A005191 -id:A005191 - cp's OEIS frontend

A124806 Number of circular n-letter words over the alphabet {0,1,2,3,4} with adjacent letters differing by at most 2.

1, 5, 19, 65, 247, 955, 3733, 14649, 57583, 226505, 891219, 3507047, 13801285, 54313277, 213745019, 841177105, 3310392415, 13027820227, 51270096661, 201769982673, 794052091767, 3124938240153, 12297982928987, 48397879544975

Offset: 0

R. H. Hardin, Dec 28 2006

Empirical: a(base, n) = a(base-1, n) + A005191(n+1) for base >= 2*floor(n/2) + 1 where base is the number of letters in the alphabet.

G. C. Greubel, Table of n, a(n) for n = 0..1000
OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less
Index entries for linear recurrences with constant coefficients, signature (5,-3,-5,1,1).

Cf. A000032, A005191, A099503, A124805, A124807.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-3*x^2-10*x^3+3*x^4+4*x^5)/((1-x-x^2)*(1-4*x+x^3)) )); // G. C. Greubel, Aug 03 2023

LinearRecurrence[{5,-3,-5,1,1}, {1,5,19,65,247,955}, 60] (* G. C. Greubel, Aug 03 2023 *)

@CachedFunction
def a(n): # a = A124806
    if (n<6): return (1,5,19,65,247,955)[n]
    else: return 5*a(n-1)-3*a(n-2)-5*a(n-3)+a(n-4)+a(n-5)
[a(n) for n in range(31)] # G. C. Greubel, Aug 03 2023

From Colin Barker, Jun 04 2017: (Start)
G.f.: (1 - 3*x^2 - 10*x^3 + 3*x^4 + 4*x^5) / ((1 - x - x^2)*(1 - 4*x + x^3)).
a(n) = 5*a(n-1) - 3*a(n-2) - 5*a(n-3) + a(n-4) + a(n-5) for n>5. (End)
a(n) = -4*[n=0] + LucasL(n-1) + 3*A099503(n) - 8*A099503(n-1). - G. C. Greubel, Aug 03 2023

A270918 Largest coefficient of (1+x+...+x^n)^(2*n).

Original entry on oeis.org

1, 2, 19, 580, 38165, 4395456, 786588243, 202384723528, 70886845397481, 32458256583753952, 18832730699014127291, 13507852690353224821652, 11738630472138500287398379, 12155701820213424461220851360, 14790850878997102285050287114419

Offset: 0

Vaclav Kotesovec, Mar 26 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..210

Cf. A077047.

Cf. A002426, A005190, A005191, A018901, A025012, A025013, A025014, A025015.

Table[Max[CoefficientList[Expand[Sum[x^k, {k, 0, n}]^(2n)], x]], {n, 0, 20}]

a(n) = vecmax(Vec((sum(k=0,n,x^k))^(2*n))); \\ Michel Marcus, Apr 01 2016

a(n) ~ exp(2) * sqrt(3/Pi) * n^(2*n - 3/2).

Typo in formula corrected by Vaclav Kotesovec, Dec 10 2021

A104314 Prime coefficient of x^(2*k) in (x^4 + x^3 + x^2 + x + 1)^k for k in A104313.

Original entry on oeis.org

5, 19, 1974442362935339179, 47705925773278538281, 234692178470218983001

Offset: 1

T. D. Noe, Mar 01 2005

nonn

a(6) > A005191(195315), if it exists. See A104313 for more information. - Jinyuan Wang, Jul 26 2021

Cf. A005191 (pentanomial coefficients), A104312 (nontrivial prime quadrinomial coefficients), A104313.

f=1; Do[f=Expand[f*(x^4+x^3+x^2+x+1)]; s=Coefficient[f, x, 2n]; If[PrimeQ[s], Print[{n, s}]], {n, 100}]

A104631 Coefficient of x^(2n+1) in the expansion of (1+x+x^2+x^3+x^4)^n.

Original entry on oeis.org

0, 1, 4, 18, 80, 365, 1686, 7875, 37080, 175725, 837100, 4004770, 19227924, 92599533, 447118140, 2163837030, 10492874384, 50972030189, 248000853348, 1208335275170, 5894873067200, 28791371852145, 140768761906190

Offset: 0

T. D. Noe, Mar 17 2005

In the triangle of pentanomial coefficients, these numbers are in the column next to the central pentanomial coefficients, A005191. Note that for n>0, n divides a(n). This divisibility property is also true for the triangle of trinomial coefficients, A027907, but apparently for no other triangle of m-nomial coefficients. The quotient a(n)/n is in A104632.

