038994, A038995, A038996, A038997.
Rows: A000012, A000225, A003462, A006095, A003463, A160869, A023000, A006096.
Transposed array: A128119.

T[, 1] = 1; T[1, ] = 1; T[n_, k_] := T[n, k] = DivisorSum[n, (n/#)^(k-1) *T[#, k-1]&]; Table[T[n-k+1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 04 2015 *)

T(n,k)={ if ( (n==1) || (k==1), 1, sumdiv(n,d, d*T(d, k-1)) ) }

T(n,1) = 1; T(1,k) = 1; T(n, k) = Sum_{d|n} d*T(d, k-1).
From Álvar Ibeas, Oct 31 2015: (Start)
T(n,k) = Sum_{d|n} (n/d)^(k-1) * T(d, k-1).
T(Product(p^e), k) = Product(Gaussian_poly[e+k-1, e]_p). (End)

A218736 a(n) = (33^n - 1)/32.

Original entry on oeis.org

0, 1, 34, 1123, 37060, 1222981, 40358374, 1331826343, 43950269320, 1450358887561, 47861843289514, 1579440828553963, 52121547342280780, 1720011062295265741, 56760365055743769454, 1873092046839544391983, 61812037545704964935440, 2039797239008263842869521

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 33 (A009977).

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 (34,-33).

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, A218737-A218753, A133853, A094028, A218723.

[n le 2 select n-1 else 34*Self(n-1)-33*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

LinearRecurrence[{34, -33}, {0, 1}, 30] (* Vincenzo Librandi, Nov 07 2012 *)

