A257714
Pentagonal numbers (A000326) that are the sum of five consecutive pentagonal numbers.
Original entry on oeis.org
44290, 487065, 97731740, 1074935965, 476036316661270, 5235848584389645, 1050611935177517000, 11555515453364758825, 5117369992623387417086890, 56285147779473003009380865, 11294033255019751129047408500, 124221295646279547914265231925
Offset: 1
44290 is in the sequence because P(172) = 44290 = 8400+8626+8855+9087+9322 = P(75)+ ... +P(79).
- Colin Barker, Table of n, a(n) for n = 1..398
- Index entries for linear recurrences with constant coefficients, signature (1,0,0,10749957122,-10749957122,0,0,-1,1).
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CoefficientList[Series[5 (29 x^8 + 275 x^7 + 60401 x^6 + 606965 x^5 - 16071841615 x^4 + 195440845 x^3 + 19448935 x^2 + 88555 x + 8858)/((1 - x) (x^2 - 322 x + 1) (x^2 + 322 x + 1) (x^4 + 103682 x^2 + 1)), {x, 0, 33}], x] (* Vincenzo Librandi, May 06 2015 *)
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Vec(-5*x*(29*x^8 +275*x^7 +60401*x^6 +606965*x^5 -16071841615*x^4 +195440845*x^3 +19448935*x^2 +88555*x +8858) / ((x -1)*(x^2 -322*x +1)*(x^2 +322*x +1)*(x^4 +103682*x^2 +1)) + O(x^100))
A257715
Pentagonal numbers (A000326) that are the sum of six consecutive pentagonal numbers.
Original entry on oeis.org
651, 354051, 196476315, 1833809355, 1017687528051, 564774036750651, 313425981747606051, 173938318056614696235, 1623451323680702588835, 900947621231988101541051, 499988268427580436128625651, 277472588498948806845840543051, 153985687725108202266731539138755
Offset: 1
651 is in the sequence because P(21) = 651 = 51+70+92+117+145+176 = P(6)+ ... +P(11).
- Colin Barker, Table of n, a(n) for n = 1..417
- Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,885289046402,-885289046402,0,0,0,-1,1).
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CoefficientList[Series[3 (17 x^10 + 6808 x^9 + 56840 x^8 + 35265352 x^7 + 19570796200 x^6 - 4188939995034 x^5 + 338617906232 x^4 + 545777680 x^3 + 65374088 x^2 + 117800 x + 217)/((1 - x) (x^10 - 885289046402 x^5 + 1)), {x, 0, 33}], x] (* Vincenzo Librandi, May 06 2015 *)
LinearRecurrence[{1,0,0,0,885289046402,-885289046402,0,0,0,-1,1},{651,354051,196476315,1833809355,1017687528051,564774036750651,313425981747606051,173938318056614696235,1623451323680702588835,900947621231988101541051,499988268427580436128625651},20] (* Harvey P. Dale, Dec 14 2015 *)
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Vec(-3*x*(17*x^10 +6808*x^9 +56840*x^8 +35265352*x^7 +19570796200*x^6 -4188939995034*x^5 +338617906232*x^4 +545777680*x^3 +65374088*x^2 +117800*x +217) / ((x -1)*(x^10 -885289046402*x^5 +1)) + O(x^100))
A259403
Pentagonal numbers (A000326) that are the sum of eleven consecutive pentagonal numbers.
Original entry on oeis.org
2882, 27676, 1114135, 10982301, 443390277, 4370895551, 176468183540, 1739605414426, 70233893626072, 692358584013426, 27952913194960545, 275556976831896551, 11125189217700638267, 109670984420510781301, 4427797355731659037150, 43648776242386459028676
Offset: 1
2882 is in the sequence because P(44) = 2882 = 92 + 117 + 145 + 176 + 210 + 247 + 287 + 330 + 376 + 425 + 477 = P(8)+ ... +P(18).
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LinearRecurrence[{1,398,-398,-1,1},{2882,27676,1114135,10982301,443390277},30] (* Harvey P. Dale, Jan 21 2017 *)
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Vec(-11*x*(16*x^4+14*x^3-5507*x^2+2254*x+262)/((x-1)*(x^2-20*x+1)*(x^2+20*x+1)) + O(x^20))
A259404
Pentagonal numbers (A000326) that are the sum of twelve consecutive pentagonal numbers.
Original entry on oeis.org
417912, 9706632, 3050311681782, 70865417283102, 22269721626195937752, 517374380230514907672, 162586828187971503638961822, 3777247909935632832763236342, 1187014240408376459988712771009992, 27576939095353370682323270116205112
Offset: 1
417912 is in the sequence because P(528) = 417912 = 32340 + 32782 + 33227 + 33675 + 34126 + 34580 + 35037 + 35497 + 35960 + 36426 + 36895 + 37367 = P(147)+ ... +P(158).
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Select[Total/@Partition[PolygonalNumber[5,Range[5*10^6]],12,1],IntegerQ[ (1+Sqrt[ 1+24#])/6]&] (* The program generates the first four terms of the sequence. To generate more, increase the Range constant but the program will take a long time to run. *) (* Harvey P. Dale, Dec 17 2020 *)
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Vec(-6*x*(377*x^4+7980*x^3-131798379*x^2+1548120*x+69652) / ((x-1)*(x^2-2702*x+1)*(x^2+2702*x+1)) + O(x^20))
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