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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A221463 T(n,k)=Number of 0..k arrays of length n with each element unequal to at least one neighbor, starting with 0.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 0, 3, 4, 2, 0, 4, 9, 12, 3, 0, 5, 16, 36, 32, 5, 0, 6, 25, 80, 135, 88, 8, 0, 7, 36, 150, 384, 513, 240, 13, 0, 8, 49, 252, 875, 1856, 1944, 656, 21, 0, 9, 64, 392, 1728, 5125, 8960, 7371, 1792, 34, 0, 10, 81, 576, 3087, 11880, 30000, 43264, 27945, 4896, 55, 0, 11
Offset: 1

Views

Author

R. H. Hardin, general recursion proved by Robert Israel in the Sequence Fans Mailing List, Jan 17 2013

Keywords

Comments

Table starts
..0.....0.......0........0.........0..........0..........0...........0
..1.....2.......3........4.........5..........6..........7...........8
..1.....4.......9.......16........25.........36.........49..........64
..2....12......36.......80.......150........252........392.........576
..3....32.....135......384.......875.......1728.......3087........5120
..5....88.....513.....1856......5125......11880......24353.......45568
..8...240....1944.....8960.....30000......81648.....192080......405504
.13...656....7371....43264....175625.....561168....1515031.....3608576
.21..1792...27945...208896...1028125....3856896...11949777....32112640
.34..4896..105948..1008640...6018750...26508384...94253656...285769728
.55.13376..401679..4870144..35234375..182191680..743424031..2543058944
.89.36544.1522881.23515136.206265625.1252200384.5863743809.22630629376

Examples

			Some solutions for n=6 k=4
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..4....2....1....3....3....2....2....1....4....2....3....1....3....2....3....3
..1....4....0....1....2....4....3....0....2....0....3....3....2....0....0....4
..0....0....3....4....2....3....2....1....0....3....2....4....3....3....4....1
..1....2....1....2....1....0....3....4....0....1....3....1....3....0....2....0
..0....3....2....3....3....1....4....0....4....2....0....0....0....4....0....3

Links

Crossrefs

Column 1 is A000045(n-1)
Column 2 is A028860(n+1)
Column 3 is A106435(n-1)
Column 4 is A094013
Column 5 is A106565(n-1)
Row 2 is A000027
Row 3 is A000290
Row 4 is A011379

Formula

Recursion for column k:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = 2*a(n-1) +2*a(n-2)
k=3: a(n) = 3*a(n-1) +3*a(n-2)
k=4: a(n) = 4*a(n-1) +4*a(n-2)
k=5: a(n) = 5*a(n-1) +5*a(n-2)
k=6: a(n) = 6*a(n-1) +6*a(n-2)
k=7: a(n) = 7*a(n-1) +7*a(n-2)
Empirical for row n:
n=2: a(k) = 1*k
n=3: a(k) = 1*k^2
n=4: a(k) = 1*k^3 + 1*k^2
n=5: a(k) = 1*k^4 + 2*k^3
n=6: a(k) = 1*k^5 + 3*k^4 + 1*k^3
n=7: a(k) = 1*k^6 + 4*k^5 + 3*k^4
n=8: a(k) = 1*k^7 + 5*k^6 + 6*k^5 + 1*k^4
n=9: a(k) = 1*k^8 + 6*k^7 + 10*k^6 + 4*k^5
n=10: a(k) = 1*k^9 + 7*k^8 + 15*k^7 + 10*k^6 + 1*k^5
n=11: a(k) = 1*k^10 + 8*k^9 + 21*k^8 + 20*k^7 + 5*k^6
n=12: a(k) = 1*k^11 + 9*k^10 + 28*k^9 + 35*k^8 + 15*k^7 + 1*k^6
n=13: a(k) = 1*k^12 + 10*k^11 + 36*k^10 + 56*k^9 + 35*k^8 + 6*k^7
n=14: a(k) = 1*k^13 + 11*k^12 + 45*k^11 + 84*k^10 + 70*k^9 + 21*k^8 + 1*k^7
n=15: a(k) = 1*k^14 + 12*k^13 + 55*k^12 + 120*k^11 + 126*k^10 + 56*k^9 + 7*k^8
Apparently then T(n,k) = sum { binomial(n-2-i,i)*k^(n-1-i) , 0<=2*i<=n-2 }.
The formula reduces to T(n,k) = [4*k^(n-1)*(1+G)^(2*n-2) +4^n] /[2^(n+1) *G *(1+G)^(n-1)] for even n and to T(n,k) = [4*k^(n-1) *(1+G)^(2*n-2) -4^n] /[2^(n+1) *G *(1+G)^(n-1)] for odd n, where G=sqrt(1+4/k). - R. J. Mathar, Jan 21 2013

