A133301 -id:A133301 - cp's OEIS frontend

A257714 Pentagonal numbers (A000326) that are the sum of five consecutive pentagonal numbers.

44290, 487065, 97731740, 1074935965, 476036316661270, 5235848584389645, 1050611935177517000, 11555515453364758825, 5117369992623387417086890, 56285147779473003009380865, 11294033255019751129047408500, 124221295646279547914265231925

Offset: 1

Colin Barker, May 05 2015

			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).

Cf. A000326, A133301, A257715, A259402, A259403, A259404.

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 *)

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))

G.f.: -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)).

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

Colin Barker, May 05 2015

			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).

Cf. A000326, A133301, A257714, A259402, A259403, A259404.

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 *)

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))

G.f.: -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)).

A259402 Pentagonal numbers (A000326) that are the sum of seven consecutive pentagonal numbers.

Original entry on oeis.org

287, 532, 17145051, 32963672, 1106094475927, 2126616990876, 71358579001465427, 137196568515066592, 4603627364594444737551, 8851099419054387781412, 296998415728087428795555787, 571019827783678204813603176, 19160555787678205016722039960967

Offset: 1

Colin Barker, Jun 26 2015

			287 is in the sequence because P(14) = 287 = 5+12+22+35+51+70+92 = P(2)+ ... +P(8).

Colin Barker, Table of n, a(n) for n = 1..416
Index entries for linear recurrences with constant coefficients, signature (1,64514,-64514,-1,1).

Cf. A000326, A133301, A257714, A257715, A259403, A259404.

LinearRecurrence[{1,64514,-64514,-1,1},{287,532,17145051,32963672,1106094475927},20] (* Harvey P. Dale, May 13 2022 *)

Vec(-7*x*(1968*x^4+1813*x^3-195857*x^2+35*x+41)/((x-1)*(x^2-254*x+1)*(x^2+254*x+1)) + O(x^20))

G.f.: -7*x*(1968*x^4+1813*x^3-195857*x^2+35*x+41) / ((x-1)*(x^2-254*x+1)*(x^2+254*x+1)).

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

Colin Barker, Jun 26 2015

			2882 is in the sequence because P(44) = 2882 = 92 + 117 + 145 + 176 + 210 + 247 + 287 + 330 + 376 + 425 + 477 = P(8)+ ... +P(18).

Colin Barker, Table of n, a(n) for n = 1..767
Index entries for linear recurrences with constant coefficients, signature (1,398,-398,-1,1).

Cf. A000326, A133301, A257714, A257715, A259402, A259404.

LinearRecurrence[{1,398,-398,-1,1},{2882,27676,1114135,10982301,443390277},30] (* Harvey P. Dale, Jan 21 2017 *)

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))

G.f.: -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)).

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

Colin Barker, Jun 26 2015

			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).

Colin Barker, Table of n, a(n) for n = 1..290
Index entries for linear recurrences with constant coefficients, signature (1,7300802,-7300802,-1,1).

Cf. A000326, A133301, A257714, A257715, A259402, A259403.

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 *)

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))

G.f.: -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))

A144797 Numbers k such that 2*k^2 + 17 is a square.

Original entry on oeis.org

2, 4, 16, 26, 94, 152, 548, 886, 3194, 5164, 18616, 30098, 108502, 175424, 632396, 1022446, 3685874, 5959252, 21482848, 34733066, 125211214, 202439144, 729784436, 1179901798, 4253495402, 6876971644, 24791187976, 40081928066, 144493632454, 233614596752

Offset: 1

Richard Choulet, Sep 21 2008

			a(1)=2 because 2*4 + 17 = 25 = 5^2.

Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Cf. A133301.

Select[Range[6000000],IntegerQ[Sqrt[2#^2+17]]&] (* Harvey P. Dale, Aug 18 2012 *)
LinearRecurrence[{0, 6, 0, -1}, 2{1, 2, 8, 13}, 30] (* Robert G. Wilson v, Dec 02 2014 *)

Vec(2*x*(1+x)*(1+x+x^2) / ((x^2+2*x-1)*(x^2-2*x-1)) + O(x^50)) \\ Colin Barker, Oct 20 2014

G.f.: 2*x*(1+x)*(1+x+x^2) / ( (x^2+2*x-1)*(x^2-2*x-1) ). - R. J. Mathar, Nov 27 2011
a(n) = 2*A077241(n-1). - R. J. Mathar, Nov 27 2011
a(n) = 6*a(n-2) - a(n-4). - Colin Barker, Oct 20 2014

Corrected by R. J. Mathar, Nov 27 2011

Editing and more terms from Colin Barker, Oct 20 2014

A144941 Numbers k such that 6*k-1 = A144796(k).