			G.f. = x + 4*x^2 + 18*x^3 + 80*x^4 + 365*x^5 + 1686*x^6 + 7875*x^7 + ... - _Michael Somos_, Aug 12 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Andrei G. Pronko, Periodic Motzkin chain: Ground states and symmetries, arXiv:2504.00835 [math-ph], 2025. See p. 16.

Cf. A035343 (triangle of pentanomial coefficients).

Cf. A104632.

P:=PolynomialRing(Integers()); [n eq 0 select 0 else Coefficients((1+x+x^2+x^3+x^4)^n)[2*n+2]: n in [0..22]]; // Bruno Berselli, Nov 17 2011

f=1; Table[f=Expand[f(x^4+x^3+x^2+x+1)]; Coefficient[f, x, 2n+1], {n, 30}]

x='x+O('x^30); concat([0], Vec(sqrt((5*x^2+2*x-1+(x+1)*sqrt(5*x^2-6*x+1))/(2*x*(1-x)*(5*x+4)*(5*x-1))))) \\ G. C. Greubel, Aug 12 2018

G.f.: sqrt((5*x^2+2*x-1+(x+1)*sqrt(5*x^2-6*x+1))/(2*x*(1-x)*(5*x+4)*(5*x-1))). - Mark van Hoeij, Nov 16 2011
From Vaclav Kotesovec, Oct 17 2012: (Start)
Recurrence: 2*(n-1)*(2*n+1)*a(n) = (19*n^2 - 19*n + 2)*a(n-1) + 5*(2*n^2 - 3*n - 1)*a(n-2) - 25*(n-2)*n*a(n-3).
a(n) ~ 5^n/(2*sqrt(Pi*n)). (End)
a(n) = n * A104632(n) for n>=0. - Michael Somos, Aug 12 2018

A124828 Number of base 7 circular n-digit numbers with adjacent digits differing by 2 or less.

Original entry on oeis.org

1, 7, 29, 103, 417, 1717, 7229, 30793, 132225, 570649, 2470769, 10719793, 46569777, 202477633, 880792193, 3832748833, 16681516545, 72613292353, 316105114817, 1376159456641, 5991281182977, 26084303730049

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A005191(n+1) for base>=2.int(n/2)+1.

See A285280 for confirmation of linear recurrence and code to produce sequence; also confirms conjectured g.f. from Colin Barker. - Ray Chandler, Aug 12 2023.

Ray Chandler, Table of n, a(n) for n = 0..99
OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less
Index entries for linear recurrences with constant coefficients, signature (7, -10, -10, 14, 4, -4).

Cf. Row 7 of A285280.

G.f.: (1 - 10*x^2 - 20*x^3 + 42*x^4 + 16*x^5 - 20*x^6) / ((1 - x)*(1 - 2*x - 2*x^2)*(1 - 4*x - 2*x^2 + 2*x^3)). - Colin Barker, Jun 03 2017

A124843 Number of base 8 circular n-digit numbers with adjacent digits differing by 2 or less.

Original entry on oeis.org

1, 8, 34, 122, 502, 2098, 8980, 38928, 170382, 750722, 3323554, 14763438, 65736004, 293186252, 1309156946, 5850527002, 26160514526, 117022825786, 523619082772, 2343388805944, 10488943094022, 46952619517170

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A005191(n+1) for base>=2.int(n/2)+1.

See A285280 for confirmation of linear recurrence and code to produce sequence; also confirms conjectured g.f. from Colin Barker. - Ray Chandler, Aug 12 2023.

Ray Chandler, Table of n, a(n) for n = 0..99
OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less
Index entries for linear recurrences with constant coefficients, signature (8, -15, -10, 29, 4, -12).

Cf. Row 8 of A285280.

G.f.: (1 - 15*x^2 - 20*x^3 + 87*x^4 + 16*x^5 - 60*x^6) / ((1 - 3*x - x^2 + 2*x^3)*(1 - 5*x + x^2 + 6*x^3)). - Colin Barker, Jun 03 2017

A124851 Number of base 9 circular n-digit numbers with adjacent digits differing by 2 or less.

Original entry on oeis.org

1, 9, 39, 141, 587, 2479, 10731, 47063, 208547, 931047, 4180239, 18849103, 85269011, 386687375, 1756855951, 7993210831, 36405316227, 165940691695, 756832203759, 3453347063599, 15762537566627, 71964915505967

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A005191(n+1) for base>=2.int(n/2)+1.

See A285280 for confirmation of linear recurrence and code to produce sequence; also confirms conjectured g.f. from Colin Barker. - Ray Chandler, Aug 12 2023.