A218736(n):=(33^n-1)/32$
makelist(A218736(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218736(n)=33^n>>5
```

From Vincenzo Librandi, Nov 07 2012: (Start)
G.f.: x/((1 - x)*(1 - 33*x)).
a(n) = 34*a(n-1) - 33*a(n-2).
a(n) = floor(33^n/32). (End)
E.g.f.: exp(x)*(exp(32*x) - 1)/32. - Stefano Spezia, Mar 24 2023

A036239 Number of 2-element intersecting families of an n-element set; number of 2-way interactions when 2 subsets of power set on {1..n} are chosen at random.

Original entry on oeis.org

0, 2, 15, 80, 375, 1652, 7035, 29360, 120975, 494252, 2007555, 8120840, 32753175, 131818052, 529680075, 2125927520, 8525298975, 34165897052, 136857560595, 548011897400, 2193792030375, 8780400395252, 35137296305115, 140596265198480

Offset: 1

N. J. A. Sloane

Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are intersecting but for which x is not a subset of y and y is not a subset of x, or 1) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x. - Ross La Haye, Jan 10 2008

Graph theory formulation. Let P(A) be the power set of an n-element set A. Then a(n) = the number of edges in the intersection graph G of P(A). The vertices of G are the elements of P(A) and the edges of G are the pairs of elements {x,y} of P(A) such that x and y are intersecting (and x <> y). - Ross La Haye, Dec 23 2017

W. W. Kokko, "Interactions", manuscript, 1983.

T. D. Noe, Table of n, a(n) for n = 1..200
Taylor Brysiewicz, Holger Eble, and Lukas Kühne, Enumerating chambers of hyperplane arrangements with symmetry, arXiv:2105.14542 [math.CO], 2021.
V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, in Russian, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, English translation, in Discrete Mathematics and Applications, 9, (1999), no. 6.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A036240, A051180-A051185, A028243, A032263.

Cf. A003462, A006516, A016127, A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009

LinearRecurrence[{10,-35,50,-24},{0,2,15,80},40] (* or *) With[{c=1/2!}, Table[ c(4^n-3^n-2^n+1),{n,40}]] (* Harvey P. Dale, May 11 2011 *)

a(n)=(4^n-3^n-2^n+1)/2 \\ Charles R Greathouse IV, Jul 25 2011

[(4^n - 2^n)/2-(3^n - 1)/2 for n in range(1,24)] # Zerinvary Lajos, Jun 05 2009

a(n) = (1/2) * (4^n - 3^n - 2^n + 1).
a(n) = 3*Stirling2(n+1,4) + 2*Stirling2(n+1,3). - Ross La Haye, Jan 10 2008
a(n) = A006516(n) - A003462(n). - Zerinvary Lajos, Jun 05 2009
From Harvey P. Dale, May 11 2011: (Start)
a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4); a(0)=0, a(1)=2, a(2)=15, a(3)=80.
G.f.: x^2*(2-5*x)/(1 - 10*x + 35*x^2 - 50*x^3 + 24*x^4). (End)
E.g.f.: exp(x)*(exp(x) - 1)^2*(exp(x) + 1)/2. - Stefano Spezia, Jun 26 2022

A000340 a(0)=1, a(n) = 3*a(n-1) + n + 1.

Original entry on oeis.org

1, 5, 18, 58, 179, 543, 1636, 4916, 14757, 44281, 132854, 398574, 1195735, 3587219, 10761672, 32285032, 96855113, 290565357, 871696090, 2615088290, 7845264891, 23535794695, 70607384108, 211822152348, 635466457069

Offset: 0

N. J. A. Sloane, Simon Plouffe

From Johannes W. Meijer, Feb 20 2009: (Start)
Second right hand column (n-m=1) of the A156920 triangle.
The generating function of this sequence enabled the analysis of the polynomials A156921 and A156925.
(End)
Partial sums of A003462, and thus the second partial sums of A000244 (3^n). Also column k=2 of A106516. - John Keith, Jan 04 2022

