Jack W Grahl - cp's OEIS frontend

A341463 a(n) = (-1)^(n+1) * (3^n+1)/2.

-1, 2, -5, 14, -41, 122, -365, 1094, -3281, 9842, -29525, 88574, -265721, 797162, -2391485, 7174454, -21523361, 64570082, -193710245, 581130734, -1743392201, 5230176602, -15690529805, 47071589414, -141214768241, 423644304722, -1270932914165, 3812798742494, -11438396227481, 34315188682442

Offset: 0

Jack W Grahl, Feb 12 2021

Shown by Tutte (he erroneously gave the negative of this sequence) to be the value of the function g(X_n), where X_n is the graph with one vertex and n loops, and g() is the extension to all graphs of the function f(G) defined on trivalent graphs by f(G) =(-1)^n.Q(G), where 2n is the number of vertices of G, and Q(G) is the number of spanning subgraphs of G such that every vertex of G is incident with 2 edges, and obeying the recursions discussed by Tutte in the article.

This sequence is given in balanced ternary representation as (-1), 1(-1), (-1)11, 1(-1)(-1)(-1), (-1)1111, 1(-1)(-1)(-1)(-1)(-1), etc.

W. T. Tutte, Some polynomials associated with graphs, Combinatorics, Proceedings of the British Combinatorial Conference. Vol. 13. Cambridge Univ. Press London, 1973.

Index entries for linear recurrences with constant coefficients, signature (-4,-3).

Absolute values give A007501.

Cf. A076040.

def a(n):
    return (-1)**(n+1) * (3 ** n + 1) // 2

a(n) = -4*a(n-1) - 3*a(n-2) for n > 1.

G.f.: -(1 + 2*x)/(1 + 4*x + 3*x^2). - Stefano Spezia, Feb 13 2021

A320505 Index of the first occurrence of n consecutive 0's in A164349.

Original entry on oeis.org

0, 2, 7, 2046

Offset: 1

Jack W Grahl, Oct 13 2018

nonn

Indexes are 0-based since A164349 has lead index 0.
The first 4 terms were found empirically and given in a comment to A164349 by Robert G Wilson.
The 5th term is 2^2059 - 3.
Each term is roughly 2 to the power of the previous term.

			Let us call A(n) the string of 0's and 1's given by applying the step "repeat and drop the last symbol" n times to A(0) = 01. Let B be the operation "drop the last symbol", e.g., B(01) = 0. Then A(n+1) = A(n) + B(A(n)) and A(n+m) = A(n) + ... B^m(A(n)). Then if A(n) contains a sequence of k consecutive 0's and no longer sequence, let p be the number of digits which follow after the end of the last occurrence of this sequence in A(n). In other words, B^p(A(n)) ends in a sequence of exactly k 0's. So A(n+p) = A(n) + ... + B^p(A(n)) also ends in a sequence of k 0's. Therefore A(n+p+1), since it concatenates A(n+p) with B(A(n+p)), which starts with a 0, contains a sequence of exactly k+1 0's. To see that no earlier sequence contains k+1 consecutive 0's, see that A(n) didn't. A new such sequence can only be formed by concatenating a prefix of A(n) with another prefix of A(n), which starts with only one 0. So such a prefix would have to end with n 0's. But by definition of p, no B^m(A(n)) for m < p ends with n 0's.
Now we can construct a(5). As can be easily checked, A(12) is made up of "01" + (2044 symbols) + "0000" + (2047 symbols). Therefore "00000" first appears in A(2047 + 12 + 1) = A(2060). The index of the last 0 is equal to the length of A(2059) = 2^2059 + 1. So the index of the first 0 is 2^2059 - 3.

Jack W Grahl, Table of n, a(n) for n = 1..5

Cf. A164349.

A318895 Number of isoclinism classes of the groups of order 2^n.

Original entry on oeis.org

1, 1, 1, 2, 3, 8, 27, 115

Offset: 0

Jack W Grahl, Sep 05 2018

The concept of isoclinism was introduced in Hall (1940) and is crucial to enumerating the groups of order p^n where p is a prime.

