A196456 -id:A196456 - cp's OEIS frontend

A196450 Number of n X 2 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 3,0,2,4,1 for x=0,1,2,3,4.

3, 6, 14, 30, 67, 146, 320, 706, 1550, 3403, 7481, 16443, 36131, 79399, 174500, 383499, 842796, 1852205, 4070612, 8945988, 19660603, 43208249, 94959144, 208692464, 458645240, 1007969053, 2215223460, 4868418432, 10699372182, 23514119454

Offset: 1

R. H. Hardin, Oct 02 2011

nonn

Every 0 is next to 0 3's, every 1 is next to 1 0's, every 2 is next to 2 2's, every 3 is next to 3 4's, every 4 is next to 4 1's.

Column 2 of A196456.

			Some solutions for n=4:
..1..0....2..2....1..0....1..1....1..0....0..1....1..0....2..2....2..2....0..0
..1..0....2..2....1..0....0..0....2..2....2..2....2..2....2..2....2..2....0..0
..1..0....0..1....1..1....0..0....2..2....2..2....2..2....1..0....0..0....0..0
..1..0....0..1....0..1....1..1....0..0....0..1....0..1....1..0....1..1....1..1

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A196456.

Empirical: a(n) = 2*a(n-1) +3*a(n-3) -2*a(n-4) -3*a(n-5) -5*a(n-6) -a(n-7).

Empirical g.f.: x*(3 + 2*x^2 - 7*x^3 - 5*x^4 - 9*x^5 - x^6) / (1 - 2*x - 3*x^3 + 2*x^4 + 3*x^5 + 5*x^6 + x^7). - Colin Barker, May 09 2018

A196451 Number of nX3 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 3,0,2,4,1 for x=0,1,2,3,4.

Original entry on oeis.org

4, 14, 40, 131, 383, 1232, 3966, 12314, 38745, 122989, 386667, 1216157, 3839910, 12101854, 38109643, 120144068, 378709117, 1193259020, 3760674157, 11852918903, 37353389828, 117718989595, 371005905259, 1169236902187, 3684870558556

Offset: 1

R. H. Hardin Oct 02 2011

nonn

Every 0 is next to 0 3's, every 1 is next to 1 0's, every 2 is next to 2 2's, every 3 is next to 3 4's, every 4 is next to 4 1's

Column 3 of A196456

			Some solutions for n=4
..0..0..1....1..0..1....0..0..0....0..2..2....1..1..0....0..0..0....0..2..2
..0..0..1....1..0..1....0..2..2....0..2..2....0..1..1....1..1..1....0..2..2
..2..2..1....1..2..2....0..2..2....0..1..1....0..2..2....1..1..1....0..0..0
..2..2..0....0..2..2....0..0..1....1..1..0....0..2..2....0..0..0....1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..200

Empirical: a(n) = 4*a(n-1) -5*a(n-2) +15*a(n-3) -28*a(n-4) +13*a(n-5) -a(n-6) -8*a(n-7) +6*a(n-8) +12*a(n-9) -46*a(n-10) +27*a(n-11) -28*a(n-12) +83*a(n-13) -5*a(n-14) +25*a(n-15) -23*a(n-16) -4*a(n-18)

A196452 Number of n X 4 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 3,0,2,4,1 for x=0,1,2,3,4.

Original entry on oeis.org

5, 30, 131, 594, 2930, 15274, 76809, 380093, 1900745, 9527317, 47791454, 239934641, 1203634551, 6035457841, 30269130212, 151821524586, 761535367044, 3819934203741, 19160599201791, 96107073057178, 482062802792429, 2417986066816592, 12128437822354667, 60835368690467514

Offset: 1

R. H. Hardin, Oct 02 2011

Every 0 is next to 0 3's, every 1 is next to 1 0's, every 2 is next to 2 2's, every 3 is next to 3 4's, every 4 is next to 4 1's.

			Some solutions for n=5:
..0..0..2..2....0..0..0..0....0..0..1..1....2..2..1..1....1..0..0..0
..0..0..2..2....0..0..0..0....2..2..2..0....2..2..0..0....1..0..0..0
..1..1..1..0....0..0..2..2....2..1..2..2....1..1..1..1....1..0..0..0
..1..2..2..0....0..0..2..2....2..0..0..2....0..0..2..2....1..2..2..0
..0..2..2..0....0..0..0..1....2..2..2..2....0..0..2..2....0..2..2..0

R. H. Hardin, Table of n, a(n) for n = 1..200

Column 4 of A196456.