			G.f. = 1 + 5*x + 18*x^2 + 58*x^3 + 179*x^4 + 543*x^5 + 1636*x^6 + ...

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See pp. 9, 18.
Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See p. 7.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 389
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
László Tóth, On Schizophrenic Patterns in b-ary Expansions of Some Irrational Numbers, arXiv:2002.06584 [math.NT], 2020. Mentions this sequence. See also Proc. Amer. Math. Soc. 148 (2020), 461-469.
Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

From Johannes W. Meijer, Feb 20 2009: (Start)
Cf. A156921, A156925, A156927, A156933. Other columns A156922, A156923, A156924.
Equals A156920 second right hand column.
Equals A142963 second right hand column divided by 2^n.
Equals A156919 second right hand column divided by 2.
(End)
Cf. A014915.
Equals column k=1 of A008971 (shifted). - Jeremy Dover, Jul 11 2021
Cf. A000340, A003462 (first differences), A106516.

[(3^(n+2)-2*n-5)/4: n in [0..30]]; // Vincenzo Librandi, Aug 15 2011

a[ -1]:=0:a[0]:=1:for n from 1 to 50 do a[n]:=4*a[n-1]-3*a[n-2]+1 od: seq(a[n],n=0..50); # Miklos Kristof, Mar 09 2005
A000340:=-1/(3*z-1)/(z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation

a[ n_] := MatrixPower[ {{1, 0, 0}, {1, 1, 0}, {1, 1, 3}}, n + 1][[3, 1]]; (* Michael Somos, May 28 2014 *)
RecurrenceTable[{a[0]==1,a[n]==3a[n-1]+n+1},a,{n,30}] (* or *) LinearRecurrence[{5,-7,3},{1,5,18},30] (* Harvey P. Dale, Jan 31 2017 *)

G.f.: 1/((1-3*x)*(1-x)^2).
a(n) = (3^(n+2) - 2*n - 5)/4.
a(n) = Sum_{k=0..n+1} (n-k+1)*3^k = Sum_{k=0..n+1} k*3^(n-k+1). - Paul Barry, Jul 30 2004
a(n) = Sum_{k=0..n} binomial(n+2, k+2)*2^k. - Paul Barry, Jul 30 2004
a(-1)=0, a(0)=1, a(n) = 4*a(n-1) - 3*a(n-2) + 1. - Miklos Kristof, Mar 09 2005
a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3). - Johannes W. Meijer, Feb 20 2009
a(-2 - n) = 3^-n * A014915(n). - Michael Somos, May 28 2014
E.g.f.: exp(x)*(9*exp(2*x) - 2*x - 5)/4. - Stefano Spezia, Nov 09 2024

A008344 a(1)=0; thereafter a(n+1) = a(n) - n if a(n) >= n otherwise a(n+1) = a(n) + n.

Original entry on oeis.org

0, 1, 3, 0, 4, 9, 3, 10, 2, 11, 1, 12, 0, 13, 27, 12, 28, 11, 29, 10, 30, 9, 31, 8, 32, 7, 33, 6, 34, 5, 35, 4, 36, 3, 37, 2, 38, 1, 39, 0, 40, 81, 39, 82, 38, 83, 37, 84, 36, 85, 35, 86, 34, 87, 33, 88, 32, 89, 31, 90, 30, 91, 29, 92, 28, 93, 27, 94, 26, 95, 25, 96, 24, 97, 23, 98

Offset: 1

N. J. A. Sloane

p^a(n) = A084110(p^(n-1)) for n>1 and p prime. - Reinhard Zumkeller, May 12 2003
For n > 1: a(A029858(n)) = A029858(n) and a(A003462(n)) = 0. - Reinhard Zumkeller, May 09 2012
Absolute first differences of A085059; abs(a(n+1)-a(n)) = n, see also A086283. - Reinhard Zumkeller, Oct 17 2014
For n>3, when a(n) = 3, a(n+1) is in A116970. - Bill McEachen, Oct 03 2023

N. J. A. Sloane, Table of n, a(n) for n = 1..29524 (Up to the 10th 0 term)
Index entries for sequences related to Recamán's sequence

Cf. A046901, A008343, A088230, A085059, A086283.
Equals A085059(n)-1.
Cf. A076042 (based on squares).

a008344 n = a008344_list !! (n-1)
a008344_list = 0 : f 0 [1..] where
   f x (z:zs) = y : f y zs where y = if x < z then x + z else x - z
-- Reinhard Zumkeller, Oct 17 2014, May 08 2012

A008344 := proc(n) option remember; if n = 0 then 0 elif A008344(n-1) >= (n-1) then A008344(n-1)-(n-1) else A008344(n-1)+(n-1); fi; end;

