A006498 -id:A006498 - cp's OEIS frontend

A215099 a(0)=0, a(1)=1, a(n) = least k>a(n-1) such that k+a(n-2) is prime.

0, 1, 2, 4, 5, 7, 8, 10, 11, 13, 18, 24, 25, 29, 34, 38, 39, 41, 44, 48, 53, 55, 56, 58, 71, 73, 78, 84, 85, 89, 94, 102, 103, 109, 120, 124, 131, 133, 138, 144, 145, 149, 162, 164, 169, 173, 178, 180, 181, 187, 192, 196, 197, 201

Offset: 0

Alex Ratushnyak, Aug 03 2012

nonn

For n>0 and (n mod 4)<2, a(n) is odd.
Same definition, but k+a(n-2) is a
Fibonacci number: A006498 except first two terms,
Lucas number: A000045 except first two terms,
Pell number: A089928(n-1),
Jacobsthal number: A215095,
factorial: A215096,
square: A194274,
cube: A215097,
triangular number: A011848(n+2),
oblong number: A215098.
Example of a related sequence definition: a(0)=0, a(1)=1, a(n) = least k>a(n-1) such that k+a(n-2) is a cube.

Iain Fox, Table of n, a(n) for n = 0..10000

Cf. A062042: a(1) = 2, a(n) = least k>a(n-1) such that k+a(n-1) is a prime.

Cf. A073627, A073628.

first(n) = my(res = vector(n, i, i-1), k); for(x=3, n, k=res[x-1]+1; while(!isprime(k+res[x-2]), k++); res[x]=k); res \\ Iain Fox, Apr 22 2019 (corrected by Iain Fox, Apr 25 2019)

from sympy import prime
prpr = 0
prev = 1
for n in range(77):
    print(prpr, end=', ')
    b = c = 0
    while c<=prev:
        c = prime(b+1) - prpr
        b+=1
    prpr = prev
    prev = c

A345067 Consider the "Quilt Tiling"; T(n, k) is the area of the tile containing the unit square whose upper right corner has coordinates (n, k); square array T(n, k) read by antidiagonals upwards, n, k > 0.

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 6, 4, 4, 6, 6, 6, 4, 6, 6, 6, 6, 1, 1, 6, 6, 15, 6, 2, 9, 2, 6, 15, 15, 15, 2, 9, 9, 2, 15, 15, 15, 15, 15, 9, 9, 9, 15, 15, 15, 15, 15, 15, 2, 9, 9, 2, 15, 15, 15, 15, 15, 15, 2, 4, 9, 4, 2, 15, 15, 15, 40, 15, 15, 6, 4, 4, 4, 4, 6, 15, 15, 40

Offset: 1

Rémy Sigrist, Jun 06 2021

The "Quilt Tiling" is described in Shectman's paper (see Links section).

All terms belong to A006498.

			Array T(n, k) begins:
  n\k|  1   2   3   4   5   6   7   8   9  10  11
  ---+---+-------+-----------+-------------------+
   1 |  1|  2   2|  6   6   6| 15  15  15  15  15|
     +-----------+           |                   |
   2 |  2|  4   4|  6   6   6| 15  15  15  15  15|
     |   |       +---+-------+                   |
   3 |  2|  4   4|  1|  2   2| 15  15  15  15  15|
     +---+---+---+---+-------+-------+-----------+
   4 |  6   6|  1|  9   9   9|  2   2|  6   6   6|
     |       +---+           +-------+           |
   5 |  6   6|  2|  9   9   9|  4   4|  6   6   6|
     |       |   |           |       +---+-------+
   6 |  6   6|  2|  9   9   9|  4   4|  1|  2   2|
     +-------+---+---+-------+-------+---+-------+
   7 | 15  15  15|  2|  4   4| 25  25  25  25  25|
     |           |   |       |                   |
   8 | 15  15  15|  2|  4   4| 25  25  25  25  25|
     |           +---+---+---+                   |
   9 | 15  15  15|  6   6|  1| 25  25  25  25  25|
     |           |       +---+                   |
  10 | 15  15  15|  6   6|  2| 25  25  25  25  25|
     |           |       |   |                   |
  11 | 15  15  15|  6   6|  2| 25  25  25  25  25|
     +-----------+-------+---+-------------------+

Rémy Sigrist, Table of n, a(n) for n = 1..10153
J. Parker Shectman, A Quilt after Fibonacci, Part 2 of 3: Cohorts, Free Monoids, and Numeration
Rémy Sigrist, Illustration of the connection between the "Quilt Tiling" and the sequences A000201 and A005206
Rémy Sigrist, PARI program for A345067

Cf. A000045, A001654, A005206, A006498, A095791, A005206.