An isoclinism exists between two groups G1 and G2 if the following holds: There is an isomorphism f between their two inner automorphism groups G1/Z(G1) and G2/Z(G2). There is an isomorphism h between their two commutator groups [G1, G1] and [G2, G2]. Lastly, f and h commute with F1 and F2, where F1 is the mapping from G1/Z(G1) x G1/Z(G1) to [G1, G1], given by a, b -> ab(a^-1)(b^-1), and F2 is defined analogously.

			There are 51 groups of order 32. These fall into 8 isoclinism classes. Therefore a(5) = 8.

P. Hall, The classification of prime-power groups, J. Reine Angew. Math. 182 (1940), 130-141.
Rodney James, M. F. Newman and E. A. O'Brien, The groups of order 128, Journal of Algebra, Volume 129, Issue 1 (1990), 136-158.
Vipul Naik, This sequence, along with other properties of groups of order 2^n

Cf. A000001, A000679. A000041 has an interpretation as the number of Abelian groups with order 2^n.

A316532 Leading least prime signatures, ordered by the underlying partitions, as in A063008.

Original entry on oeis.org

1, 6, 30, 36, 210, 180, 2310, 216, 900, 1260, 30030, 1080, 6300, 13860, 510510, 1296, 5400, 7560, 44100, 69300, 180180, 9699690, 6480, 27000, 37800, 83160, 485100, 900900, 3063060, 223092870, 7776, 32400, 45360, 189000, 264600, 415800, 1081080, 5336100

Offset: 0

Jack W Grahl, Jul 06 2018

The sequence A063008 gives the least number with each prime signature, ordered by the underlying partition. This sequence is a subsequence which only includes those prime signatures M for which M/2 is not a prime signature, the so-called 'leading' least prime signatures.
This sequence is therefore constructed by taking the partitions first in increasing order of their sum, then in decreasing order of the first term, then decreasing order of the second term, etc. We drop all partitions, except the empty partition, where the first term and the second term are different. Then we map (m1, m2, m3, ..., mk) to 2^m1 * 3^m2 * ... * pk^mk to give the terms of this sequence.
The sequence A062515 had a description which suggested that it had been confused with this sequence. They are the same leading least prime signatures, but in a different order, given by a different construction using integer partitions.

			The first few partitions are [], [1,1], [1,1,1], [2,2], [1,1,1,1]. So the first few terms are 1, 2 * 3 = 6, 2 * 3 * 5 = 30, 2^2 * 3^2 = 36, 2 * 3 * 5 * 7 = 210.

Subsequence of A063008. A re-ordering of A062515, also of A056153. Cf A025487.

primes :: [Integer]
primes = 2 : 3 : filter (\a -> all (not . divides a) (takeWhile (\x -> x <= a `div` 2) primes)) [4..]
divides :: Integer -> Integer -> Bool
divides a b = a `mod` b == 0
partitions :: [[Integer]]
partitions = concat $ map (partitions_of_n) [0..]
partitions_of_n :: Integer -> [[Integer]]
partitions_of_n n = partitions_at_most n n
partitions_at_most :: Integer -> Integer -> [[Integer]]
partitions_at_most _ 0 = [[]]
partitions_at_most 0 _ = []
partitions_at_most m n = concat $ map (\k -> map ([k] ++) (partitions_at_most k (n-k))) ( reverse [1..(min m n)])
prime_signature :: [Integer] -> Integer
prime_signature p = product $ zipWith (^) primes p
seq :: [Integer]
seq = map prime_signature $ filter compare_first_second partitions
    where
  compare_first_second p
        | length p == 0 = True
        | length p == 1 = False
        | otherwise = p!!0 == p!!1

A267795 Integers n such that n, 2n, 3n ... 10n contain almost equally many copies of each base 10 digit.

Original entry on oeis.org

1, 9, 109, 909, 10909, 90909, 1090909, 9090909, 13431958, 25834963, 32973507, 38296415, 45096237, 51546969, 94845303, 96237045, 109090909, 113431958, 126084879, 132868745, 132875488, 133595248, 134319558, 134755956, 134758658, 137584878, 143865844, 153584878

Offset: 1

Jack W Grahl, Jan 20 2016

Here 'almost equally many' means that the most common digit appears only once more than the least common.

			The first 10 multiples of 109 are 109, 218, 327, 436, 545, 654, 763, 872, 981, 1090. Every digit appears 3 times except for '1' which appears 4 times. It is clear that all numbers of the form 10909..0909 and 90909..0909 appear in the list, and it seems likely that these are the only members.

