A159359 -id:A159359 - cp's OEIS frontend

A159355 Number of n X n arrays of squares of integers summing to 4.

5, 135, 1836, 12675, 58941, 211925, 635440, 1663821, 3921325, 8495531, 17179020, 32795295, 59626581, 103962825, 174792896, 284660665, 450710325, 695946991, 1050740300, 1554600411, 2258257485, 3226077405, 4538848176, 6296973125, 8624108701, 11671286355

Offset: 2

R. H. Hardin, Apr 11 2009

Each array either has four 1's or one 4, and all other elements 0. - Robert Israel, Jun 19 2018

R. H. Hardin, Table of n, a(n) for n=2..100
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A014626, A159359.

Maple
```
seq(binomial(n^2,4)+n^2, n=2..100);
```

Vec(x^2*(5 + 90*x + 801*x^2 + 591*x^3 + 252*x^4 - 88*x^5 + 37*x^6 - 9*x^7 + x^8) / (1 - x)^9 + O(x^40)) \\ Colin Barker, Jun 19 2018

Empirical: n^2*(n^2+1)*(n^4-7*n^2+18)/24. - R. J. Mathar, Aug 11 2009
From Robert Israel, Jun 19 2018: (Start)
Empirical formula confirmed.
a(n) = binomial(n^2,4)+n^2 = A014626(n^2).
(End)
From Colin Barker, Jun 19 2018: (Start)
G.f.: x^2*(5 + 90*x + 801*x^2 + 591*x^3 + 252*x^4 - 88*x^5 + 37*x^6 - 9*x^7 + x^8) / (1 - x)^9.
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) for n>10.
(End)

A159363 Number of n X n arrays of squares of integers summing to 6.

Original entry on oeis.org

12, 336, 9688, 184000, 1969212, 14039088, 75099360, 324796176, 1192537500, 3844187424, 11144826264, 29583574384, 72891000364, 168494340000, 368541092736, 768025638240, 1533632745708, 2948331631152, 5478589599000, 9873410641248, 17307337994716, 29583198551632

Offset: 2

R. H. Hardin, Apr 11 2009

All such sequences have holonomic recurrences (cf. comment in A159359). - Georg Fischer, Feb 17 2022

R. H. Hardin, Table of n, a(n) for n = 2..100
Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).

Cf. A159355-A159446.

C:=binomial; seq(n^2*C(n^2-1,2)+C(n^2,6),n=2..22); # Georg Fischer, Feb 18 2022

RecurrenceTable[{a[n-1] * (600*n+600*n^2-206*n^3-206*n^4-71*n^5-71*n^6+14*n^7+14*n^8-n^9-n^10) + a[n] * (-672-232*n+2424*n^2-2090*n^3+492*n^4+203*n^5-125*n^6-30*n^7+40*n^8-11*n^9+n^10) == 0, a[2]==12}, a[n], {n,2,20}] (* Georg Fischer, Feb 18 2022 *)

Empirical g.f.: -4*x^2*(1+x)*(3 + 42*x + 1522*x^2 + 18686*x^3 + 42654*x^4 + 18686*x^5 + 1522*x^6 + 42*x^7 + 3*x^8)/(-1+x)^13. - Vaclav Kotesovec, Nov 30 2012

A159367 Number of n X n arrays of squares of integers summing to 7.

Original entry on oeis.org

4, 540, 18720, 531300, 8583300, 86748088, 623757696, 3483871560, 16023245700, 63174296020, 219752181792, 688950636972, 1978607887620, 5271705817200, 13162584962560, 31050835145616, 69671782148868, 149524133455500

Offset: 2

R. H. Hardin Apr 11 2009

nonn

All such sequences have holonomic recurrences (cf. comment in A159359). - Georg Fischer, Feb 17 2022

R. H. Hardin, Table of n, a(n) for n=2..100
Index entries for linear recurrences with constant coefficients, signature (15, -105, 455, -1365, 3003, -5005, 6435, -6435, 5005, -3003, 1365, -455, 105, -15, 1).