Empirical: a(n) = 7*a(n-1) -15*a(n-2) +38*a(n-3) -84*a(n-4) +194*a(n-5) -250*a(n-6) -1059*a(n-7) -66*a(n-8) -910*a(n-9) +1845*a(n-10) +25120*a(n-11) +10190*a(n-12) -38236*a(n-13) +68320*a(n-14) +10899*a(n-15) -251830*a(n-16) -12937*a(n-17) -47142*a(n-18) -553200*a(n-19) +69561*a(n-20) +533489*a(n-21) -1642449*a(n-22) -436401*a(n-23) +6042242*a(n-24) +3515060*a(n-25) -7528796*a(n-26) -1508098*a(n-27) +18034521*a(n-28) +1036374*a(n-29) -20794117*a(n-30) +15310099*a(n-31) +32164418*a(n-32) -19996088*a(n-33) -49523594*a(n-34) +23771811*a(n-35) +42630527*a(n-36) -14333853*a(n-37) -49762720*a(n-38) -5236367*a(n-39) +14631189*a(n-40) -2601831*a(n-41) -19899517*a(n-42) -5183657*a(n-43) -1258284*a(n-44) -3972327*a(n-45) -1773169*a(n-46) +2977093*a(n-47) +1341855*a(n-48) -2141053*a(n-49) -508619*a(n-50) -426819*a(n-51) -38656*a(n-52) -355272*a(n-53) +146662*a(n-54) -41336*a(n-55) +30846*a(n-56) -21828*a(n-57) +5950*a(n-58) -410*a(n-59) -213*a(n-60) -93*a(n-61) +26*a(n-62) +3*a(n-63) -2*a(n-64) for n>65.

A196453 Number of nX5 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 3,0,2,4,1 for x=0,1,2,3,4.

Original entry on oeis.org

8, 67, 383, 2930, 27320, 249402, 2032584, 15952022, 128543371, 1060801262, 8803139813, 72685286254, 596527548372, 4887671652328, 40110935638661, 329698087638675, 2710883854573401, 22279902843871853, 183037488877957616

Offset: 1

R. H. Hardin Oct 02 2011

nonn

Every 0 is next to 0 3's, every 1 is next to 1 0's, every 2 is next to 2 2's, every 3 is next to 3 4's, every 4 is next to 4 1's

Column 5 of A196456

			Some solutions for n=4
..1..1..1..1..0....1..0..0..1..1....2..2..2..2..0....0..1..1..0..0
..0..0..0..1..1....1..1..1..1..0....2..0..0..2..0....0..1..2..2..0
..0..0..0..2..2....0..1..2..2..0....2..2..0..2..0....1..1..2..2..0
..1..1..1..2..2....0..1..2..2..0....0..2..2..2..1....1..0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..200

A196454 Number of nX6 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 3,0,2,4,1 for x=0,1,2,3,4.

Original entry on oeis.org

12, 146, 1232, 15274, 249402, 3707781, 46887621, 578160718, 7394096617, 96338727461, 1258078944815, 16370595753240, 212274372772624, 2750178418606822, 35652972674573032, 462472339632802800

Offset: 1

R. H. Hardin Oct 02 2011

nonn

Every 0 is next to 0 3's, every 1 is next to 1 0's, every 2 is next to 2 2's, every 3 is next to 3 4's, every 4 is next to 4 1's

Column 6 of A196456

			Some solutions for n=4
..2..2..2..2..2..0....0..0..1..1..0..0....0..0..2..2..1..0....1..0..0..1..2..2
..2..1..1..1..2..0....1..2..2..2..2..2....1..1..2..2..1..1....1..1..1..1..2..2
..2..0..0..0..2..0....1..2..0..0..1..2....2..2..0..0..0..1....0..2..2..0..1..1
..2..2..2..2..2..0....0..2..2..2..2..2....2..2..0..0..0..1....0..2..2..1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..200

A196455 Number of n X 7 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 3,0,2,4,1 for x=0,1,2,3,4.

Original entry on oeis.org

17, 320, 3966, 76809, 2032584, 46887621, 926499600, 17962317556, 351103877665, 6945098996988, 139065692664815, 2795329197283554, 56072977068447935, 1121420878698363003, 22398808824228997399, 447457890891091698310

Offset: 1

R. H. Hardin, Oct 02 2011

nonn

Every 0 is next to 0 3's, every 1 is next to 1 0's, every 2 is next to 2 2's, every 3 is next to 3 4's, every 4 is next to 4 1's.

Column 7 of A196456.