Lars Blomberg, Table of n, a(n) for n = 1..4000

Cf. A038365.

def f(n):
  """ This returns True iff n is in the sequence """
  l = [ n * i for i in range(1, 11) ]
  s = "".join(str(i) for i in l)
  c = [ s.count(str(j)) for j in range(10) ]
  return min(c) >= max(c) - 1
for n in range(1, 10000000):
  if f(n):
    print(n, end=', ')

a(7)-a(28) from Lars Blomberg, Aug 11 2016

A256535 The largest number of T-tetrominoes that fit within an n X n square.

Original entry on oeis.org

0, 0, 1, 4, 5, 8, 11, 16, 19, 24, 29, 36, 41, 48, 55, 64, 71, 80, 89, 100, 109, 120, 131, 144, 155, 168, 181, 196, 209, 224, 239, 256, 271, 288, 305, 324, 341, 360, 379, 400, 419, 440, 461, 484, 505, 528, 551, 576, 599, 624, 649, 676, 701, 728, 755, 784, 811

Offset: 1

Jack W Grahl, Sep 15 2015

No T-tetromino fits in a 1 X 1 or 2 X 2 square: a(1)=a(2)=0. A single T-tetromino can be placed in a 3 X 3 square, and must occupy the center square. Four T-tetrominos fit within a 4 X 4 square with no spaces left over, in a rotationally symmetric tiling: a(4)=4.
For n = 4m, it is obvious that a(n) = n^2/4, by repeating the construction for n=4. For n = 4m + 2, it can be shown, using a chessboard coloring, that a(n) < n^2/4. By tiling an L-shaped strip of width 2 in a manner that can be indefinitely extended, one can show that a(n) = n^2/4 - 1.
Shuxin Zhan proved that it is not possible to tile a square of side 4m+1 or 4m+3 with T-tetrominos and a single monomino. Thus there must be at least 5 empty squares in any partial tiling by T-tetrominos. This bound is achieved for tilings in 5 X 5, 7 X 7, 9 X 9 and 11 X 11 squares. Robert Hochberg proved that for n > 11, there must be either 5 or 9 empty squares. He conjectured that 5 is always enough.
Jack W Grahl proved that, for squares, 5 monominos are always sufficient. This means that the sequence is given by n^2/4, (n^2-1)/4-1, n^2/4-1, (n^2-1)/4-1, for n = 4m, n = 4m+1, n = 4m+2 and n = 4m+3, respectively (which the exception of a(1) = 0), and generating function x^3*(-1-2*x+2*x^2-2*x^3+x^4) / ( (1+x)*(x^2+1)*(x-1)^3 ). - Jack W Grahl, Jul 25 2018

			The optimal tiling for a 4 X 4 square is:
   AAAB
   DABB
   DDCB
   DCCC
This forms the building block of a solution for all n a multiple of four.
For n=7 a solution is given by:
   ABBBCCC
   AABD/CE
   A/DDDEE
       F/E
       FFG
       FGG
       //G
with 5 empty squares, and the 4 X 4 square in the lower left filled in as above.
For n=6, a tiling of the excess after removing a 4 X 4 square shows us how optimal solutions can be generated for any even number that is not a multiple of 4:
   //ABBB
   /AAABC
       CC
       DC
       DD
       D/
The pairs A&B and C&D can be extended in the manner of a frieze. A nice solution for 9 X 9 does not include tilings of smaller even squares:
   ABBBCDDDE
   AABFCCDEE
   AGFFCHHHE
   GGGFI/HJ/
   KKKIIIJJJ
   /KL//MNNN
   OLLLPMMNQ
   OORPPMSQQ
   ORRRPSSSQ

