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.

Showing 1-3 of 3 results.

A070212 Number of 5 X 5 pandiagonal magic squares with sum n.

Original entry on oeis.org

1, 10, 55, 220, 715, 2001, 4995, 11385, 24090, 47905, 90376, 162955, 282490, 473110, 768570, 1215126, 1875015, 2830620, 4189405, 6089710, 8707501, 12264175, 17035525, 23361975, 31660200, 42436251, 56300310, 73983205, 96354820, 124444540, 159463876, 202831420, 256200285, 321488190
Offset: 0

Views

Author

Sharon Sela (sharonsela(AT)hotmail.com), May 07 2002

Keywords

Comments

In contrast to other definitions, a magic square may contain here any nonnegative integers, not necessarily distinct. For example, the 10 solutions for n = 1 are the 10 permutation matrices of size 5 X 5 which are pandiagonal in the sense that any of the 10 (principal or broken) diagonals has exactly one 1 and four 0's. - M. F. Hasler, Oct 23 2018

Links

Crossrefs

Cf. A027567, A014820, A111158, A053494.

Programs

  • GAP
    a:=[1, 10, 55, 220, 715, 2001, 4995, 11385, 24090];;  for n in [10..36] do a[n]:=9*a[n-1]-36*a[n-2]+84*a[n-3]-126*a[n-4]+126*a[n-5]-84*a[n-6]+36*a[n-7]-9*a[n-8]+a[n-9]; od; a; # Muniru A Asiru, Oct 23 2018
  • Maple
    seq(coeff(series(-(x^4+x^3+x^2+x+1)/(x-1)^9,x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 23 2018
  • Mathematica
    LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,10,55,220,715,2001,4995,11385,24090},40] (* Harvey P. Dale, Mar 13 2018 *)
  • PARI
    apply( A070212(n)=1/8064*(n+4)*(n+3)*(n+2)*(n+1)*(n^2+5*n+8)*(n^2+5*n+42), [0..20]) \\ Edited by M. F. Hasler, Oct 23 2018

Formula

a(n) = (1/8064) * (n+4)*(n+3)*(n+2)*(n+1)*(n^2+5n+8)*(n^2+5n+42).
G.f.: -(x^4+x^3+x^2+x+1) / (x-1)^9. [Colin Barker, Dec 10 2012]

Extensions

More terms from Benoit Cloitre, May 12 2002
More terms from M. F. Hasler, Oct 23 2018

A111086 Number of 3 X 3 X 3 X 3 magic cubes with magic sum 3n.

Original entry on oeis.org

1, 153, 6297, 82161, 582377, 2823169, 10577681, 32908425, 88984025, 215645185, 478631121, 988480025, 1922282689, 3552547017, 6284626217, 10704205425, 17636581137, 28219457161, 43991281193, 66997065953, 99914018553, 146199131313, 210261368801, 297660801977
Offset: 0

Views

Author

N. J. A. Sloane, Oct 12 2005

Keywords

Links

Crossrefs

Cf. A111158.

Formula

G.f.:= r(t)/s(t), where
r = t^54+150*t^51+5837*t^48+63127*t^45+331124*t^42+1056374*t^39+2326380*t^36+3842273*t^33+5055138*t^30+5512456*t^27+5055138*t^24+3842273*t^21+2326380*t^18+1056374*t^15+331124*t^12+63127*t^9+5837*t^6+150*t^3+1 and
s = (t^3+1)^4*(t^12+t^9+t^6+t^3+1)*(1-t^3)^9*(t^6+t^3+1).

Extensions

This paper also gives a g.f. for the number of 5 X 5 magic squares with magic sum n (A111158). - N. J. A. Sloane.

A216039 Number of 6 by 6 magic squares with line sum n.

Original entry on oeis.org

1, 96, 14763, 957936, 33177456, 718506720, 10837963166, 122793273216, 1103391397593, 8187061491760, 51724720525317, 284976371277888, 1395347280436638, 6165194801711616, 24889894891691712, 92768491235726640, 321987367305139071, 1048378447871747424, 3222195250935497833, 9398840830661453088
Offset: 0

Views

Author

Guoce Xin, Aug 30 2012

Keywords

Examples

			For n = 1, there are a(1) = 96 order 6 permutation matrices with exactly one 1 in each of the two diagonals.

Links

Crossrefs

Cf. A111158.