			Some solutions for n=4
..2..2..2..1..1..1..0....0..1..1..0..0..1..1....0..1..1..0..0..0..0
..2..0..2..0..0..2..2....0..2..2..2..2..2..0....2..2..1..0..0..0..0
..2..2..2..1..1..2..2....0..2..0..0..1..2..0....2..2..1..0..0..2..2
..0..0..0..1..1..0..0....1..2..2..2..2..2..1....0..0..1..1..1..2..2

R. H. Hardin, Table of n, a(n) for n = 1..95

Cf. A196456.

A194657 Decimal expansion of (4Pi^6log(2) - 90Pi^4zeta(3) + 1350Pi^2zeta(5) - 5715*zeta(7))/1536.

Original entry on oeis.org

1, 1, 7, 5, 7, 5, 8, 3, 4, 0, 7, 2, 3, 3, 2, 4, 8, 2, 0, 6, 2, 4, 2, 9, 0, 6, 7, 9, 4, 9, 1, 4, 7, 5, 8, 4, 3, 3, 4, 1, 6, 4, 3, 8, 9, 9, 8, 1, 6, 2, 9, 0, 8, 8, 8, 6, 9, 5, 3, 0, 2, 4, 7, 6, 4, 9, 1, 9, 1, 2, 8, 4, 2, 7, 1, 5, 5, 9, 4, 7, 1, 1, 8, 2, 6, 8, 8, 8, 9, 0, 0, 3, 1, 4, 1, 1, 5, 9, 4, 4, 7, 1, 9, 9, 4

Offset: 0

Seiichi Kirikami, Sep 01 2011

The absolute value of the integral {x=0..Pi/2} x^5*log(sin(x )) dx or (d^5/da^5 (integral {x=0..Pi/2} sin(ax)*log(sin(x )) dx)) at a=0. The absolute value of m=2 of (-1)^(m+1)*(sum {n=1..infinity} (limit {a -> 0} (d^(2m+1)/da^(2m+1) ((1-cos((a+2n)*Pi/2))/n/(a+2n)))))-(pi/2)^2(m+1)*log(2)/2/(m+1).

			0.11757583407233248206...

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 1.441.2

Cf. A173623, A173624, A193716, A193717, A196456.

RealDigits[ N[(4 Pi^6*Log[2]-90 Pi^4*Zeta[3]+1350 Pi^2*Zeta[5]-5715 Pi^2*Zeta[7])/1536,150]][[1]]

Equals (4*A092732*A002162-90*A092425*A002117+1350*A002388*A013663-5715*A013665)/1536.

A196449 Number of n X n 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 3,0,2,4,1 for x=0,1,2,3,4.

Original entry on oeis.org

1, 6, 40, 594, 27320, 3707781, 926499600, 539621963877, 739769209147928

Offset: 1

R. H. Hardin Oct 02 2011

nonn

Every 0 is next to 0 3's, every 1 is next to 1 0's, every 2 is next to 2 2's, every 3 is next to 3 4's, every 4 is next to 4 1's

Diagonal of A196456

			Some solutions for n=4
..1..0..0..1....1..0..0..0....1..0..0..1....1..0..0..1....1..0..0..1
..1..2..2..2....1..2..2..0....2..2..0..1....1..2..2..1....1..2..2..2
..0..2..0..2....0..2..2..0....2..2..0..1....0..2..2..0....0..2..0..2
..1..2..2..2....0..1..1..0....0..0..0..1....0..0..1..1....0..2..2..2

cp's OEIS Frontend

A196450 Number of n X 2 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 3,0,2,4,1 for x=0,1,2,3,4.

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A196451 Number of nX3 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 3,0,2,4,1 for x=0,1,2,3,4.

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A196452 Number of n X 4 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 3,0,2,4,1 for x=0,1,2,3,4.

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A196453 Number of nX5 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 3,0,2,4,1 for x=0,1,2,3,4.

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A196454 Number of nX6 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 3,0,2,4,1 for x=0,1,2,3,4.

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A196455 Number of n X 7 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 3,0,2,4,1 for x=0,1,2,3,4.

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A194657 Decimal expansion of (4*Pi^6*log(2) - 90*Pi^4*zeta(3) + 1350*Pi^2*zeta(5) - 5715*zeta(7))/1536.

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A196449 Number of n X n 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 3,0,2,4,1 for x=0,1,2,3,4.

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A194657 Decimal expansion of (4Pi^6log(2) - 90Pi^4zeta(3) + 1350Pi^2zeta(5) - 5715*zeta(7))/1536.