Colin Barker, Table of n, a(n) for n = 1..1000
Jack W Grahl, Every square can be tiled with T-tetrominos and no more than 5 monominos, arXiv:1807.09201 [math.CO], 2018.
Robert Hochberg, The gap number of the T-tetromino arxiv:1403.6730, [math.CO], June 2014.
Shuxin Zhan, Tiling a deficient rectangle with t-tetrominoes, Penn State Mathematics Advanced Study Semesters REU, August 2012.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Delete[Flatten[ Table[{4n^2, 4n^2 + 2n - 1, 4n^2 + 4n, 4n^2 + 6n + 1}, {n, 0, 14}]], 2] (* or *)
CoefficientList[ Series[1 + (x^4 - 2x^3 - 2x + 1)/((x - 1)^3 (x^3 + x^2 + x + 1)), {x, 0, 58}], x] (* Robert G. Wilson v, Jul 25 2018 *)
LinearRecurrence[{2,-1,0,1,-2,1},{0,0,1,4,5,8,11},60] (* Harvey P. Dale, Aug 11 2024 *)

concat([0,0], Vec(x^3*(1 + 2*x - 2*x^2 + 2*x^3 - x^4) / ((1 - x)^3*(1 + x)*(1 + x^2)) + O(x^60))) \\ Colin Barker, May 24 2019

From Jack W Grahl, Jul 25 2018: (Start)
a(4m) = 4m^2;
a(4m+1) = 4m^2 + 2m - 1;
a(4m+2) = 4m^2 + 4m;
a(4m+3) = 4m^2 + 6m + 1.
(End)
From Colin Barker, May 24 2019: (Start)
G.f.: x^3*(1 + 2*x - 2*x^2 + 2*x^3 - x^4) / ((1 - x)^3*(1 + x)*(1 + x^2)).
a(n) = (-7 + 3*(-1)^n + 2*(-i)^n + 2*i^n + 2*n^2) / 8 for n>1, where i=sqrt(-1).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>7.
(End)

A236972 The number of partitions of n into at least 5 parts from which we can form every partition of n into 5 parts by summing elements.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 2, 3, 3, 4, 7, 8, 9, 13, 14, 24, 29, 35, 38

Offset: 1

Jack W Grahl, Feb 02 2014

The corresponding partitions with 2 in the definition instead of 5 are the complete partitions, which A126796 counts.
The qualifier 'into at least 5 parts' is only relevant for n = 1, 2, 3 or 4. It is included because otherwise the condition would be vacuously true for all partitions of 1, 2, 3 and 4. It seems neater to consider that there are no partitions of 1, 2, 3 or 4 of this form.
What is the limit for large n of the proportion of partitions of n for which this holds, or this sequence divided by A000041?

			The valid partitions of 11 are all those which contain only 1's, 2's and 3's, with no more than one 3 and no more than three 2's or 3's. This is because every partition of 11 into 5 parts contains at least one element 3 or more, and at least 3 elements 2 or more. There are 7 such partitions, therefore a(11) = 7.

A000041 counts partitions, A126796 counts complete partitions - the case for partitions into 2 instead of 5, A236970 and A236971 are the cases for 3 and 4 respectively.

A236971 Number of partitions of n into at least 4 parts from which we can form every partition of n into 4 parts by summing elements.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 3, 3, 6, 7, 8, 11, 19, 21, 26, 31, 52, 66, 76, 88, 134, 169, 215, 251, 358, 412, 517, 639, 899, 1065, 1242, 1496, 2072, 2482, 2930, 3449, 4677, 5566

Offset: 1

Jack W Grahl, Feb 02 2014

The corresponding partitions with 2 in the definition instead of 3 are the complete partitions, which A126796 counts.
The qualifier 'into at least 4 parts' is only relevant for n = 1, 2 or 3. It is included because otherwise the condition would be vacuously true for all partitions of 1, 2 and 3. It seems neater to consider that there are no partitions of 1, 2 or 3 of this form.
What is the limit for large n of the proportion of partitions of n for which this holds, or this sequence divided by A000041?

			The valid partitions of 7 are (2, 2, 1, 1, 1), (2, 1, 1, 1, 1, 1) and (1, 1, 1, 1, 1, 1, 1). Given any partition of 7 into 4 parts, we can express these four parts as disjoint sums of elements from these partitions. For the third one this is trivial, for the second one because one element of the partition must be at least 2, for the third because in fact two elements of the partition must be at least 2. So a(7) = 3.

A000041 counts partitions, A126796 counts complete partitions - the case for partitions into 2 instead of 4, A236970 and A236972 are the cases for 3 and 5 respectively.

a(30)-a(38) from Willy Van den Driessche, Oct 22 2019

A236970 The number of partitions of n into at least 3 parts from which we can form every partition of n into 3 parts by summing elements.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 5, 6, 7, 13, 16, 19, 29, 38, 49, 72, 84, 108, 155, 195, 234, 331, 410, 501, 672, 824, 1006, 1341, 1621, 1981, 2583, 3111, 3740, 4846, 5819, 6957, 8787, 10582, 12606, 15840, 18762, 22386, 27851, 32934, 38824, 47961, 56633, 66577, 81168, 95612

Offset: 1

Jack W Grahl, Feb 02 2014

nonn

The corresponding partitions with 2 in the definition instead of 3 are the complete partitions, which are counted by A126796.