Formula

G.f.: (x^138+99*x^137+15057*x^136+1002806*x^135+36140317*x^134+823860011*x^133+13197261179*x^132+159778881431*x^131+1540197926928*x^130+12283604989433*x^129+83443844586997*x^128
+493826644119635*x^127+2591895971809073*x^126+12239625173465375*x^125+52618101897021930*x^124
+207948182505922572*x^123+761697282842373791*x^122+2603936594202983265*x^121
+8357520624415623570*x^120+25313244131813040492*x^119+72673216612249799707*x^118
+198540029295827265030*x^117+517913155627899876744*x^116+1293950334879519037064*x^115
+3104565556800370034675*x^114+7170548645642540233444*x^113+15977552472766155842750*x^112
+34412717940513453504180*x^111+71769782821380635837621*x^110+145167679454737704278880*x^109
+285189004474854548554157*x^108+544883332503752228347324*x^107
+1013692519414068545966383*x^106+1838319814003865364502115*x^105
+3253035784774708879439262*x^104+5622314253334154424175766*x^103
+9498907763273239021574685*x^102+15700357961071728256043309*x^101
+25406320589195514110356366*x^100+40277791473075750762252075*x^99
+62597197699253178187339298*x^98+95425280193517651890574674*x^97
+142766762407648666487568356*x^96+209732150155458679271033099*x^95
+302678001784712603830421513*x^94+429303207319389562327707454*x^93
+598674963030494000816618195*x^92+821156092631443052249172731*x^91
+1108206045308608891199410839*x^90+1472032087920610932242371227*x^89
+1925075439230166802560415829*x^88+2479329488091630543216144069*x^87
+3145503368703854928491254853*x^86+3932062984462037001968113054*x^85
+4844201407852058337442332388*x^84+5882809249486653844574028923*x^83
+7043530583232146694988816214*x^82+8315998814445857390844541404*x^81
+9683347293907738803126233896*x^80+11122080015097990434647761713*x^79
+12602367905141556425711508726*x^78+14088806780184052230859053795*x^77
+15541636034748392591830628113*x^76+16918375811338196658691711642*x^75
+18175798884655835561351408187*x^74+19272116367842845200134757907*x^73
+20169228060755970451363952559*x^72+20834872558688610557869003806*x^71
+21244511627696474156825956913*x^70+21382798694422310755770332936*x^69
+21244511627696474156825956913*x^68+20834872558688610557869003806*x^67
+20169228060755970451363952559*x^66+19272116367842845200134757907*x^65
+18175798884655835561351408187*x^64+16918375811338196658691711642*x^63
+15541636034748392591830628113*x^62+14088806780184052230859053795*x^61
+12602367905141556425711508726*x^60+11122080015097990434647761713*x^59
+9683347293907738803126233896*x^58+8315998814445857390844541404*x^57
+7043530583232146694988816214*x^56+5882809249486653844574028923*x^55
+4844201407852058337442332388*x^54+3932062984462037001968113054*x^53
+3145503368703854928491254853*x^52+2479329488091630543216144069*x^51
+1925075439230166802560415829*x^50+1472032087920610932242371227*x^49
+1108206045308608891199410839*x^48+821156092631443052249172731*x^47
+598674963030494000816618195*x^46+429303207319389562327707454*x^45
+302678001784712603830421513*x^44+209732150155458679271033099*x^43
+142766762407648666487568356*x^42+95425280193517651890574674*x^41
+62597197699253178187339298*x^40+40277791473075750762252075*x^39
+25406320589195514110356366*x^38+15700357961071728256043309*x^37
+9498907763273239021574685*x^36+5622314253334154424175766*x^35
+3253035784774708879439262*x^34+1838319814003865364502115*x^33
+1013692519414068545966383*x^32+544883332503752228347324*x^31
+285189004474854548554157*x^30+145167679454737704278880*x^29
+71769782821380635837621*x^28+34412717940513453504180*x^27
+15977552472766155842750*x^26+7170548645642540233444*x^25
+3104565556800370034675*x^24+1293950334879519037064*x^23
+517913155627899876744*x^22+198540029295827265030*x^21
+72673216612249799707*x^20+25313244131813040492*x^19+8357520624415623570*x^18
+2603936594202983265*x^17+761697282842373791*x^16+207948182505922572*x^15
+52618101897021930*x^14+12239625173465375*x^13+2591895971809073*x^12
+493826644119635*x^11+83443844586997*x^10+12283604989433*x^9+1540197926928*x^8
+159778881431*x^7+13197261179*x^6+823860011*x^5
+36140317*x^4+1002806*x^3+15057*x^2+99*x+1)*(x-1)^3/((x^4-1)^5*(x^8-1)^2*(x^3-1)^5*(x^9-1)*(x^5-1)^4*(x^6-1)^6*(x^7-1)^3*(x^10-1)) [typos corrected by Georg Fischer, Apr 17 2020]
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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