A258129
Octagonal numbers (A000567) that are the sum of three consecutive octagonal numbers.
Original entry on oeis.org
698901, 5102520783381, 37252493940331837461, 271973082264557457061125141, 1985621622943208359132836202790421, 14496630316026749501691464257547633057301, 105837027604506739193825102426073141683789429781, 772695182809023513889440668692977953487035688873891861
Offset: 1
698901 is in the sequence because Oct(483) = 698901 = 231296 + 232965 + 234640 = Oct(278) + Oct(279) + Oct(280).
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CoefficientList[Series[-21*x*(x^2 -844482*x +33281)/((x-1)*(x^2 -7300802*x +1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 18 2017 *)
LinearRecurrence[{7300803,-7300803,1},{698901,5102520783381,37252493940331837461},20] (* Harvey P. Dale, Sep 16 2018 *)
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Vec(-21*x*(x^2 -844482*x +33281)/((x-1)*(x^2 -7300802*x +1)) + O('x^20))
A258130
Octagonal numbers (A000567) that are the sum of ten consecutive octagonal numbers.
Original entry on oeis.org
1045, 1325345, 1910970885, 2755618515265, 3973599987865685, 5729928426883626945, 8262552817966202013445, 11914595433578836419585185, 17180838352667864150839647765, 24774756989951626526674352316385, 35725182398671892783600265200403845
Offset: 1
1045 is in the sequence because Oct(19) = 1045 = 1+8+21+40+65+96+133+176+225+280 = Oct(1) + ... + Oct(10).
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CoefficientList[Series[-95 x (63 x^2 - 1922 x + 11)/((x - 1) (x^2 - 1442 x + 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 18 2017 *)
LinearRecurrence[{1443,-1443,1},{1045,1325345,1910970885},20] (* Harvey P. Dale, Jul 25 2019 *)
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Vec(-95*x*(63*x^2-1922*x+11)/((x-1)*(x^2-1442*x+1)) + O('x^20))
A258131
Octagonal numbers (A000567) that are the sum of eleven consecutive octagonal numbers.
Original entry on oeis.org
49665, 348161, 19701781, 138502485, 7841194625, 55123576321, 3120775694421, 21939044808725, 1242060885120385, 8731684710231681, 494337111502154261, 3475188575627335765, 196744928316972210945, 1383116321414969338241, 78303987133043437737301
Offset: 1
49665 is in the sequence because Oct(129) = 49665 = 3400 + 3605 + 3816 + 4033 + 4256 + 4485 + 4720 + 4961 + 5208 + 5461 + 5720 = Oct(34) + ... + Oct(44).
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CoefficientList[Series[-11*x*(95*x^4 -64*x^3 -37550*x^2 +27136*x +4515)/((x-1)*(x^2 -20*x +1)*(x^2 +20*x +1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 18 2017 *)
Select[Total/@Partition[Table[n(3n-2),{n,5*10^6}],11,1],IntegerQ[(Sqrt[1+3#]+1)/3]&] (* Harvey P. Dale, Aug 31 2018 *)
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Vec(-11*x*(95*x^4 -64*x^3 -37550*x^2 +27136*x +4515)/((x-1)*(x^2 -20*x +1)*(x^2 +20*x +1)) + O('x^20))
A258132
Octagonal numbers (A000567) that are the sum of fifteen consecutive octagonal numbers.
Original entry on oeis.org
4715040, 8463840, 1122749669280, 2015496399840, 267373851637578960, 479974343542849680, 63672943775553479639280, 114302050117965712710960, 15163202176330482896520455040, 27220118818712616412771202880, 3610995292612020914167620625112640
Offset: 1
4715040 is in the sequence because Oct(1254) = 4715040 = 300833 + 302736 + 304645 + 306560 + 308481 + 310408 + 312341 + 314280 + 316225 + 318176 + 320133 + 322096 + 324065 + 326040 + 328021 = Oct(317) + ... + Oct(331).
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CoefficientList[Series[-240*x*(7*x^4 +4*x^3 -449376*x^2 +15620*x +19646)/((x-1)*(x^2 -488*x +1)*(x^2 +488*x +1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 18 2017 *)
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Vec(-240*x*(7*x^4 +4*x^3 -449376*x^2 +15620*x +19646)/((x-1)*(x^2 -488*x +1)*(x^2 +488*x +1)) + O('x^20))
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