Empirical: a(n)=n^2*(n-1)*(n+1)*(n^2-2)*(n^2-3)*(n^6-15*n^4+74*n^2+720)/5040. [From R. J. Mathar, Aug 11 2009]

A159371 Number of n X n arrays of squares of integers summing to 8.

Original entry on oeis.org

6, 675, 34830, 1347525, 32145930, 460513662, 4464289944, 32292364770, 186464336850, 900743450145, 3764484413118, 13954005203463, 46750424936670, 143665627355100, 409707743053920, 1094582930018724, 2760817157366382

Offset: 2

R. H. Hardin Apr 11 2009

nonn

All such sequences have holonomic recurrences (cf. comment in A159359). - Georg Fischer, Feb 17 2022

R. H. Hardin, Table of n, a(n) for n=2..100
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Empirical: n^2*(n^2-1)*(n^12-27*n^10+295*n^8+15*n^6-10016*n^4+35652*^2-15120)/40320. [From R. J. Mathar, Aug 11 2009]

A159375 Number of n X n arrays of squares of integers summing to 9.

Original entry on oeis.org

16, 766, 61184, 3112500, 105851488, 2138413851, 27990555776, 262835331687, 1909384608000, 11319915386120, 56916060868096, 249702337698346, 976762617522160, 3464394870851125, 11290721919375872, 34177386571594701

Offset: 2

R. H. Hardin, Apr 11 2009

nonn

All such sequences have holonomic recurrences (cf. comment in A159359). - Georg Fischer, Feb 17 2022

R. H. Hardin, Table of n, a(n) for n = 2..100
Index entries for linear recurrences with constant coefficients, signature (19, -171, 969, -3876, 11628, -27132, 50388, -75582, 92378, -92378, 75582, -50388, 27132, -11628, 3876, -969, 171, -19, 1).

Empirical g.f.: -x-(1+x)*x*(1 - 4*x + 637*x^2 + 47760*x^3 + 2021602*x^4 + 54462984*x^5 + 548532899*x^6 + 2125377516*x^7 + 3360726010*x^8 + 2125377516*x^9 + 548532899*x^10 + 54462984*x^11 + 2021602*x^12 + 47760*x^13 + 637*x^14 - 4*x^15 + x^16)/(-1+x)^19. - Vaclav Kotesovec, Nov 30 2012

a(n) = binomial(n^2,1) + multinomial(n^2,1,2,(n^2-3)) + multinomial(n^2,1,5,n^2-6) + binomial(n^2,9) = (1/362880)*n^18 - (1/10080)*n^16 + (13/8640)*n^14 - (1/240)*n^12 - (1091/17280)*n^10 + (251/480)*n^8 - (95209/90720)*n^6 + (1213/2520)*n^4 + (10/9)*n^2 corresponding to the ways of obtaining 9 as a sum of n^2 squares: 9 + (n^2-1)*0, 2*4 + 1 + (n^2-3)*0, 4 + 5*1 + (n^2 - 6)*0, and 9*1 + (n^2 - 9)*0. - Robert Israel, Dec 18 2023

A159383 Number of n X n arrays of squares of integers summing to 11.

Original entry on oeis.org

12, 1584, 152688, 13648200, 846679356, 32762864904, 779081474880, 12381739674840, 143119261158300, 1283949949655736, 9379210663695600, 57819891720389760, 309087991429311276, 1463326913027026800

Offset: 2

R. H. Hardin Apr 11 2009

nonn

All such sequences have holonomic recurrences (cf. comment in A159359). - Georg Fischer, Feb 17 2022

R. H. Hardin, Table of n, a(n) for n=2..100
Index entries for linear recurrences with constant coefficients, signature (23, -253, 1771, -8855, 33649, -100947, 245157, -490314, 817190, -1144066, 1352078, -1352078, 1144066, -817190, 490314, -245157, 100947, -33649, 8855, -1771, 253, -23, 1).