PARI
```
See Links section.
```

T(n, k) = T(k, n).
T(n, n) = A130312(n+1)^2.
T(n, 1) = A001654(A095791(n)+1).
T(n, k) is the square of a Fibonacci number for n = 1+A005206(k+1)..A000201(k).

A349840 The number of compositions of n using elements from the set {1,3,5,7,8}.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 22, 35, 56, 91, 147, 238, 385, 623, 1009, 1632, 2640, 4272, 6912, 11184, 18096, 29280, 47377, 76657, 124033, 200690, 324723, 525413, 850136, 1375549, 2225686, 3601235, 5826920, 9428155

Offset: 0

Michael A. Allen, Dec 05 2021

Number of ways to tile an n-board (an n X 1 array of 1 X 1 cells) using squares, trominoes, pentominoes, heptominoes, and octominoes.
Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-1,0,2,4,6,7} for all i=1,...,n.
a(n) gives the sums of the antidiagonals of A349839.

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), North-Holland, Amsterdam, 1970, pp. 755-770.

Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics, 4(1) (2010), 119-135.
K. Edwards and Michael A. Allen, Strongly restricted permutations and tiling with fences, Discrete Applied Mathematics, 187 (2015), 82-90.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,1,0,1,1).

Sums of antidiagonals of triangles in the same family as A349839: A000045, A006498, A079962, A349843.

CoefficientList[Series[1/(1-x-x^3-x^5-x^7-x^8),{x,0,35}],x]

a(n) = a(n-1) + a(n-3) + a(n-5) + a(n-7) + a(n-8) + delta(n,0), a(n<0)=0 (where delta(i,j) is the Kronecker delta).
a(n) = a(n-1) + a(n-2) + a(n-8) - a(n-9) - a(n-10) + delta(n,0) - delta(n,2), a(n<0)=0.
G.f.: 1/(1-x-x^3-x^5-x^7-x^8).

A349843 Expansion of (1 - x^2)/((1 - x^10)*(1 - x - x^2)).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 56, 90, 145, 235, 380, 615, 995, 1610, 2605, 4215, 6821, 11036, 17856, 28892, 46748, 75640, 122388, 198028, 320416, 518444, 838861, 1357305, 2196165, 3553470, 5749635, 9303105

Offset: 0

Michael A. Allen, Dec 13 2021

The number of compositions of n using elements from the set {1,3,5,7,9,10}.
Number of ways to tile an n-board (an n X 1 array of 1 X 1 cells) using squares, trominoes, pentominoes, heptominoes, nonominoes, and decominoes.
Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-1,0,2,4,6,8,9} for all i=1,...,n.
a(n) gives the sums of the antidiagonals of A349841.

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), North-Holland, Amsterdam, 1970, pp. 755-770.

Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics, 4(1) (2010), 119-135.
K. Edwards and M. A. Allen, Strongly restricted permutations and tiling with fences, Discrete Applied Mathematics, 187 (2015), 82-90.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,1,0,1,0,1,1).

Sums of antidiagonals of triangles in the same family as A349841: A000045, A006498, A079962, A349840.

CoefficientList[Series[1/(1-x-x^3-x^5-x^7-x^9-x^10), {x, 0, 35}], x]

a(n) = a(n-1) + a(n-3) + a(n-5) + a(n-7) + a(n-9) + a(n-10) + delta(n,0), a(n<0)=0.
a(n) = a(n-1) + a(n-2) + a(n-10) - a(n-11) - a(n-12) + delta(n,0) - delta(n,2), a(n<0)=0.
G.f.: 1/(1-x-x^3-x^5-x^7-x^9-x^10).