a[1]=0; a[n_] := a[n]=If[a[n-1]>=n-1, a[n-1]-n+1, a[n-1]+n-1]
Transpose[ NestList[ If[First[#]>=Last[#],{First[#]-Last[#],Last[#]+1}, {First[#]+Last[#],Last[#]+1}]&,{0,1},80]][[1]] (* Harvey P. Dale, Jun 20 2011 *)
s = 0; Table[If[s < n, s = s + n, s = s - n], {n, 0, 80}] (* Horst H. Manninger, Dec 03 2018 *)

a(n) = my(expo = logint(2*n+1, 3), res = n - (3^expo-1)/2); if(res==0, 0, if(res%2, (3^expo-res)/2, 3^expo-1+res/2)) \\ Jianing Song, May 25 2021

This is a concatenation S_0, S_1, S_2, ... where S_i = [b_0, b_1, ..., b_{3^(i+1)-1}] with b_0 = 0, b_{2j-1} = k+1-j, b_{2j} = 2k+j (j=1..k), k=(3^(i+1)-1)/2. E.g. S_0 = [0, 1, 3], S_1 = [0, 4, 9, 3, 10, 2, 11, 1, 12].
a((3^n-1)/2) = 0; a((3^n-1)/2 + 2k-1) = (3^n+1)/2 - k for 1 <= k <= (3^n-1)/2; a((3^n-1)/2 + 2k) = 3^n - 1 + k for 1 <= k < (3^n-1)/2. - Benoit Cloitre, Jan 09 2003 [Corrected by Jianing Song, May 25 2021]
a(n) = (n-1+a(n-1)) mod (2*(n-1)). - Jon Maiga, Jul 09 2021

Name edited by Dmitry Kamenetsky, Feb 14 2017

A058481 a(n) = 3^n - 2.

Original entry on oeis.org

1, 7, 25, 79, 241, 727, 2185, 6559, 19681, 59047, 177145, 531439, 1594321, 4782967, 14348905, 43046719, 129140161, 387420487, 1162261465, 3486784399, 10460353201, 31381059607, 94143178825, 282429536479, 847288609441

Offset: 1

Vladeta Jovovic, Nov 26 2000

a(n) = number of 2 X n binary matrices with no zero rows or columns.
a(n)^2 + 2*a(n+1) + 1 is a square number, i.e., a(n)^2 + 2*a(n+1) + 1 = (a(n)+3)^2: for n=2, a(2)^2 + 2*a(3) + 1 = 7^2 + 2*25 + 1 = 100 = (7+3)^2; for n=3, a(3)^2 + 2*a(4) + 1 = 25^2 + 2*79 + 1 = 784 = (25+3)^2. - Bruno Berselli, Apr 23 2010
Sum of n-th row of triangle of powers of 3: 1; 3 1 3; 9 3 1 3 9; 27 9 3 1 3 9 27; ... . - Philippe Deléham, Feb 24 2014
a(n) = least k such that k*3^n + 1 is a square. Thus, the square is given by (3^n-1)^2. - Derek Orr, Mar 23 2014
Binomial transform of A058481: (1, 6, 12, 24, 48, 96, ...) and second binomial transform of (1, 5, 1, 5, 1, 5, ...). - Gary W. Adamson, Aug 24 2016
Number of ordered pairs of nonempty sets whose union is [n]. a(2) = 7: ({1,2},{1,2}), ({1,2},{1}), ({1,2},{2}), ({1},{1,2}), ({1},{2}), ({2},{1,2}), ({2},{1}). If "nonempty" is omitted we get A000244. - Manfred Boergens, Mar 29 2023

			G.f. = x + 7*x^2 + 25*x^3 + 79*x^4 + 241*x^5 + 727*x^6 + 2185*x^7 + 6559*x^8 + ...
a(1) = 1;
a(2) = 3 + 1 + 3 = 7;
a(3) = 9 + 3 + 1 + 3 + 9 = 25;
a(4) = 27 + 9 + 3 + 1 + 3 + 9 + 27 = 79; etc. - _Philippe Deléham_, Feb 24 2014

G. C. Greubel, Table of n, a(n) for n = 1..1000
John Elias, Illustration: Union of Triple-Sierpinski Triangles
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A055602, A024206 (unlabeled case), A055609, A058482, A000244.

Cf. A003462, A007051, A034472, A024023, A067771, A029858, A134931, A115099, A100774, A079004.

A058481:=n->3^n-2; seq(A058481(n), n=1..30); # Wesley Ivan Hurt, Mar 24 2014

a=1;lst={a};Do[a=a*3+4;AppendTo[lst,a],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 25 2008 *)
3^Range[30]-2  (* Harvey P. Dale, Mar 28 2011 *)
LinearRecurrence[{4, -3}, {1, 7}, 25] (* G. C. Greubel, Aug 25 2016 *)

a(n)=3^n-2 \\ Charles R Greathouse IV, Feb 06 2017

{a(n) = if( n<1, 0, 3^n - 2)}; /* Michael Somos, Feb 17 2017 */

Number of m X n binary matrices with no zero rows or columns is Sum_{j=0..m} (-1)^j*C(m, j)*(2^(m-j)-1)^n.