A106568 Expansion of 4*x/(1 - 4*x - 4*x^2).

Original entry on oeis.org

0, 4, 16, 80, 384, 1856, 8960, 43264, 208896, 1008640, 4870144, 23515136, 113541120, 548225024, 2647064576, 12781158400, 61712891904, 297976201216, 1438756372480, 6946930294784, 33542746669056, 161958707855360, 782005818097664, 3775858103812096, 18231455687639040
Offset: 0

Views

Author

Roger L. Bagula, May 30 2005

Keywords

Comments

This sequence is part of a class of sequences with the properties: a(n) = m*(a(n-1) + a(n-2)) with a(0) = 0 and a(1) = m, g.f.: m*x/(1 - m*x - m*x^2), and have the Binet form m*(alpha^n - beta^n)/(alpha - beta) where 2*alpha = m + sqrt(m^2 + 4*m) and 2*beta = p - sqrt(m^2 + 4*m). - G. C. Greubel, Sep 06 2021

Links

Crossrefs

Cf. A000129, A057087, A009013.
Sequences of the form a(n) = m*(a(n-1) + a(n-2)): A000045 (m=1), A028860 (m=2), A106435 (m=3), A094013 (m=4), A106565 (m=5), A221461 (m=6), A221462 (m=7).

Programs

  • Magma
    [n le 2 select 4*(n-1) else 4*(Self(n-1) +Self(n-2)): n in [1..41]]; // G. C. Greubel, Sep 06 2021
  • Maple
    A106568 := n -> ifelse(n=0, 0, 4^(n)*hypergeom([(1-n)/2, 1-n/2], [1-n], -1)):
seq(simplify(A106568(n)), n = 0..24);  # Peter Luschny, Mar 30 2025
  • Mathematica
    LinearRecurrence[{4,4}, {0,4}, 40] (* G. C. Greubel, Sep 06 2021 *)
  • Sage
    [2^(n+1)*lucas_number1(n,2,-1) for n in (0..40)] # G. C. Greubel, Sep 06 2021

Formula

a(n) = 4 * A057087(n).
a(n) = A094013(n+1). - R. J. Mathar, Aug 24 2008
From Philippe Deléham, Sep 19 2009: (Start)
a(n) = 4*a(n-1) + 4*a(n-2) for n > 2; a(0) = 0, a(1)=4.
G.f.: 4*x/(1 - 4*x - 4*x^2). (End)
G.f.: Q(0) - 1, where Q(k) = 1 + 2*(1+2*x)*x + 2*(2*k+3)*x - 2*x*(2*k+1 +2*x+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 04 2013
a(n) = 2^(n+1)*A000129(n). - G. C. Greubel, Sep 06 2021
a(n) = 4^n*hypergeom([(1-n)/2, 1-n/2], [1-n], -1) for n > 0. - Peter Luschny, Mar 30 2025

Extensions

Edited by N. J. A. Sloane, Apr 30 2006
Simpler name using o.g.f. by Joerg Arndt, Oct 05 2013
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