A387020 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,0,5} for all i=1,...,n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 16, 20, 25, 34, 46, 67, 94, 130, 175, 231, 305, 400, 540, 729, 999, 1363, 1855, 2510, 3370, 4531, 6070, 8180, 11026, 14921, 20197, 27322, 36940, 49820, 67204, 90528, 122091, 164686, 222344, 300316, 405574, 547768, 739291, 997794, 1346130

Offset: 0

Michael A. Allen, Aug 13 2025

			a(7) = 2: 1234567, 6712345.
a(8) = 3: 12345678, 17823456, 67123458.

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), North-Holland, Amsterdam, 1970, pp. 755-770.

Vladimir Baltić, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics, 4(1) (2010), 119-135.
Kenneth Edwards and Michael A. Allen, Strongly restricted permutations and tiling with fences, Discrete Applied Mathematics, 187 (2015), 82-90.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,3,-2,2,-1,1,0,0,-3,1,-2,0,0,0,0,1).

Sequences for numbers of permutations such that p(i)-i is in {-2,0,d} for d=1,..,8: A000930, A006498, A080000, A224809, A387020, A224808, A387021, A224811.

CoefficientList[Series[(1 - 2*x^7 - x^9 + x^14)/(1 - x - 3*x^7 + 2*x^8 - 2*x^9 + x^10 - x^11 + 3*x^14 - x^15 + 2*x^16 - x^21),{x,0,51}],x]
LinearRecurrence[{1, 0, 0, 0, 0, 0, 3, -2, 2, -1, 1, 0, 0, -3, 1, -2, 0, 0, 0, 0, 1}, {1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 16, 20, 25, 34, 46, 67, 94, 130}, 52]

a(n) = a(n-1) + 3*a(n-7) - 2*a(n-8) + 2*a(n-9) - a(n-10) + a(n-11) - 3*a(n-14) + a(n-15) - 2*a(n-16) + a(n-21) for n >= 21.

G.f.: (1 - 2*x^7 - x^9 + x^14)/((1 - x)*(1 - x + x^2 - 2*x^3 + x^4 - x^5 - x^7 + x^10)*(1 + x + x^3 + 2*x^4 + x^5 + 2*x^6 + 2*x^7 + x^8 + x^9 + x^10)).

A387021 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,0,7} for all i=1,...,n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 17, 21, 25, 30, 36, 46, 59, 81, 109, 153, 207, 277, 361, 463, 589, 743, 949, 1211, 1589, 2083, 2773, 3670, 4861, 6388, 8344, 10848, 14019, 18166, 23479, 30556, 39762, 52049, 68125, 89345, 117034, 153078, 199979, 260572, 339546, 441669, 575341

Offset: 0

Michael A. Allen, Aug 13 2025

			a(9)=2: 123456789, 891234567.

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), North-Holland, Amsterdam, 1970, pp. 755-770.

V. Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics, 4(1) (2010), 119-135.
Kenneth Edwards and Michael A. Allen, Strongly restricted permutations and tiling with fences, Discrete Applied Mathematics, 187 (2015), 82-90.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,4,-3,3,-2,2,-1,1,0,0,-6,3,-6,2,-3,0,0,0,0,4,-1,3,0,0,0,0,0,0,-1).

Sequences for numbers of permutations such that p(i)-i is in {-2,0,d} for d=1,...,8: A000930, A006498, A080000, A224809, A387020, A224808, A387021, A224811.

CoefficientList[Series[(1 - 3*x^9 - 2*x^11 - x^13 + 3*x^18 + 2*x^20 - x^27)/ (1 - x - 4*x^9 + 3*x^10 - 3*x^11 + 2*x^12 - 2*x^13 + x^14 - x^15 + 6*x^18 - 3*x^19 + 6*x^20 - 2*x^21 + 3*x^22 - 4*x^27 + x^28 - 3*x^29 + x^36),{x,0,55}],x]
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 4, -3, 3, -2, 2, -1, 1, 0, 0, -6, 3, -6, 2, -3, 0, 0, 0, 0, 4, -1, 3, 0, 0, 0, 0, 0, 0, -1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 17, 21, 25, 30, 36, 46, 59, 81, 109, 153, 207, 277, 361, 463, 589, 743, 949, 1211, 1589, 2083, 2773}, 56]