Empirical G.f.: -12*x^2*(1+x)*(1 + 108*x + 9833*x^2 + 866490*x^3 + 46525328*x^4 + 1327261872*x^5 + 16745609840*x^6 + 97277340534*x^7 + 275802359702*x^8 + 390874554984*x^9 + 275802359702*x^10 + 97277340534*x^11 + 16745609840*x^12 + 1327261872*x^13 + 46525328*x^14 + 866490*x^15 + 9833*x^16 + 108*x^17 + x^18)/(-1+x)^23. - Vaclav Kotesovec, Nov 30 2012

A159386 Number of n X n arrays of squares of integers summing to 12.

Original entry on oeis.org

8, 1857, 232740, 26296475, 2128426860, 110964458710, 3533207162352, 73077185537370, 1067559034014900, 11805270721428855, 104131103837358468, 762072014041423813, 4768129728862470880, 26106675760931007900, 127411116947126838720, 562536757736562399012

Offset: 2

R. H. Hardin, Apr 11 2009

nonn

All such sequences have holonomic recurrences (cf. comment in A159359). - Georg Fischer, Feb 17 2022

R. H. Hardin, Table of n, a(n) for n=2..100
Index entries for linear recurrences with constant coefficients, signature (25, -300, 2300, -12650, 53130, -177100, 480700, -1081575, 2042975, -3268760, 4457400, -5200300, 5200300, -4457400, 3268760, -2042975, 1081575, -480700, 177100, -53130, 12650, -2300, 300, -25, 1).

Empirical G.f.: -x^2*(1+x)*(8 + 1649*x + 187066*x^2 + 20829609*x^3 + 1515837476*x^4 + 63614656244*x^5 + 1276374119248*x^6 + 12185198155972*x^7 + 58755893406228*x^8 + 149482468806702*x^9 + 204117353324396*x^10 + 149482468806702*x^11 + 58755893406228*x^12 + 12185198155972*x^13 + 1276374119248*x^14 + 63614656244*x^15 + 1515837476*x^16 + 20829609*x^17 + 187066*x^18 + 1649*x^19 + 8*x^20)/(-1+x)^25. - Vaclav Kotesovec, Nov 30 2012

A159389 Number of n X n arrays of squares of integers summing to 13.

Original entry on oeis.org

16, 1962, 350240, 48299450, 5030081280, 346589305664, 14664677168512, 394240010602320, 7283982145272800, 99396267512446410, 1059813625787601696, 9216296747192215226, 67545604711093622960, 427985285624487838800

Offset: 2

R. H. Hardin Apr 11 2009

nonn

All such sequences have holonomic recurrences (cf. comment in A159359). - Georg Fischer, Feb 17 2022

R. H. Hardin, Table of n, a(n) for n=2..100
Index entries for linear recurrences with constant coefficients, signature (27, -351, 2925, -17550, 80730, -296010, 888030, -2220075, 4686825, -8436285, 13037895, -17383860, 20058300, -20058300, 17383860, -13037895, 8436285, -4686825, 2220075, -888030, 296010, -80730, 17550, -2925, 351, -27, 1).

Empirical G.f.: -2*x^2*(1+x)*(8 + 757*x + 150684*x^2 + 19591732*x^3 + 1902144428*x^4 + 111467309309*x^5 + 3357053495448*x^6 + 49669712820552*x^7 + 379085540031904*x^8 + 1572597871783886*x^9 + 3663335172842872*x^10 + 4854870652776840*x^11 + 3663335172842872*x^12 + 1572597871783886*x^13 + 379085540031904*x^14 + 49669712820552*x^15 + 3357053495448*x^16 + 111467309309*x^17 + 1902144428*x^18 + 19591732*x^19 + 150684*x^20 + 757*x^21 + 8*x^22)/(-1+x)^27. - Vaclav Kotesovec, Nov 30 2012

A159392 Number of n X n arrays of squares of integers summing to 14.