The qualifier 'into at least 3 parts' is only relevant for n = 1 or 2. It is included because otherwise the condition would be vacuously true for all partitions of 1 and 2. It seems neater to consider that there are no partitions of 1 and 2 of this form.

			The valid partitions of 5 are (2,1,1,1) and (1,1,1,1,1). Given any partition of 5 into 3 parts, it contains one part of at least 2. Therefore we can make any partition of 5 into 3 parts by joining (2,1,1,1) into three sums. (3, 1, 1) is not a valid partition, since (2,2,1) is a partition of 5 into 3 parts which cannot be made by summing elements from (3,1,1). Therefore a(5) = 2.

Jack W Grahl, Haskell code for generating this sequence

Cf. A000041. A126796 is the case for 2 instead of 3, A236971 and A236972 are the cases for 4 and 5.

ric[p_, {x_,y_}] := If[x==0, If[y > Total[p], False, y==0 || AnyTrue[ Reverse@ Union[p], y>=# && ric[ DeleteCases[p, #, 1, 1], {0, y-#}] &]], If[x >= Total[p], False, AnyTrue[ Reverse@ Union@ p, x>=# && ric[ DeleteCases[p, #, 1, 1], {x-#, y}] &]]]; chk[p_] := AllTrue[ Rest /@ IntegerPartitions[Plus @@ p, {3}], ric[p,#] &]; a[n_] := Length@ Select[ IntegerPartitions[n, {3, Infinity}], chk]; Array[a, 24] (* Giovanni Resta, Jul 18 2018 *)

A229037 The "forest fire": sequence of positive integers where each is chosen to be as small as possible subject to the condition that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form an arithmetic progression.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 4, 4, 1, 1, 2, 1, 1, 2, 2, 4, 4, 2, 4, 4, 5, 5, 8, 5, 5, 9, 1, 1, 2, 1, 1, 2, 2, 4, 4, 1, 1, 2, 1, 1, 2, 2, 4, 4, 2, 4, 4, 5, 5, 8, 5, 5, 9, 9, 4, 4, 5, 5, 10, 5, 5, 10, 2, 10, 13, 11, 10, 8, 11, 13, 10, 12, 10, 10, 12, 10, 11, 14, 20, 13

Offset: 1

Jack W Grahl, Sep 11 2013

Added name "forest fire" to make it easier to locate this sequence. - N. J. A. Sloane, Sep 03 2019
This sequence and A235383 and A235265 were winners in the best new sequence contest held at the OEIS Foundation booth at the 2014 AMS/MAA Joint Mathematics Meetings. - T. D. Noe, Jan 20 2014
See A236246 for indices n such that a(n)=1. - M. F. Hasler, Jan 20 2014
See A241673 for indices n such that a(n)=2^k. - Reinhard Zumkeller, Apr 26 2014
The graph (for up to n = 10000) has an eerie similarity (why?) to the distribution of rising smoke particles subjected to a lateral wind, and where the particles emanate from randomly distributed burning areas in a fire in a forest or field. - Daniel Forgues, Jan 21 2014
The graph (up to n = 100000) appears to have a fractal structure. The dense areas are not random but seem to repeat, approximately doubling in width and height each time. - Daniel Forgues, Jan 21 2014
a(A241752(n)) = n and a(m) != n for m < A241752(n). - Reinhard Zumkeller, Apr 28 2014
The indices n such that a(n) = 1 are given by A236313 (relative spacing) up to 19 terms, and A003278 (directly) up to 20 terms. If just placing ones, the 21st 1 would be n=91. The sequence A003278 fails at n=91 because the numbers filling the gaps create an arithmetic progression with a(27)=9, a(59)=5 and a(91)=1. Additionally, if you look at indices n starting at the first instance of 4 or 5, A003278/A236313 provide possible indices for a(n)=4 or a(n)=5, with some indexes instead filled by numbers < (4,5). A003278/A236313 also fail to predict indices for a(n)=4 or a(n)=5 by the ~20th term. - Daniel Putt, Sep 29 2022