a(n) = a(n-1) + 4*a(n-9) - 3*a(n-10) + 3*a(n-11) - 2*a(n-12) + 2*a(n-13) - a(n-14) + a(n-15) - 6*a(n-18) + 3*a(n-19) - 6*a(n-20) + 2*a(n-21) - 3*a(n-22) + 4*a(n-27) - a(n-28) + 3*a(n-29) - a(n-36) for n >= 36.

G.f.: (1 - 3*x^9 - 2*x^11 - x^13 + 3*x^18 + 2*x^20 - x^27)/ (1 - x - 4*x^9 + 3*x^10 - 3*x^11 + 2*x^12 - 2*x^13 + x^14 - x^15 + 6*x^18 - 3*x^19 + 6*x^20 - 2*x^21 + 3*x^22 - 4*x^27 + x^28 - 3*x^29 + x^36).

A089931 a(n) = 3a(n-1) + 3a(n-3) + a(n-4).

Original entry on oeis.org

1, 3, 9, 30, 100, 330, 1089, 3597, 11881, 39240, 129600, 428040, 1413721, 4669203, 15421329, 50933190, 168220900, 555595890, 1835008569, 6060621597, 20016873361, 66111241680, 218350598400, 721163036880, 2381839709041

Offset: 0

Paul Barry, Nov 15 2003

Michael De Vlieger, Table of n, a(n) for n = 0..1927
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (3,0,3,1).

Cf. A006498.

LinearRecurrence[{3,0,3,1},{1,3,9,30},30] (* Harvey P. Dale, Jun 16 2015 *)

a(n) = ((3 + sqrt(13)^n(11 + 3*sqrt(13))/13 + (3 - sqrt(13)^n(11 - 3*sqrt(13))/13)*2^(-1 - n) + 2(-1)^n/13;
a(n) = (-i)^n*Sum_{k=0..floor(n/2)} U(n-2k, 3i/2) where i = sqrt(-1).
G.f.: -1 / ( (1+x^2)*(x^2+3*x-1) ). - R. J. Mathar, Feb 14 2015

A106408 Triangle, read by rows, where T(1,1) = 1; T(2,1) = T(2,2) = 2; for n > 2, T(n,n) = T(n-1,n-1) + T(n-2,n-2); T(n+1,n) = 2 * T(n,n); for all other entries, T(n,k) = T(n-1,k) + T(n-2,k).

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 5, 6, 6, 5, 8, 10, 9, 10, 8, 13, 16, 15, 15, 16, 13, 21, 26, 24, 25, 24, 26, 21, 34, 42, 39, 40, 40, 39, 42, 34, 55, 68, 63, 65, 64, 65, 63, 68, 55, 89, 110, 102, 105, 104, 104, 105, 102, 110, 89, 144, 178, 165, 170, 168, 169, 168, 170, 165, 178, 144

Offset: 1

Gerald McGarvey, May 28 2005

Row sums are A004798 (convolution of Fibonacci numbers 1,2,3,5,... with themselves). Central numbers of the rows are A006498 (a(n) = a(n-1)+a(n-3)+a(n-4)). First column and main diagonal are Fibonacci numbers 1,2,3,5,... First subdiagonal are 2*Fibonacci numbers. T(n,k) = F(n-k+2)*F(k+1) where F(m) is the m-th Fibonacci number. For the antidiagonal sums b(n): b(1) = 1, b(2) = 2, then b(n) = b(n-1) + b(n-2) + F(floor((n+3)/2)).

T(n,k) is the number of Boolean intervals of the form [s_k,w] in the weak order on S_n, for a fixed simple reflection s_k. - Bridget Tenner, Jan 16 2020

			Triangle begins
   1;
   2,  2;
   3,  4,  3;
   5,  6,  6,  5;
   8, 10,  9, 10,  8;

B. E. Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.

Cf. A000045, A004798, A006498.