Original entry on oeis.org

24, 2520, 503616, 85380600, 11267944488, 1008474419568, 56159712530496, 1957557182156496, 45750088895603400, 771069955155892920, 9947917198190930112, 102886031599392144792, 883927680158797591800

Offset: 2

R. H. Hardin Apr 11 2009

nonn

All such sequences have holonomic recurrences (cf. comment in A159359). - Georg Fischer, Feb 17 2022

R. H. Hardin, Table of n, a(n) for n=2..100
Index entries for linear recurrences with constant coefficients, signature (29, -406, 3654, -23751, 118755, -475020, 1560780, -4292145, 10015005, -20030010, 34597290, -51895935, 67863915, -77558760, 77558760, -67863915, 51895935, -34597290, 20030010, -10015005, 4292145, -1560780, 475020, -118755, 23751, -3654, 406, -29, 1).

Empirical G.f.: -24*x^2*(1+x)*(1 + 75*x + 18270*x^2 + 2969695*x^3 + 371519352*x^4 + 29402869921*x^5 + 1270115156506*x^6 + 27861646979401*x^7 + 320243742405791*x^8 + 2035253623371844*x^9 + 7457245326412232*x^10 + 16147921368666408*x^11 + 20880695398301008*x^12 + 16147921368666408*x^13 + 7457245326412232*x^14 + 2035253623371844*x^15 + 320243742405791*x^16 + 27861646979401*x^17 + 1270115156506*x^18 + 29402869921*x^19 + 371519352*x^20 + 2969695*x^21 + 18270*x^22 + 75*x^23 + x^24)/(-1+x)^29. - Vaclav Kotesovec, Nov 30 2012

A159397 Number of n X n arrays of squares of integers summing to 16.

Original entry on oeis.org

5, 3987, 960197, 243293450, 49328395731, 7138321890616, 666179701320556, 38558976229027926, 1439078695947198175, 37055970405356439240, 701629710103618241661, 10289171538319337074541, 121736692023067368010505

Offset: 2

R. H. Hardin Apr 11 2009

nonn

All such sequences have holonomic recurrences (cf. comment in A159359). - Georg Fischer, Feb 17 2022

R. H. Hardin, Table of n, a(n) for n=2..100
Index entries for linear recurrences with constant coefficients, signature (33, -528, 5456, -40920, 237336, -1107568, 4272048, -13884156, 38567100, -92561040, 193536720, -354817320, 573166440, -818809200, 1037158320, -1166803110, 1166803110, -1037158320, 818809200, -573166440, 354817320, -193536720, 92561040, -38567100, 13884156, -4272048, 1107568, -237336, 40920, -5456, 528, -33, 1).

Empirical G.f.: -x - x*(1+x)*(1 -29*x + 4379*x^2 + 821431*x^3 + 212904294*x^4 + 41572005795*x^5 + 5592296001394*x^6 + 449779113080002*x^7 + 19634897290562828*x^8 + 461755603301498981*x^9 + 6109738979241997636*x^10 + 47748225885297619913*x^11 + 229005554540179223339*x^12 + 691824816742731558421*x^13 + 1336833271423720702097*x^14 + 1664132445913632199036*x^15 + 1336833271423720702097*x^16 + 691824816742731558421*x^17 + 229005554540179223339*x^18 + 47748225885297619913*x^19 + 6109738979241997636*x^20 + 461755603301498981*x^21 + 19634897290562828*x^22 + 449779113080002*x^23 + 5592296001394*x^24 + 41572005795*x^25 + 212904294*x^26 + 821431*x^27 + 4379*x^28 - 29*x^29 + x^30)/(-1+x)^33. - Vaclav Kotesovec, Nov 30 2012

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A159355 Number of n X n arrays of squares of integers summing to 4.

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A159363 Number of n X n arrays of squares of integers summing to 6.

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A159367 Number of n X n arrays of squares of integers summing to 7.

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A159371 Number of n X n arrays of squares of integers summing to 8.

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A159375 Number of n X n arrays of squares of integers summing to 9.

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A159383 Number of n X n arrays of squares of integers summing to 11.

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A159386 Number of n X n arrays of squares of integers summing to 12.

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A159389 Number of n X n arrays of squares of integers summing to 13.

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A159392 Number of n X n arrays of squares of integers summing to 14.

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A159397 Number of n X n arrays of squares of integers summing to 16.

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