Giovanni Resta, Alois P. Heinz, and Charles R Greathouse IV, Table of n, a(n) for n = 1..100000 (1..1000 from Resta, 1001..10000 from Heinz, and 10001..100000 from Greathouse)
Jean Bickerton, Colored plot of 200000 terms; color is difference from previous term in sequence
Xan Gregg, Enhanced scatterplot of 10000 terms [In this plot, the points have been made translucent to reduce the information lost to overstriking, and the point size varies with n in an attempt to keep the density comparable.]
OEIS, Pin plot of 200 terms and scatterplot of 10000 terms
Sébastien Palcoux, On the first sequence without triple in arithmetic progression (version: 2019-08-21), MathOverflow
Sébastien Palcoux, Table of n, a(n) for n = 1..1000000
Sébastien Palcoux, Density plot of the first 1000000 terms
Reddit User "garnet420", Colored plot of 16 million terms; horizontal divisions are 1000000; vertical divisions are 25000
Reddit User "garnet420", B/W plot of 16 million terms; horizontal divisions are 1000000; vertical divisions are 25000
N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)
N. J. A. Sloane and Brady Haran, Amazing Graphs II (including Star Wars), Numberphile video (2019)
Index entries for sequences with interesting graphs or plots
Index entries for non-averaging sequences

Cf. A094870, A362942 (partial sums).
For a variant see A309890.
A selection of sequences related to "no three-term arithmetic progression": A003002, A003003, A003278, A004793, A005047, A005487, A033157, A065825, A092482, A093678, A093679, A093680, A093681, A093682, A094870, A101884, A101886, A101888, A140577, A185256, A208746, A229037.

import Data.IntMap (empty, (!), insert)
a229037 n = a229037_list !! (n-1)
a229037_list = f 0 empty  where
   f i m = y : f (i + 1) (insert (i + 1) y m) where
     y = head [z | z <- [1..],
                   all (\k -> z + m ! (i - k) /= 2 * m ! (i - k `div` 2))
                       [1, 3 .. i - 1]]
-- Reinhard Zumkeller, Apr 26 2014

a[1] = 1; a[n_] := a[n] = Block[{z = 1}, While[Catch[ Do[If[z == 2*a[n-k] - a[n-2*k], Throw@True], {k, Floor[(n-1)/2]}]; False], z++]; z]; a /@ Range[100] (* Giovanni Resta, Jan 01 2014 *)

step(v)=my(bad=List(),n=#v+1,t); for(d=1,#v\2,t=2*v[n-d]-v[n-2*d]; if(t>0, listput(bad,t))); bad=Set(bad); for(i=1,#bad, if(bad[i]!=i, return(i))); #bad+1
first(n)=my(v=List([1])); while(n--, listput(v, step(v))); Vec(v) \\ Charles R Greathouse IV, Jan 21 2014

A229037_list = []
for n in range(10**6):
    i, j, b = 1, 1, set()
    while n-2*i >= 0:
        b.add(2*A229037_list[n-i]-A229037_list[n-2*i])
        i += 1
        while j in b:
            b.remove(j)
            j += 1
    A229037_list.append(j) # Chai Wah Wu, Dec 21 2014

a(n) <= (n+1)/2. - Charles R Greathouse IV, Jan 21 2014

cp's OEIS Frontend

User: Jack W Grahl

A341463 a(n) = (-1)^(n+1) * (3^n+1)/2.

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A320505 Index of the first occurrence of n consecutive 0's in A164349.

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A318895 Number of isoclinism classes of the groups of order 2^n.

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A316532 Leading least prime signatures, ordered by the underlying partitions, as in A063008.

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Haskell

A267795 Integers n such that n, 2n, 3n ... 10n contain almost equally many copies of each base 10 digit.

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A256535 The largest number of T-tetrominoes that fit within an n X n square.

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A236972 The number of partitions of n into at least 5 parts from which we can form every partition of n into 5 parts by summing elements.

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A236971 Number of partitions of n into at least 4 parts from which we can form every partition of n into 4 parts by summing elements.

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A236970 The number of partitions of n into at least 3 parts from which we can form every partition of n into 3 parts by summing elements.

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