Original entry on oeis.org

1, 36, 753, 41348, 868769, 47715364, 1002558481, 55063488516, 1156951618113, 63543218031908, 1335121164743729, 73328818545333124, 1540728667162644961, 84621393058096392996, 1777999546784527541073, 97653014260224692184068

Offset: 1

Richard Choulet, Sep 26 2008

Also the index of a pentagonal number which is equal to the sum of two consecutive pentagonal numbers. - Colin Barker, Dec 22 2014

			a(1) = 1 because 6*1 - 1 = 5 = A144796(1).

Colin Barker, Table of n, a(n) for n = 1..653
Index entries for linear recurrences with constant coefficients, signature (1,1154,-1154,-1,1).

Cf. A133301, A144796, A144797.

a:=[1,36,753,41348,868769];; for n in [6..30] do a[n]:=a[n-1] +1154*a[n-2]-1154*a[n-3]-a[n-4]+a[n-5]; od; a; # G. C. Greubel, Mar 16 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( x*(1+ 35*x-437*x^2+205*x^3+4*x^4)/((1-x)*(1-34*x+x^2)*(1+34*x+x^2)) )); // G. C. Greubel, Mar 16 2019

LinearRecurrence[{1,1154,-1154,-1,1},{1,36,753,41348,868769},30] (* Harvey P. Dale, Dec 27 2018 *)

Vec(-x*(1+35*x-437*x^2+205*x^3+4*x^4) / ((x-1)*(x^2-34*x+1)*(x^2+34*x+1)) + O(x^30)) \\ Colin Barker, Dec 22 2014

a=(x*(1+ 35*x-437*x^2+205*x^3+4*x^4)/((1-x)*(1-34*x+x^2)*(1+34*x +x^2))).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Mar 16 2019

For the odd and even indices respectively the same recurrence is obtained: a(n+2) = 1154*a(n+1) - a(n) - 192.
We also have a(n+2) = 577*a(n+1) - 96 + 68*sqrt((72*a(n)^2-24*a(n)-32)).
G.f.: x*(1 + 35*x - 437*x^2 + 205*x^3 + 4*x^4) / ((1-x)*(1 - 34*x + x^2)*(1 + 34*x + x^2)). - R. J. Mathar, Nov 27 2011

a(6) corrected and sequence extended by R. J. Mathar, Nov 27 2011

A144942 Expansion of x^2(3x^3+145x^2-507x-25) / ((x-1)(x^2-34x+1)(x^2+34x+1)).

Original entry on oeis.org

0, 25, 532, 29237, 614312, 33739857, 708915900, 38935766125, 818088334672, 44931840368777, 944073229295972, 51851304849802917, 1089459688519217400, 59836360864832197825, 1257235536477947584012, 69051108586711506487517, 1450848719635862992732832

Offset: 1

Richard Choulet, Sep 26 2008

Also the index of the first of two consecutive pentagonal numbers whose sum is also a pentagonal number. - Colin Barker, Dec 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (1,1154,-1154,-1,1).

Cf. A133301, A144796, A144797.

CoefficientList[Series[x (3 x^3 + 145 x^2 - 507 x - 25)/((x - 1) (x^2 - 34 x + 1) (x^2 + 34 x + 1)), {x, 0, 20}], x] (* Vincenzo Librandi, Oct 20 2014 *)
LinearRecurrence[{1,1154,-1154,-1,1},{0,25,532,29237,614312},20] (* Harvey P. Dale, Jun 16 2025 *)

concat(0, Vec(x^2*(3*x^3+145*x^2-507*x-25)/((x-1)*(x^2-34*x+1)*(x^2+34*x+1)) + O(x^20))) \\ Colin Barker, Oct 20 2014

a(6) corrected, and more terms from Colin Barker, Oct 20 2014

Edited, name changed by Colin Barker, Oct 20 2014

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A257714 Pentagonal numbers (A000326) that are the sum of five consecutive pentagonal numbers.

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A257715 Pentagonal numbers (A000326) that are the sum of six consecutive pentagonal numbers.

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A259402 Pentagonal numbers (A000326) that are the sum of seven consecutive pentagonal numbers.

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A259403 Pentagonal numbers (A000326) that are the sum of eleven consecutive pentagonal numbers.

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A259404 Pentagonal numbers (A000326) that are the sum of twelve consecutive pentagonal numbers.

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A144797 Numbers k such that 2*k^2 + 17 is a square.

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A144941 Numbers k such that 6*k-1 = A144796(k).

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A144942 Expansion of x^2(3x^3+145x^2-507x-25) / ((x-1)(x^2-34x+1)(x^2+34x+1)).