Edwin Hermann - cp's OEIS frontend

A384759 Number of legal arrangements in pawn-only chess on an n X n board where no pieces have been taken and no piece attacks another piece.

0, 3, 2031, 728174, 247646098, 91880342535, 38818192375310, 18907485764545412, 10626953883068264472, 6866760686250915376779, 5073038373153476636807709, 4259014676256866422905669602, 4038463837000965678262091166880, 4299625631242136963071149921577615, 5111407212497576694797045579672852791

Offset: 4

Edwin Hermann, Jun 09 2025

nonn

The number of ways of arranging n pawns of each color on an n X n board such that no pawn threatens another, each file contains one pawn of each color, none of the pawns are passed pawns, and each pawn is placed between row 2 and row n-1 inclusive.

There is no requirement that the arrangements counted here can actually be achieved via a sequence of legal chess moves.

			The a(5) = 3 positions are:
  . . . . .    . . . . .    . . . . .
  b b b b b    . b . b .    b . b . b
  . . . . .    b w b w b    w b w b w
  w w w w w    w . w . w    . w . w .
  . . . . .    . . . . .    . . . . .

Cf. A035290, A294240.

MkTfrMtx(n)={my(m=binomial(n,2), M=matrix(m,m)); for(i=1,n-1, for(j=i+1,n, for(p=1,n-1, for(q=p+1,n, if(q<>i+1&&j<>p+1, M[binomial(n-i,2)+(j-i), binomial(n-p,2)+(q-p)]=1) )))); M}
a(n)={my(M=MkTfrMtx(n-2)); vecsum(M^(n-1)*vectorv(#M,i,1))} \\ Andrew Howroyd, Jun 15 2025

a(9) onwards from Andrew Howroyd, Jun 15 2025

A383200 Number of positive integers with n digits in which adjacent digits differ by at most 5.

Original entry on oeis.org

9, 74, 610, 5020, 41317, 340050, 2798709, 23034169, 189577752, 1560278726, 12841536934, 105689495131, 869854553902, 7159149960981, 58921836913893, 484943447787706, 3991222267830858, 32848892512931768, 270355712339865433, 2225104276073281126, 18313239977617203949

Offset: 1

Edwin Hermann, Apr 19 2025

Index entries for linear recurrences with constant coefficients, signature (8,3,-9,-1,2).

Cf. A235163, A383197, A383198, A383199, A383201, A383202.

Rest[CoefficientList[Series[-x*(2*x^4-x^3-9*x^2+2*x+9)/((x+1)*(2*x^4-3*x^3-6*x^2+9*x-1)),{x,0,21}],x]] (* or *) LinearRecurrence[{8,3,-9,-1,2},{9,74,610,5020,41317},21] (* James C. McMahon, May 08 2025 *)

G.f.: -x*(2*x^4-x^3-9*x^2+2*x+9)/((x+1)*(2*x^4-3*x^3-6*x^2+9*x-1)). - Alois P. Heinz, Apr 26 2025

a(11)-a(21) from Alois P. Heinz, Apr 26 2025

A383202 Number of positive integers with n digits in which adjacent digits differ by at most 7.

Original entry on oeis.org

9, 86, 813, 7693, 72786, 688661, 6515721, 61648078, 583279341, 5518660133, 52214449434, 494023669525, 4674173312097, 44224391459894, 418426247682381, 3958913146568317, 37457003208767394, 354397037125653845, 3353104871295311673, 31725187008033469918

Offset: 1

Edwin Hermann, Apr 19 2025

Index entries for linear recurrences with constant coefficients, signature (9,5,-6).

Cf. A235163, A383197, A383198, A383199, A383200, A383201.

Rest[CoefficientList[Series[-x*(6*x^2-5*x-9)/(6*x^3-5*x^2-9*x+1),{x,0,20}],x]] (* or *) LinearRecurrence[{9,5,-6},{9,86,813},20] (* James C. McMahon, May 07 2025 *)

G.f.: -x*(6*x^2-5*x-9)/(6*x^3-5*x^2-9*x+1). - Alois P. Heinz, Apr 26 2025

a(11)-a(20) from Alois P. Heinz, Apr 26 2025

A383201 Number of positive integers with n digits in which adjacent digits differ by at most 6.