Ray Chandler, Table of n, a(n) for n = 0..99
OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less
Index entries for linear recurrences with constant coefficients, signature (9, -21, -7, 50, -4, -30, 4, 4).

Cf. Row 9 of A285280.

G.f.: (1 - 21*x^2 - 14*x^3 + 150*x^4 - 16*x^5 - 150*x^6 + 24*x^7 + 28*x^8) / ((1 + x)*(1 - 4*x + 2*x^2)*(1 - 6*x + 5*x^2 + 8*x^3 - 4*x^4 - 2*x^5)). - Colin Barker, Jun 02 2017

A124852 Number of base 10 circular n-digit numbers with adjacent digits differing by 2 or less.

Original entry on oeis.org

1, 10, 44, 160, 672, 2860, 12482, 55198, 246712, 1111372, 5037174, 22940158, 104870790, 480863770, 2210197754, 10178143810, 46942294232, 216761695840, 1001878336772, 4634206919128, 21448419453382, 99316222901062

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A005191(n+1) for base>=2.int(n/2)+1.

See A285280 for confirmation of linear recurrence and code to produce sequence; also confirms conjectured g.f. from Colin Barker. - Ray Chandler, Aug 12 2023.

Ray Chandler, Table of n, a(n) for n = 0..99
OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less
Index entries for linear recurrences with constant coefficients, signature (11, -39, 39, 37, -65, 4, 16, -1, -1).

Cf. Row 10 of A285280.

G.f.: (1 - x - 27*x^2 + 27*x^3 + 201*x^4 - 313*x^5 + 8*x^6 + 112*x^7 - 7*x^8 - 9*x^9) / ((1 - x)*(1 + x)*(1 - 4*x + x^2 + x^3)*(1 - 7*x + 11*x^2 - x^4)). - Colin Barker, Jun 01 2017

A124857 Number of base 11 circular n-digit numbers with adjacent digits differing by 2 or less.

Original entry on oeis.org

1, 11, 49, 179, 757, 3241, 14233, 63333, 284877, 1291697, 5894119, 27031653, 124481521, 575160311, 2664800299, 12374329729, 57568895517, 268238883291, 1251429223153, 5844466935453, 27318547433927, 127784523940077

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A005191(n+1) for base>=2.int(n/2)+1.

See A285280 for confirmation of linear recurrence and code to produce sequence. - Ray Chandler, Aug 12 2023.

Ray Chandler, Table of n, a(n) for n = 0..99
OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less
Index entries for linear recurrences with constant coefficients, signature (11, -36, 12, 105, -77, -102, 66, 35, -15, -3, 1).

Cf. Row 11 of A285280.

A124858 Number of base 12 circular n-digit numbers with adjacent digits differing by 2 or less.

Original entry on oeis.org

1, 12, 54, 198, 842, 3622, 15984, 71468, 323042, 1472022, 6751064, 31123148, 144092684, 669468708, 3119587196, 14572658668, 68216250402, 319893194558, 1502357897232, 7064711394284, 33257109397452, 156701323391972

Offset: 0

R. H. Hardin, Dec 28 2006

[Empirical] a(base,n)=a(base-1,n)+A005191(n+1) for base>=2.int(n/2)+1.

See A285280 for confirmation of linear recurrence and code to produce sequence. - Ray Chandler, Aug 12 2023.

Ray Chandler, Table of n, a(n) for n = 0..99
OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less
Index entries for linear recurrences with constant coefficients, signature (12, -45, 30, 134, -160, -141, 170, 59, -60, -9, 6).

Cf. Row 12 of A285280.

cp's OEIS Frontend

A124806 Number of circular n-letter words over the alphabet {0,1,2,3,4} with adjacent letters differing by at most 2.

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A270918 Largest coefficient of (1+x+...+x^n)^(2*n).

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A104314 Prime coefficient of x^(2*k) in (x^4 + x^3 + x^2 + x + 1)^k for k in A104313.

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A104631 Coefficient of x^(2n+1) in the expansion of (1+x+x^2+x^3+x^4)^n.

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A124828 Number of base 7 circular n-digit numbers with adjacent digits differing by 2 or less.

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A124843 Number of base 8 circular n-digit numbers with adjacent digits differing by 2 or less.

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A124851 Number of base 9 circular n-digit numbers with adjacent digits differing by 2 or less.

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A124852 Number of base 10 circular n-digit numbers with adjacent digits differing by 2 or less.

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A124857 Number of base 11 circular n-digit numbers with adjacent digits differing by 2 or less.

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