G.f.: (1+x+y+x*y)/((1-x-x^2)*(1-y-y^2)) [U coordinates] - N. J. A. Sloane, Jun 01 2005

A189101 Expansion of g.f. 1/(1-(x+x^2+x^3+x^4+x^6+x^7)).

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 30, 59, 115, 225, 441, 863, 1689, 3307, 6474, 12673, 24809, 48567, 95075, 186120, 364352, 713261, 1396290, 2733399, 5350944, 10475089, 20506194, 40143239, 78585017, 153839228, 301158021, 589551538, 1154115087, 2259313307, 4422866209

Offset: 0

N. J. A. Sloane, Apr 19 2011

Compositions of n into parts !=5 and <=7. - Joerg Arndt, Jun 06 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300
D. Applegate, M. LeBrun, and N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,0,1,1).

This sequence is the next in the series after A000931, A006498, A079976, A079968.

makelist(coeff(taylor(1/(1-(x+x^2+x^3+x^4+x^6+x^7)), x, 0, n), x, n), n, 0, 34);  /* Bruno Berselli, Jun 05 2011 */

Vec(1/(1-(x+x^2+x^3+x^4+x^6+x^7))+O(x^99)) \\ Charles R Greathouse IV, Feb 26 2014

A215095 a(0)=0, a(1)=1, a(n) = least k>a(n-1) such that k+a(n-2) is a Jacobsthal number.

Original entry on oeis.org

0, 1, 3, 4, 8, 17, 35, 68, 136, 273, 547, 1092, 2184, 4369, 8739, 17476, 34952, 69905, 139811, 279620, 559240, 1118481, 2236963, 4473924, 8947848, 17895697, 35791395, 71582788, 143165576, 286331153, 572662307, 1145324612, 2290649224, 4581298449, 9162596899

Offset: 0

Alex Ratushnyak, Aug 03 2012

nonn

Same definition, but k+a(n-2) is a
Fibonacci number: A006498 except first two terms,
Lucas number: A000045 except first two terms,
Pell number: A089928(n-1),
factorial: A215096,
square: A194274,
cube: A215097,
triangular number: A011848(n+2),
oblong number: A215098,
prime number: A215099.
Example of a related sequence definition: a(0)=0, a(1)=1, a(n) = least k>a(n-1) such that k+a(n-2) is a cube.

Cf. A001045, A006498, A089928, A000045.

prpr = 0
prev = 1
jac = [0]*10000
for n in range(10000):
    jac[n] = prpr
    curr = prpr*2 + prev
    prpr = prev
    prev = curr
prpr, prev = 0, 1
for n in range(1, 44):
    print(prpr, end=', ')
    b = c = 0
    while c<=prev:
        c = jac[b] - prpr
        b+=1
    prpr = prev
    prev = c

Conjecture: G.f. (x+2*x^2)/(1-x-x^2-x^3-2*x^4). - David Scambler, Aug 06 2012

cp's OEIS Frontend

A215099 a(0)=0, a(1)=1, a(n) = least k>a(n-1) such that k+a(n-2) is prime.

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A345067 Consider the "Quilt Tiling"; T(n, k) is the area of the tile containing the unit square whose upper right corner has coordinates (n, k); square array T(n, k) read by antidiagonals upwards, n, k > 0.

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A349840 The number of compositions of n using elements from the set {1,3,5,7,8}.

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A349843 Expansion of (1 - x^2)/((1 - x^10)*(1 - x - x^2)).

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A387020 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,0,5} for all i=1,...,n.

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A387021 Number of permutations (p(1),p(2),...,p(n)) of (1,2,...,n) such that p(i)-i is in {-2,0,7} for all i=1,...,n.

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A089931 a(n) = 3*a(n-1) + 3*a(n-3) + a(n-4).

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A106408 Triangle, read by rows, where T(1,1) = 1; T(2,1) = T(2,2) = 2; for n > 2, T(n,n) = T(n-1,n-1) + T(n-2,n-2); T(n+1,n) = 2 * T(n,n); for all other entries, T(n,k) = T(n-1,k) + T(n-2,k).

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A089931 a(n) = 3a(n-1) + 3a(n-3) + a(n-4).