Original entry on oeis.org

9, 81, 724, 6472, 57851, 517112, 4622299, 41317257, 369321783, 3301249634, 29508817638, 263769909867, 2357755102376, 21075220921085, 188384678470177, 1683910560899833, 15051939468415328, 134544486519385896, 1202650255852445247, 10750107085908359068

Offset: 1

Edwin Hermann, Apr 19 2025

Index entries for linear recurrences with constant coefficients, signature (8,9,-5,-4).

Cf. A235163, A383197, A383198, A383199, A383200, A383202.

G.f.: -x*(4*x^3+5*x^2-9*x-9)/(4*x^4+5*x^3-9*x^2-8*x+1). - Alois P. Heinz, Apr 26 2025

a(11)-a(20) from Alois P. Heinz, Apr 26 2025

A383198 Number of positive integers with n digits in which adjacent digits differ by at most 3.

Original entry on oeis.org

9, 54, 328, 2000, 12202, 74458, 454366, 2772710, 16920138, 103253214, 630091042, 3845059318, 23464039746, 143186649814, 873780342786, 5332145758694, 32538816680050, 198564450196598, 1211717109125762, 7394366670845606, 45123286657530514, 275359755529253142

Offset: 1

Edwin Hermann, Apr 19 2025

Index entries for linear recurrences with constant coefficients, signature (7,-4,-10,6).

Cf. A235163, A383197, A383199, A383200, A383201, A383202.

LinearRecurrence[{7,-4,-10,6},{9, 54, 328, 2000},22] (* or *) Rest[CoefficientList[Series[(-x*(10*x^3-14*x^2-9*x+9)/(6*x^4-10*x^3-4*x^2+7*x-1)),{x,0,22}],x]] (* James C. McMahon, May 01 2025 *)

G.f.: -x*(10*x^3-14*x^2-9*x+9)/(6*x^4-10*x^3-4*x^2+7*x-1). - Alois P. Heinz, Apr 23 2025

More terms from Alois P. Heinz, Apr 23 2025

A383199 Number of positive integers with n digits in which adjacent digits differ by at most 4.

Original entry on oeis.org

9, 65, 475, 3465, 25282, 184463, 1345887, 9819916, 71648478, 522764591, 3814216651, 27829445433, 203050351876, 1481504383412, 10809413614854, 78868091114176, 575440631436879, 4198553757680021, 30633661742154286, 223510591001999469, 1630787227154056312

Offset: 1

Edwin Hermann, Apr 19 2025

Index entries for linear recurrences with constant coefficients, signature (7,3,-6,-1,1).

Cf. A235163, A383197, A383198, A383200, A383201, A383202.

Rest[CoefficientList[Series[-x*(x^4-x^3-7*x^2+2*x+9)/(x^5-x^4-6*x^3+3*x^2+7*x-1),{x,0,21}],x]] (* or *) LinearRecurrence[{7,3,-6,-1,1},{9,65,475,3465,25282},21] (* James C. McMahon, May 01 2025 *)

G.f.: -x*(x^4-x^3-7*x^2+2*x+9)/(x^5-x^4-6*x^3+3*x^2+7*x-1). - Alois P. Heinz, Apr 23 2025

More terms from Alois P. Heinz, Apr 23 2025

A383197 Number of positive integers with n digits in which adjacent digits differ by at most 2.

Original entry on oeis.org

9, 41, 188, 867, 4010, 18574, 86096, 399225, 1851529, 8587802, 39833891, 184770640, 857073208, 3975623218, 18441391129, 85542653145, 396800342804, 1840608838251, 8537899488042, 39604141848678, 183708898915088, 852157340908409, 3952841397780937, 18335763176322738

Offset: 1

Edwin Hermann, Apr 19 2025

Index entries for linear recurrences with constant coefficients, signature (7,-11,0,1).