From Mohammad K. Azarian, Jan 14 2009: (Start)
G.f.: 1/(1-3*x)-2/(1-x)+1.
E.g.f.: e^(3*x)-2*(e^x)+1. (End)
a(n) = 3*a(n-1) + 4 (with a(1)=1). - Vincenzo Librandi, Aug 07 2010
a(n) = 4*a(n-1) - 3*a(n-2). - G. C. Greubel, Aug 25 2016

More terms from Larry Reeves (larryr(AT)acm.org), Dec 04 2000

A022167 Triangle of Gaussian binomial coefficients [ n,k ] for q = 3.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 13, 13, 1, 1, 40, 130, 40, 1, 1, 121, 1210, 1210, 121, 1, 1, 364, 11011, 33880, 11011, 364, 1, 1, 1093, 99463, 925771, 925771, 99463, 1093, 1, 1, 3280, 896260, 25095280, 75913222, 25095280, 896260, 3280, 1

Offset: 0

N. J. A. Sloane

The coefficients of the matrix inverse are apparently given by T^(-1)(n,k) = (-1)^n*A157783(n,k). - R. J. Mathar, Mar 12 2013

			Triangle begins:
  1;
  1,    1;
  1,    4,      1;
  1,   13,     13,        1;
  1,   40,    130,       40,        1;
  1,  121,   1210,     1210,      121,        1;
  1,  364,  11011,    33880,    11011,      364,      1;
  1, 1093,  99463,   925771,   925771,    99463,   1093,    1;
  1, 3280, 896260, 25095280, 75913222, 25095280, 896260, 3280, 1;

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

T. D. Noe, Rows n = 0..50 of triangle, flattened
R. Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
Index entries for sequences related to Gaussian binomial coefficients

Columns k=0..3 give A000012, A003462, A006100, A006101.

Cf. A006117 (row sums).

A022167 := proc(n,m)
        A027871(n)/A027871(n-m)/A027871(m) ;
end proc:
seq(seq(A022167(n,m),m=0..n),n=0..10) ; # R. J. Mathar, Nov 14 2011

p[n_] := Product[3^k-1, {k, 1, n}]; t[n_, m_] := p[n]/(p[n-m]*p[m]); Table[t[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jan 10 2014, after R. J. Mathar *)
Table[QBinomial[n, k, 3], {n, 0, 10}, {k, 0, n}] // Flatten
(* or, after Vladimir Kruchinin, using S for qStirling2: *)
S[n_, k_, q_] /; 1 <= k <= n := S[n-1, k-1, q] + Sum[q^j, {j, 0, k-1}]* S[n-1, k, q]; S[n_, 0, ] := KroneckerDelta[n, 0]; S[0, k, ] := KroneckerDelta[0, k]; S[, , ] = 0;
T[n_, k_] /; n >= k := Sum[Binomial[n, j]*S[n-j, n-k, q]*(q-1)^(k-j) /. q -> 3, {j, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 08 2020 *)

T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - Peter A. Lawrence, Jul 13 2017
T(n,k) = Sum_{j=0..k} C(n,j)*qStirling2(n-j,n-k,3)*(2)^(k-j),j,0,k), n >= k, where qStirling2(n,k,3) is triangle A333143. - Vladimir Kruchinin, Mar 07 2020
G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 3^j - 1. - Seiichi Manyama, May 09 2025

cp's OEIS Frontend

A132469 a(n) = (2^(5*n) - 1)/31.

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A218721 a(n) = (18^n-1)/17.

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A218753 a(n) = (49^n - 1)/48.

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A160870 Array read by antidiagonals: T(n,k) is the number of sublattices of index n in generic k-dimensional lattice (n >= 1, k >= 1).

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A218736 a(n) = (33^n - 1)/32.

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A036239 Number of 2-element intersecting families of an n-element set; number of 2-way interactions when 2 subsets of power set on {1..n} are chosen at random.

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