Cf. A235163, A383198, A383199, A383200, A383201, A383202.

G.f.: -x*(2*x^3-22*x+9)/(x^4-11*x^2+7*x-1). - Alois P. Heinz, Apr 21 2025

a(15)-a(24) from Alois P. Heinz, Apr 21 2025

A383079 Number of n-digit positive integers where all pairs of consecutive digits have a difference of at least 7.

Original entry on oeis.org

9, 9, 22, 48, 109, 244, 549, 1233, 2771, 6226, 13990, 31435, 70634, 158713, 356625, 801329, 1800570, 4045844, 9090929, 20427132, 45899349, 103134901, 231742019, 520719590, 1170046298, 2629070167, 5907467042, 13273957953, 29826312781, 67019116473, 150590587774

Offset: 1

Edwin Hermann, Apr 15 2025

Index entries for linear recurrences with constant coefficients, signature (2,1,-1).

Cf. A052268, A100062, A383070, A383071, A383072, A383077, A383078.

G.f.: x*(4*x^3-5*x^2-9*x+9)/(x^3-x^2-2*x+1). - Alois P. Heinz, Apr 15 2025

More terms from Alois P. Heinz, Apr 15 2025

A383078 Number of n-digit positive integers where all pairs of consecutive digits have a difference of at least 6.

Original entry on oeis.org

9, 16, 50, 140, 407, 1168, 3367, 9691, 27908, 80354, 231374, 666211, 1918282, 5523469, 15904199, 45794312, 131859470, 379674208, 1093228315, 3147825472, 9063802211, 26098178315, 75146709476, 216376326214, 623030800330, 1793945691511, 5165460748322, 14873351444633

Offset: 1

Edwin Hermann, Apr 15 2025

Index entries for linear recurrences with constant coefficients, signature (2,3,-1,-1).

Cf. A052268, A100062, A383070, A383071, A383072, A383077.

Cf. A147704.

G.f.: x*(2*x^4+x^3-9*x^2-2*x+9)/((x+1)*(x^3-3*x+1)). - Alois P. Heinz, Apr 15 2025

More terms from Alois P. Heinz, Apr 15 2025

A383077 Number of n-digit positive integers where all pairs of consecutive digits have a difference of at least 5.

Original entry on oeis.org

9, 25, 95, 325, 1152, 4035, 14191, 49840, 175126, 615251, 2161615, 7594445, 26681890, 93742420, 329348786, 1157113228, 4065328917, 14282870761, 50180539756, 176301151441, 619405374858, 2176179878029, 7645653483902, 26861757974676, 94374410640201, 331569117405852

Offset: 1

Edwin Hermann, Apr 15 2025

Index entries for linear recurrences with constant coefficients, signature (3,3,-4,-1,1).

Cf. A052268, A100062, A383070, A383071, A383072.

G.f.: -x*(x^4+x^3-7*x^2-2*x+9)/(x^5-x^4-4*x^3+3*x^2+3*x-1). - Alois P. Heinz, Apr 15 2025

More terms from Alois P. Heinz, Apr 15 2025

cp's OEIS Frontend

User: Edwin Hermann

A384759 Number of legal arrangements in pawn-only chess on an n X n board where no pieces have been taken and no piece attacks another piece.

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A383200 Number of positive integers with n digits in which adjacent digits differ by at most 5.

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A383202 Number of positive integers with n digits in which adjacent digits differ by at most 7.

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A383201 Number of positive integers with n digits in which adjacent digits differ by at most 6.

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A383198 Number of positive integers with n digits in which adjacent digits differ by at most 3.

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A383199 Number of positive integers with n digits in which adjacent digits differ by at most 4.

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A383197 Number of positive integers with n digits in which adjacent digits differ by at most 2.

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A383079 Number of n-digit positive integers where all pairs of consecutive digits have a difference of at least 7.

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A383078 Number of n-digit positive integers where all pairs of consecutive digits have a difference of at least 6.

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